ESS1VE
TICAL EX
FIRST SERII:
v
PROGRESSIVE
MATHEMATICAL
EXERCISES
FOR HOME WORK.
FIRST SERIF.s.
l:v
A. T. KICHARDSON, M.A.
H >
SENIOR MATHEMATICAL MASTER AT THB ISI.B OF WIOHT COLLKOK
FORMERLY SCHOLAR OF HERTFORD COLLEGE, OXFORD.
<*
'
MACMILLAX AND CO.
AND NEW YORK.
1891.
PREFACE.
THE present collection of Exercises, gathered from many
sources, is one which has accumulated through several
years, and consists of papers set weekly or bi-weekly to
boys of all ages during that time. They serve to recall
back work, and keep boys always ready for the examina-
tion. The First Series contains 261 papers, about half the
total number, and commences with exercises in Arithmetic
suitable to boys who have gone through the First Four
Rules, Simple and Compound, and are beginning Fractions;
and Algebraical Exercises consisting chiefly of Numerical
Values, Addition, and Subtraction. From these onward,
the exercises rise in difficulty by careful gradations,
reaching Cube Root and Compound Interest in Arith-
metic, and Quadratic Equations in Algebra, at the end
of the First Series.
The Second Series is a continuation of the First, and
includes problems in Higher Algebra, Logarithms, Trigo-
nometry, and easy Mechanics, and Analytical Geometry.
vi MATHEMATICAL EXERCISES.
The problems in the First Series are all arranged with a
view to cover the whole of the back work up to the point
where the student is working, in about four or live consecu-
tive papers. Here and there a question may be found
which appears to be beyond the average of the whole paper.
These are inserted purposely, and are always very simple, in
order that boys may be encouraged to look up methods they
have not yet reached, and so to find that a little research
enables them to do a new sort of question, which is not
so hard as it looks. I have found a three-fold advantage
in the plan. First, it points out the boys who have
mathematical tastes, and a desire to excel in the science ;
secondly, a new rule so learned is rarely forgotten ;
and thirdly, it greatly increases the boys' interest, and
gives a real zest to learning, when they find they can work
out something new by their own efforts, without having
been first shown the way. Instances of this occur in
Exercises LVIII. 5; LXIII. 8; CIII. 7; CLXII. 5;
CCXLV. 4, and elsewhere.
The relative standards of the different branches of
Mathematics have been arranged chiefly with a view to
the Local, and Army and Navy Examinations, but it will
be found that the papers may be so selected as to be
suitable for almost any standards that usually occur.
Euclid Riders begin at Ex. CLXXIII., and Quadratic
Equations at Ex. CCXVIII.
One of the most important things in Mathematics is
accuracy. I have therefore indicated in the earlier papers
PREFACE. vii
constantly, and occasionally in later ones, a simple method
by which the working may be readily and shortly tested ;
and which, though not infallible, will usually detect
mistakes. See Ex. LXIX. 7, et passim.
To facilitate the correcting of papers, answers are given
with especial fulness, and as they have all been worked
over at least twice, they will be found to contain few,
if any, mistakes. I shall be very grateful to any one
who will point out any I may have missed. In a collection
of upwards of 6000 problems, there may easily be some
that have escaped notice.
A. T. RICHARDSON.
ISLE OF WIGHT COLLEGE,
July, 1891.
MATHEMATICAL EXERCISES.
I.
1. Find the G.C.M. (or H.C.F.) of 16650, 10730, and
1961.
2. Multiply £1267 4s. 2d. by 335.
3. Add together
Acres. Roods. Poles. Sq. Yds.
41 2 17 14
27 1 31 27
53 3 23 18
and subtract from their sum 22 acres 35 poles.
4. Eeduce 2 Ibs. 3 oz. 5 dwts. to grains.
5. Add together f , £, f, and £.
6. When a = 4, b = 2, c=l, find the value of
7. Simplify a- {6- (c-d)} + [b- {a + (c - d - a)}]
8. Add together
, a-b+2d, -a+2c-5d, and
II.
1. Find the L.C.M. of 8, 18, 28, 38, 48.
2. Divide £876 14s. 3£d. by 49.
3. Multiply 10 tons 14 cwt. 2 qrs. 5 Ibs. by 576.
« A
2 MATHEMATICAL EXERCISES.
4. If I can buy 8 articles for £1 10s., what should I have
to pay for 12?
5. Subtract 8^ from 16f
6. When a = 4, 6 = 2, c = 0, find the value of
.
0
7. Subtract a3 + 2a2o-o3 from a4-a3+2ao2.
8. Multiply 3z2 + 2zy + 2/2 by 2x-y.
III.
1. Find the G.C.M. (or H.C.F.) of 1914 and 26361.
2. Multiply £247 13s. 5£d. by 64.
3. Divide 34 acres 2 roods 20 perches by 44.
4. Reduce 1468935 inches to miles, furlongs, poles, etc.
5. Multiply together 15, ^ ^-, and •&.
6. Find the numerical values of
a3 - 3a26 + 3ao2 - b3 and a2 - 1 Oao + 9o2,
when a = 5, and 6 = 4.
7. Simplify 5x - [a - {x + 2a - (3rc - 7a) }].
8. Add together
2x + 3y + 4z, 62, 2z - 2y, and 7x + z-y.
IV.
1. Find the L.C.M. of 15, 27, 60, 108, 56.
2. Divide £118332 19s. 5£d. by 321.
3. Multiply 5 tons 10 cwt. 3 qrs. 14 Ibs. 7 oz. by 112.
4. Reduce l^ff to its lowest terms, and express ^y- as
a mixed number.
5. Divide 2f by 1^.
6. Find the value of
- x - 2
(1) when x = 7, (2) when a; = 2.
MATHEMATICAL EXERCISES.
7. Subtract
Sz3 + 4a;2y - 7xf + lOy* from 4z3 + 4xf - 2x*y + 4y3.
8. Multiply xs + 3z2 + 4z + 2 by a2 - 2z + 2.
V.
1. Find the G.C.M. (or H.C.F.) of 7511 and 41107.
2. Multiply £7048 11s. 3£d. by 92.
3. Divide 3 tons 5 cwt. 2 qrs. by 131.
4. Reduce 2426031 pints to quarters, bushels, pecks, etc.
5. Add together 5^, -j^-, |, and -|.
6. When x = 4, y = 5, z = 6, find the value of
x-y + z x+y+z
7. Simplify 1 -[1 - {1 - (1 - ITT)}].
8. Add together
and 7d - 4a + 5& - 6c.
From the result subtract
VI.
1. Find the L.C.M. of 6, 32, 198, 24, and 112.
2. Divide £4896 8s. 2d. by 371.
3. If the area of a field be 3 roods 15 perches 2 yards 2 feet,
find the area of 6 fields, each 4 times as large.
4. If 35 men can build a wall 175 feet long in 8 days, how
many men would build a wall 325 feet long in the same
time ?
5. Subtract ^ from -|.
6. Find the value of
3m+2a- 4(& + c),
when m - a, and a - 4, b = 6, c = 3.
4 MATHEMATICAL EXERCISES.
7. Subtract 4a3-6a25 + 253 from 7a25-6a£3.
8. Multiply &-k* + k-l by &2-l.
VII.
1. Find the G.C.M. (or H.C.F.) of 6409 and 7395.
2. Multiply £483 5s. 3£d. by 999.
3. Divide 342 qrs. 4 bus. 2 pks. by 63.
4. Reduce 7 cwt. 1 qr. 16 Ibs. 8 oz. to drams.
5. Multiply together 1^-, 2£f, 2f|, 1^, and fi
6. Find the value of
a + 3 5 - (6 - 3q) 2a - (Bx - 1)
when a=l, z = 5.
7. Simplify a-b- 2{a- 3J- (4a- 205)}.
8. Add together
a - b + c, 2a + 26 - 3c, - 3a + 45 + 4c,
and subtract the sum from
4a + 55 - 5c.
VIII.
1. Find the L.C.M. of 7, 14, 21, 28, 35, 63.
2. Divide £16544 5s. 6d. by 66.
3. Multiply 6 oz. 4 dwt. 6 grs. Troy by 123.
4. Eeduce -WV/ to its lowest terms, and then to a mixed
number.
5. Divide lf£ by ^.
6. Find the value of the following, when a = 2, 5 = 3,
a2 + 52
7. Subtract
2a-3{a-5-a} from 25 -3(5 -(a -5)}.
8. Multiply 3z2-zy-2y2 by a;2
MATHEMATICAL EXERCISES.
IX.
1. Find the G.C.M. (or H.C.F) of 5203 and 3741.
2. Multiply £87 9s. 4£d. by 999.
3. Divide 28 Ib. 3 oz. 10 dwt. 3 grs. by 19.
What kind of weight is this ?
4. Keduce 897334 oz. to tons, etc.
5. Add together 1^, 2^, 2^.
6. When a = 10, b = 3, c = 7, find the value of
b + c , f 3c 4k
7. Simplify 2[4as- {2y+ (2x-y)-(a; +
8. Add together
4y - 4z + 6a, 5a-5y + 7x, and x-3a-y.
X.
1. Find the G.C.M. of 15625 and 2625.
2. Multiply £53 2s. 7£d. by 126.
3. What are the squares of 16, 18, and 21, and the square
roots of 4, 16, 36, and 121?
4. Reduce 13 acres 12 perches to square yards.
5. Multiply together 2^, Iff, |«, and 2Jf
6. Ifa=l, b = 3, c = 4,d = 6, e = 2,f=0,
find the value of
a&c + 4bd + ec -fd.
7. Simplify 3(a + &) + 4(a-6).
8. Add together
and 2z3-az2.
XL
1. Find the L.C.M. of 6, 8, 11, 16, 20.
2. Divide £16544 by 55.
3. Multiply 5 oz. 7 drs. 2 scru. 15 grs. by 35.
6 MATHEMATICAL EXERCISES.
4. If 22 yards cost 17s. 4d., what will 154 yards cost?
5. Subtract 11^ from 15j^.
6. When y - 3, find the value of
7. Subtract
ab2 — a2b + be2 — b2c from b2c — be2 - ab'2 + ac2.
8. Multiply 2x + 3y by 4#2 - 6xy + 9y2.
XII.
1. Find the G.C.M. of 22341 and 256583.
2. Multiply £348 15s. 6|d. by 98.
3. Divide 580 tons 10 cwt. 2 qrs. by 154.
4. Reduce 976000 ounces to tons, cwts., etc.
5. Multiply together lOfi and 2§ and -^.
6. When a = 4, b = 2, d = 0, find the value of
7. Simplify k- {21 -(3m + 21- k)}.
8. Add together
r+3s-4£, 5s-3r+2t, and 2t-r-7s.
XIII.
1. Find the L.C.M. of 876 and 864.
2. How many times is £12 15s. 6d. contained in £1022?
3. Multiply 10 cwt. 2 qrs. 12 Ibs. by 16, and divide the
result by 5.
4. Reduce ^silloo0 ^° ^s lowest terms, and to a mixed
number.
5. Divide 12^ by 3^.
6. If a- 6, b = 3, c = 2, d = 0, find the value of
MATHEMATICAL EXERCISES.
7. Add together
a - 3& + 4c - d, b-3c + 4d-a
and d - 3a + 46 - c.
8. Subtract
a?b-ab* + 2o6-3a2 from a2Z> - 2afe3 + a3 - a2.
XIV.
1. Find the G.C.M. of 106499 and 256583.
2. Multiply £891 19s. lid. by 524.
3. Divide £1779 4s. 3|d. by 81.
4. Reduce 17 acres 14 perches to square yards.
5. Add together 4-^-, f , 2T5^, and J. •
6. Find the value of
cfibcd - aWcd + abfid - abed2,
when a = 4, 6 = 3, c = 2, d=l.
7. Simplify x-[2x + {3x - (4a; - 5a? -
8. Add together
and 4a;% - xy2 - 6y2z.
XV.
1. Find the L.C.M. of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
2. Divide £1689 11s. 4£d. by 83.
3. What is the least number by which 272 must be multi-
plied, so that the number produced will be exactly
divisible by 238 ?
4. If 4s. 6d. be the amount of taxes paid for £1, what must
be paid for £400 ?
5. Add together 3T6^, ^3T, 2£, and 4^.
6. When k = 5, I = 10, m = 2, find the value of
$ MATHEMATICAL EXERCISES.
7. Subtract v*x + 2xy2 - Svxy from v3 - 8vxy + 2xy*.
8. Multiply ax + by + cz by ax-by + cz.
XVI.
1. Find the G.C.M. of 9691 and 459001.
2. Multiply £250 13s. 4d. by 65.
3. Divide sixty-three thousand six hundred and ninety-three
millions, six hundred and ninety-three thousand, six
hundred and thirty, by seven millions, seven thousand,
and seven.
4. In 23568 grains, how many Ibs., oz., dwts., etc.
5 Multiply 13| by 3^-.
6. When a = 6, b - 4, c = 2, find the value of
(a - b + c)2 - abc + aW.
7. Simplify c-[c- (c-(c-c-c)}].
8. Add together
0^-9, a2- 2x* + 12&-60, and a^
XVII.
1. Find the G.C.M. of 360 and 1120.
2. Multiply £12 10s. 6|d. by 1296.
3. Divide 7050 hhd. 1 kild. 9 galls. (Ale) by 27.
4. Eeduce 3 tons 2 cwt. 2 qrs. 7 Ib. to ounces.
5. From 30^ take 24T6T.
6. If k = 4, a — 3, m = 2, prove that
kam = ka?- 2am -(a- mf + 1.
7. Simplify a - {4& + 2(a - b + c)} - 3 . o^~c.
8. Add together
3a + 4c-8/, 6a-2c + 2/, /2-4a + c, and 2c2-5a-/2.
XVIII.
1. Find the L.C.M. of 68, 41, and 82.
MATHEMATICAL EXERCISES. 9
2. Divide £21579 15s. by 7506.
3. Multiply 3 ac. 1 ro. 7 perches 11 sq. yds. by 74.
4. If £5 last 6 weeks, how long will £100 last 1
5. Subtract 22^ from 30^.
6. If k= 3, 1 = 4, m= 1, find the value of
Em - k(l + m) + (l- m)(k + 1).
7. From
8a?y + 2x2y* - Bxf + f
subtract z3 + 8x5y - 2x2y2 - 3xy* + 2y*.
8. Multiply a?-ab + W by a + b;
and subtract a from b.
XIX.
1. Find the G.C.M. of 5547 and 147008443.
2. Multiply £74 17s. 8|d. by 297.
3. Divide the sum of 2 oz. 3 drams 7 gr.; 12 oz. 4 drams
19 gr.; and 6 oz. 6 drams 1 scr. 8 grs.; by 144.
4. In 758372 grains Troy how many Ibs., oz., etc.
5. Multiply together 6£, T5T, A> and If.
6. When a - 3, b = 2, c = 0, find the value of
4
b 2a
a?b -ab2 + abc-a? + 2 + ~5~"
7. Simplify
1 - (1 - a) + ( 1 - a + a2) - ( 1 - a + a2 - a3).
8. Add together
and
XX.
1. Find the L.C.M. of 868 and 854.
2. Divide £824 Is. lOfd. by 59.
3. Multiply 45 cwt. 3 qrs. 21 Ibs. by 49.
10 MATHEMATICAL EXERCISES.
4. Eeduce f ff § to its lowest terms.
5. Divide 10/F by 9^.
6. If x = 6, y = 2, 2= 1, find the value of
xs + y3 + z? — Sxyz.
7. Subtract the sum of
a + 2x - y + 245 and 2x + y - 2a
from 3a + ^b + 3x + %y-
8. Multiply 3a2 + a5-52 by a2-2a5-352.
XXI.
1. Find the G.C.M. of 6281 and 326041.
2. Multiply 14s. ll£d. by 5069^.
3. Divide 243 days 2 hrs. 11 min. 49 sec. by 17.
4. Eeduce 12 tons 3 qrs. 17 Ibs. 4 oz. to ounces.
5. Add together 6|, 2|, ^ and 2J.
6. If a = 3, 5 = 4, c = 0, find the value of
do doc d J)
4 ~"T2 0"
7. Simplify
a - [25 - 3a + (2a - { 3c - 5} + a) - 25] + 2a.
8. Add together
and add 3a to 45.
XXII.
1. Find the L.C.M. of 72, 12, 3, 7, 48.
2. A country postman walks 16 mi. 3 fur. a day ; how far
does he walk in a year, omitting 52 Sundays ?
3. Divide 432 ac. 1 ro. 34 po. by 302.
4. Eeduce 8^~| and 4|-| to improper fractions.
5. Divide 4| by 2| ; and simplify 7| -s- 9.
MATHEMATICAL EXERCISES. 11
6. If a = 4, b = 8, c = 3, d = 2, x = 5, find the value of
3a+5b + c + 2cdx - abd.
7. From 7x2-x-y take 7z2 -2x-y2;
and subtract a from 5.
8. Add together
y - y2, - 1 3z2 + 2y2 + 2Qxy, 2Qx2 - 3xy,
XXIII.
1. Eeduce to their lowest terms f^f, and
2. Multiply 8s. 8d. by 1028.
3. Divide 15 st. 12 Ib. by 3f.
4. Reduce 7593 sq. in. to sq. yds.
5. Arrange the following in order of magnitude
6. Find the sum of
1 Oa2& - 2ab + I Ob2, and 2ab + 5a?b.
7. When x = 5, y = 3, and 2 = 4, prove that
(a;2 + ?/2 - z) x (a;2 - #2)
is numerically equal to
x4 - x2z + y2z - y*.
8. Simplify
XXIV.
1. Find the G.C.M. (or H.C.F.) of 5775 and 3773.
2. Multiply £2 14s. 7£d. by 984.
3. Find the number of days (i.) between Lady Day and
Michaelmas, (ii.) between Midsummer and Christmas.
4. Reduce 3051 pks. to bushels.
12 MATHEMATICAL EXERCISES.
5. Multiply together 5£, if, if, and 8;
and subtract 2£ from 5.
6. Take 7x2-7xy + 7y2 from 5x2 - 3xy + f ;
and add 4 to x.
7. When a = 2, & = 4, c = 5, find the value of
10
3c + a 2c + \a
8. Simplify b-7+d- (d + (b-2d)- 3c} + 26.
XXV.
1. Find the G.C.M. (or H.C.F.) of 48849 and 59133.
2. What is the value of 1546 acres at £2 15s. each ?
3. How often is 2 fur. 18 po. contained in 10 mi. 5 fur.
30 po. ?
4. Reduce 495 pks. to qrs.
5. Simplify 2| x 1$ x 2| x £ ; and | of li-f 2f.
6. From the sum of the first five of the following expressions,
take the sum of the last four : —
2ab - 5&2 + 10a2 ; 2a2 - '6ab + b2 ; 6ab
How may the working be shortened ?
7. Subtract
- 5z2 - 3xy - 2y2 - 7 from 2x2 + 3xy-
8. Find the value of
xy-{6-(z-xy)}- (3xy - z),
when x=l,y = 5,z = 7.
XXVI
1. Find the L.C.M. of 36, 5, 48, 9, 54, 8, 7.
2. Divide 45 Ib. Troy by 16.
MATHEMATICAL EXERCISES. 13
A scuttle of coals is charged 6d., when coals are 28s.
a ton. How much ought the scuttle to hold ?
4. Multiply 8 yd. 1 ft. 9 in. by 100.
5. Simplify llf-^; and 2f-lj|.
6. Find the sum of
a?x + aW - 2oa3, a;4 +15, 2as?-3a?x2, 2a?x-±aa?,
3x* + asx, 15 + 5ax5, - 2a*x + tax5 -10.
7. Eeduce to its simplest form : —
3/- {3? - (x - 3z) + 3/} - (3<7 - 3*) + 60.
8. If a = 2, b - 3, show that
(a2 - 2aZ> + 262)(a2 + 2ab + 262)
is numerically equal to
a4 + 4&4.
XXVII.
1. Find the greatest number which will divide each of the
three numbers 2622, 2793, 2736, without remainder.
2. Find the value of 246 gallons at 9s. If d. each.
3. How often is 17 dwt. 8 grs. contained in 7 Ibs. 2 oz.
13dwt. 8 grs.]
4. How many seconds are there in a lunar month ?
(29 days 12 hrs. 44 min. 3 sec.)
5. From the sum of £ and -| subtract the sum of £ and f .
6. From the sum of the first four of these quantities take
the sum of the last four : —
x2 + f - z, 2x2 - z, 3z2 - 2y2 + 3z,
3x2 - 2f, 2x2 - 3z, - x2 + 2y2 + 1 0&
How may the working be shortened 1
7. From
1 5 - 3a% + 4a#2 - 5y3 take 17-1 Oz2y - 4zy2 - 5f.
8. When a = 4, b = 5, c = 6, find the value of
14 MATHEMATICAL EXERCISES.
XXVIII.
1. What is the least number which, when divided by 3, 4, 5,
7, leaves in each case a remainder of 1 1
2. How many hours are there in the sixth part of a leap-year1?
3. Multiply 4 lea. 2 mi. 6 fur. by 11^.
4. Eeduce to their lowest terms (i.) f^f-, (ii.) ^^.
5. Show that the fraction f^f is greater than £, and less
than £.
6. Find the sum of
7. Giving any numerical values you like to a and b, show
that
(2a3 + 4&2)2
is numerically equal to
8. From 1 2bx -3cx + 4dx - e take 1 2bx - 1 3cx + 4e.
XXIX.
1. Find the G.C.M. of
13x17x19, 17x19x21, and 19x21x13.
2. Multiply £4 13s. 2*d. by 279.
3. A room containing 120 people is 92 sq. j^ds. 2 sq. ft. in
area. How much space is that for each person ?
4. Add together 7 tons 18 cwt. 3 qr. 27 Ib. 11 oz. 13 dr.;
9 cwt. 2 qr. 15 Ib. 8 oz. 9 dr. ; 3 tons 1 qr. 13 Ib. 12 oz.
4 dr.; 69 tons 12 cwt. 24 Ib. 7 oz. 11 dr.; and 11 cwt.
2 qr. 6 Ib. 13 oz. 1 dr.; and from the result subtract
7 cwt. 2 qr. 19 Ib. 5 oz. 15 dr.
5. What must be subtracted from |- to make it equal to ^ ?
6. Simplify ax + (bx - ex) - (ax - bx) - (cx + bx).
7. Subtract 2d% + e2/ - jP from b2c - d2e + e2/ ';
and subtract x from a2.
MATHEMATICAL EXERCISES. 15
8. When x = 2, y = 3, z = 4, find the value of
XXX.
1. Find the L.C.M. of 52, 12, 4, 13, 5, 26.
2. Divide 5 cwt. by 7 ; and 1 ton by 8.
3. If a postman walks 12 miles a day for 6 days in the week,
and is paid 24s. a week, how much is that per mile ?
4. Multiply 23 mi. 7 fur. 18 po. by 65.
5. Simplify (i.) 8 - 5|, (ii.) 20 - 19j§.
6. From 2a3x - 2a?xz + 5aa? + 4s4 + 20
subtract - $a?x - 2a?x2 - Sax5 + 4x*- 20.
7. Add together
12bx - Sex + 4dx - e, 12bx- I3cx +
and
8. Simplify
XXXI.
1. Find the G.C.M. (or H.C.F.) of 66150 and 145800.
2. Find the price of 3764 things at £4 11s. 8d. each.
3. When the moon is eclipsed the earth is between it and
the sun ; find the distance of the sun from the moon at
an eclipse, the distance of the moon from the earth
being two hundred and thirty-eight thousand, seven
hundred and ninety-three miles, and the distance of
the earth from the sun ninety-one million, four hundred
and thirty thousand miles.
4. Reduce 375250 sq. Ik. to acres; and 72 sq. ch. to acres.
5. Simplify 3^- of ilf of 9| of 1 of 21^.
16 MATHEMATICAL EXERCISES.
6. Keduce to its simplest form —
3 - x - (2x - 4) + (x - 4) - (5 + 3x) + x.
7. Find the value of
4X3 - 2z2a - 4za2 - a3, less 5a?x + 2z3 - 2oz2 - 4.
8. When b = 1, d = 2, show that
x (2&2 -
XXXII.
1. Find the L.C.M. of 4, 7, 18, 8, 25, 9.
2. Divide £763 8s. Ifd. by 3.
3. Add together
Cub. Yd. Cub. Ft. Cub. In.
431 26 534
259 13 883
864 2 1428
737 20 1000
844 14 663
119 10 545
4. Reduce to their lowest terms ^57, and
5. Divide 3£ by £f ; and simplify 12|-r 7^.
6. Find the sum of
3a2 + 2a&-52, 3a&-5, 5a2-2ai- 10Z>2-5,
2ai-3, 7a2+2a6-5Z>2-3, 7
7. Test the result of the last sum by putting a = 2, b = 3
8. Take
4m%i + 2mn? + nz - 4 from 4m% - 2wm2 - n3 - 4.
XXXIII.
1. Find the G.C.M. (or H.C.F.) of 5544 and 2304.
2. What would a man's income amount to in a year at
£2 3s. lOd. per day ?
3. What is the average produce per acre, if 37 acres yield
317 qrs. 7 bush. 3 pk?
MATHEMATICAL EXERCISES. 17
4. Eeduce 5640 ft. to miles, and 73482 in. to miles.
5. Add together (i.) £ + § + £, (ii.) 3f + 5£ + £.
6. Simplify
Sax - {2bx - (ax - 5) + 4} + 3bz.
7. Add together
c4 + d5 + c*d5 - 4, - Sc4 + 4 - 3d5, 5d5 - 3C4 4- 2c*,
18c*-8, d5, 3d5
8. When y = 5, a = 4, find the value of
y-g
XXXIV.
1. Multiply £10 16s. 8d. by 112.
2. Divide 53 acres 1 rood by 683.
3. Reduce 10 tons 7 cwt. 4 qrs. 2| Ib. to ounces.
4. If 6 things cost £15, what will 5 cost ?
5. Find (i.) the square of 13,
(ii. ) the cube of 8,
(iii.) the square root of 121.
6. If a = 2, b = 6, c = 4|, d = 0, find the value of
ad2 - ±bc + 2(Z>2 - d2) -(ac + b).
7. Add together
4a3-2a2 + 6a, 3a4-2a+6, 9a2-2, and
8. Multiply x2-xy + y2 by xz + xy + y2,
and divide a by a.
XXXV.
1. Find the L.C.M. of 25, 75, 125, 150, 180.
2. Divide £5678 15s. 9d. by 121.
3. Multiply 5 tons 2 cwt. 3 Ibs. by 98.
4. If a man pays an income tax of 7d. in every £, how
much would he pay on £375 15s.1?
18 MATHEMATICAL EXERCISES.
5. If you had to give someone T of a certain sum, explain
how you would find out the amount to give them.
What are the "numerator" and "denominator" of a
fraction 1
6. Find the value of
a-b + c ad -be
a-b-c bd + ac'
if a=l, 6 = 2, c = 4, d = 0.
7. Add together
2(a -f b) + 3(x + y),
3(a + b) - 2(x + y),
and a + b -3(x + y).
8. Multiply a2 + 2ax + 3a2 by z2 - 2az 4- a2.
XXXVI.
1. Find the G.C.M. of 111540, 42336, and 67392.
2. Multiply £38 4s. 2d. by 84|.
3. Divide 5 acres 3 roods 7 perches 88 sq. yards by 11.
4. Eeduce 13400520 oz. to tons, cwts., etc.
5. Multiply together
13 07 <->f 3 9 anrl A2* nf 1 °
T5> *Tlf 01 d^T5 ana *¥S OI ^T-
6. If a = 3, 6 = 2, c = 4, d = 0, find the value of
(a2 - &)<*; and of Ja*b - ab* + $d - c3 + o63 + b.
7. Simplify Sa- 2(36- 3(c -d)}.
8. Add together
7c-4^+e, 6c + 3d!-5e, and 4e-12c;
and find by how much a is greater than b.
XXXVII.
1. Find the L.C.M. of 14, 85, 105, 77.
2. Divide £4536 17s. 7£d. by 821.
MATHEMATICAL EXERCISES. 19
3. If 137 ibs. of tallow cost £2 11s. 4|d., what is the price
of a ton ?
4. Reduce to their lowest terms, and to mixed numbers if
possible,
W^ and Mfc and ftftff
5. Divide lOff by
6. Find the value of
when a=3, 6 = 2, c = 3, d = 4, « = 5,/=l.
7. Subtract
1 Oa? - 7a% + 6z*/2 - y8 from 5s3 + 9z2«/ -
8. Multiply a2 + ft2 + c2 - a& - ew - c& by a + b + c.
XXXVIII.
1. Find the L.C.M. of 3, 17, 9, 51, 34.
2. Divide 8 cwt. 19 Ibs. by 65.
3. A cistern is supplied by a pipe which will empty it in
8 hours ; if 48 gal. 2 qt. 1 pt. run through the pipe in
1 hour, how many gallons does the cistern hold ?
4. If a ton of potatoes costs £7, what is the price of 24 Ibs. ?
5. Subtract 1^5 from 4f ;
and simplify 7| — 2 £.
6. When a = 0, b = 2, x = 3, find the value of
„ ,
3a?x + -T — — ao« - o4 + or + ate.
7. Subtract 2/2-3#2 from
8. Multiply ys + yz + y + l by
XXXIX.
1. What is the greatest number by which, when 1593 and
1925 are divided, 2 and 1 are left as remainders?
20 MATHEMATICAL EXERCISES.
2. From 365 days 6 hrs. subtract 365 days 5 hrs. 48 min.
46 sec.; and multiply the result by 400.
3. Multiply 8 oz. 15 dwt. 20 gr. by 18£.
4. Reduce to whole or mixed numbers
/i \ 4055 /: j \ 913
(*:) 27 > A11-; -g?--
5. By what must 11| be divided, to make it equal to 1^?
6. Simplify
3b - (2c - d) + 26 + (3c + 4d)
and - -(4&-(3c+d)}.
7. From 6a2 + 2a&-362 take - 4a2 - Bab + 452.
8. When x = 5, y — 3 shew that
is equal to 8xy.
XL.
1. Eeduce '9 and '0075 to vulgar fractions in their lowest
terms.
2. Multiply 3 yd. 2 ft. 4 in. by 37.
3. Subtract 13-0003 from 167;
and multiply 312-14 by 1-012.
4. Add together 2TV, 3|, -^ and 1^- ;
and divide 7f| by 4^.
5. If 67 yards of cloth cost .£36 17s., how many can be
bought for £111 7s. 6d. ?
6. If a = 2, b = 3, find the value of
a4
7. Add together
&-$ + &>
8. Simplify
(2 + z)(8 + a:)-22-(f + a;2);
and write in its simplest form,
"a divided by b"
MATHEMATICAL EXERCISES. 21
XLL
1. Find the G.C.M. of 10395 and 16819.
2. Reduce -345 and 3-45 to vulgar fractions.
3. Reduce 1 Ib. 2 oz. 3 dwt. to grains Troy.
4. Subtract 13^- from 16|;
and multiply together 6f| of 2^-, |f , and 4|
5. Add together 328-405, -006, 1-27, and 30-567 ;
and divide 10-209 by -41.
6. Multiply z2 + 2ao;+3a2 by x2 - 2ax + 2a2.
7. Divide
3&4-5a3& + 3fl4-a&3-12a2Z>2 by 3a?-b* + ab.
8. By how much is 80 greater than 40 + x 1
XLII.
1. Find the L.C.M. of 16, 24, 18, 49, 105, 30.
2. Reduce 562493 oz. to tons, etc.
3. Find the value of 57625 things at £1 11s. 7d. per score.
4. Subtract 34|£ from lOOf ;
and multiply together 8|, 45|4, ITS, T5V> an(^ 2£.
5. Reduce 4 bushels 1 peck 1 gallon to the fraction of
7 bushels.
6. From the sum of
2-z + a;2, 3x - 4 - 3z2, 5 + 2z2, and 7x-ic2,
subtract l-llz-3o;2.
7. Divide a3 + as2 - 6^ - 6a3 by ' x - 2a.
8. Prove that
(a - 6)3 + (b- c)3 + (c - a)3 = 3(a - b)(b - c)(c - a).
XLIII.
1. Find the L.C.M. of 153, 204, and 221.
2. Reduce 8868097 cub. in. to cub. yds. ft. and in.
3. Find the price of 334£ articles at 13s. 5d. each.
22 MATHEMATICAL EXERCISES.
4. Subtract 2||- from 5T2^ ;
and multiply together 5£ of 2T2¥, £, and 1^.
5. What fraction is 1 Ib. Troy of 1 Ib. Avoirdupois ?
6. Subtract k2l-3U* + P from -&3 + 4H-
7. Multiply Xi-x3y + x2y2-xy3 + y* by xz-
8. What do you mean by factors ?
Factorize a5-Z>5, a2
XLIII. («).
1. Reduce g-ro- and -^- to decimals.
2. If 27 vats each contain 1574 hhds. 62 gals. 3 qts. 2 gills,
find the contents of the whole (Wine measure).
3. Multiply 641 -2 by 1-037;
and subtract 32-816 from 40-0041.
4. Add together 4|, T8T, 2|, and 1 J ;
and divide 3|| by 2f>
5. If 2 cwt. 1 qr. 14 Ib. of iron cost 16s. 2fd., what is
the price of a ton ?
6. Find the value of
(a3 - 3a26 + 3aZ>2 - Z>3)(a2 - Wab + 962),
when a = 5, & = 4.
7. Simplify
{z(:e + a) - a(a - a)} x {x(x -a)- a(a - x)}.
8. Solve the equations : —
(i.) 3z + f-2(4z-6f) = 7(5z-6).
(ii.) 3z-6 = 4a; + 8.
Prove your results.
XLIV.
1. Find the G.C.M. of 52, 78, and 416.
2. Reduce -03 and .36 to vulgar fractions in their lowes
terms.
MATHEMATICAL EXERCISES. 23
3. How many ounces are there in 1 ton 3 cwt. 2 qr. 9 lb.?
4. Subtract 2T7^ from 5j\- ;
and multiply together 3||, 18f, T\, and 7f.
5. Add together 1338-465, l:096, 2-001, and 231-0103;
and divide 9-462 by -83.
6. Find the sum of
x3 + 3a?y+3xy'* + y3, x3 - 3x2y + 3xy* - ys,
and x3 - 3xzy - y3.
7. Divide a4 + 464 by a2-2a& + 262.
8. To the number x add 2, multiply the sum by 2, subtract
2 from this product, and divide the remainder by 2.
What is the result ?
XLV.
1. Find the G.C.M. of 8214, 1110, and 1702.
2. Find the price of 327| articles at 13s. 3d. each.
3. Multiply together 10|, ^, f|, and 1£;
and subtract 7T8T from 10T^.
4. Reduce § of £ of f of £1 to the fraction of £5 12s. 6d.
5. If 8 miles 2 furlongs of railway cost £165028 12s. 6d.,
find the cost of 22 miles 7 furlongs.
6. Multiply -a?-db-ca by -a + b-c.
7. Divide
a5 + 5a46 + aW - 1 Oa263 + 1 2a64 - 965 by a2 + 2ab - 3&2.
8. Factorize
13+1, ^
XLVI.
1. Find the L.C.M. of 11704 and 101080.
2. Reduce 10 ac. 3 po. 10|- sq. yds. to sq. inches.
3. Find the value of 675| yds. at £2 14s. 4d. per yd.
24 MATHEMATICAL EXERCISES.
4. Subtract 2f from 20| ;
and multiply T*T by !•£.
5. Eeduce Is. &|d. to the fraction of £1.
6. Subtract xz-z?y2 + y3 from
7. Divide x6 - 20a3z3 + 343a6 by
and divide the quotient by
x2 + 4ax + 7a2.
8. Prove that
(a - 2)3 - 2(a - 2)2 + 3(a - 2) - 4 = a3 - 8a2 + 23a - 26.
Are there any factors of a? + b-, or of a2-62?
Prove your answer.
XLVIL
1. Reduce 3| and |^ to decimals.
2. What weight will remain after taking 15 parcels weighing
3 qr. 13 Ib. 12 oz. 9 dr. each from a heap of 2 tons
14 Ib?
3. Subtract 28-4532 from 320-6 ;
and multiply 4-2635 by -068.
4. Explain exactly what is the meaning of f and of 2$.
5. Divide each of the above fractions by the other, in turn
also find their sum.
6. If a = l, & = 2, c = 0, d = 4, find the value of
{62 - (c + a)2} - {d - (a + F+^)} {d - (b + c~^)}.
7. Multiply
9a2 + 4y2 + z2 - 2yz - 3xz - Qxy by 3x+2y + z.
8. Solve the equations —
(i.) 6(x
Show that your answers satisfy the equations.
MATHEMATICAL EXERCISES. 25
XLVIII.
1. Find the G.C.M. (or H.C.F.) of 282660 and 40299.
2. To what vulgar fractions are the following equivalent: —
(i.) 15-4, (ii.) -714285 ?
3. Eeduce 5838297 oz. to tons.
4. Subtract 78f£ from 80^, and -849 from 8-02362.
5. Add together 8-376, -06703, 37-04, and 28-064015;
and divide -064665 by -0000135.
6. Find the sum and product of
3z2 - 4zy + 2y2, and 3z2 + ±xy.
7. Divide
6s4 + 6a%2 + 6y* + 5x3y + 5xf by 3xy + 2z2 + 2f.
8. There are two numbers whose product is 180. One is
x, what is the other ?
XLIX.
1. Eeduce to their lowest terms (i.) ff {**, (ii.) -flff .
2. A man earns 3s. 6d. a day ; what will he earn in 313 days 1
3. Simplify (i.)23x8x2i)
(ii.)T\x^x51xix|.
4. Find the value of (i.) 8£ of 10|d.,
(ii.) 4i£ of half a crown.
5. Express as decimals
5 15 25 226 37S 14263 11
TT5> Ttf> iinr> TUTTJ 1W5> ' 1 0 0 0 '» 1000'
6. Find the value of (2y + 0)3.
7. Divide a;5 -15^-36 by
8. Factorize 9a262-l; «2-7a; + 6; a;6 -64;
and show that the square of a + b is not a? + 62.
L.
1. Divide 29 sq. yd. 5 sq. ft. 72 sq. in. by 62.
2. Find the value of 376^ articles at £1 5s. ll^d. each.
26 MATHEMATICAL EXERCISES.
3. Add together f + 1 + f + J ;
4. Simplify
¥
5. Supply the term wanting in this proportion : —
91:117::( ) : 144.
6. Write down the squares of
2a + 3b; 4a-35; 5dt-4a; and 5a + 5b.
Add the results together.
7. Divide
1 2a4 + 22a3& - 4a2&2 + 1 Sab3 - 364 by 2a2 + 4a5 - 62.
8. Multiply a2 + 62 + c2-a6 + ac + &c by a + b-c.
LI.
1. Reduce f£ and ±£g- to decimal fractions. What is the
meaning of *3 ?
2. How many times can £4 2s. 3|d. be subtracted from
£300, and what will finally remain ?
3. Subtract 992-8456 from 1081-02;
and multiply together 4-283 and 43-0276.
4. Add together 8$, 6|, ||, and f £f ;
and divide 2^ by 6^.
5. If 37 things cost £8 11s. l|d. how many can be bought
for £35 3s. 1
6. If a = 4, 6 = 1, c = - 1, d = 0, find the value of
7. Multiply a + b-- by a-b- -.
c c
8. Solve the equations—
(i.) 3x-2(a;+l) = 4-i(2aj-4)
(ii) g+H_3g + 5 3g-8 ,
^ ; 22 2 4
Prove your results.
MATHEMATICAL EXERCISES. 27
LIT.
1. Find the G.C.M. (or H.C.F.) of 1019527 and 1231845.
2. Add together 23'54, 156-213, and 2-0004.
3. Reduce 5462764 sq. feet to acres, etc.
4. Bring 5-68 and 1-9 to vulgar fractions.
5. Divide 8-4605 by -248 to 3 places of decimals.
6. Add together
b + %c- ^a, c + fa - 1&, and a + $ - %c,
and from the result subtract
*a-b-%c.
7. Divide the product of
xz + ax + b, x2 + bx + a, x + a, and x + b,
by x2 + (a + b)x + ab.
8. What is meant by the square, and the cube, of a quantity ?
Give the square of a + b, and the cube of ab2.
LIII.
1. Find the G.C.M. (or H.C.F.) of 75582, and 42237.
2. Find the value of 112 things at £10 16s. 8d. each.
3. Reduce to a simple fraction : —
TS "SIT
4. Reduce £3 15s. 6d. to the fraction of £4 18s. 9d; and
find the ratio of 2J sovereigns to 2| guineas.
5. Reduce to vulgar fractions : —
•5, -25, -75, -12, -64, -375, -05.
6. From (3z-2y)(3a;-y) take (3z + 27/)2- 23xy.
7. Divide oty - o?y* + 8xy - 1Q by
8. When a = 2, b = j, show that
is equal to Sab.
28 MATHEMATICAL EXERCISES.
LIV.
1. Find the L.C.M. of 4608 and 9216.
2. In 31500 grains Troy, how many Ibs. and oz. Avoirdupois
are there 1
3. Find the cost of 5842 things at 6|d. each ; and at 3^d.
each.
4. What is the difference between 12 multiplied by |, and
12 divided by |?
5. Find what fraction 258 days 9 hrs. 19 min. 57 sees, is of
446 days 7 hrs. 23 min. 33 sees.
6. From the product of
take 2&5
7. Divide a3 - a?b + abz - ac2 + 6c2 - J3 by a2 + 52-c2.
8. Three factors of
amount to (x - b)3 when multiplied together ; find
the fourth.
LV.
1. Reduce to the form of decimals 177T^\ and 1J.
2. Multiply the difference between 12 cwt. 3 qr. 17 Ib. 10 oz.
and 5 cwt. 2 qr. 23 Ib. 11 oz. by 528.
3. Subtract 234-73201 from 436-237;
and multiply 211-0313 by 2-353.
4. Add together 4^, 3f, 1^, and £-;
and divide 2^ by 1^.
5. If 69 Ib. of salt cost 7s. 2£d., what will 4 cwt. 1 qr. 7 Ib.
cost ?
6. Find the value of
when a = 5 =
MATHEMATICAL EXERCISES. 29
7. Multiply x2 - (a + b)x + ab by x-c.
8. Solve the equations —
(i.) 30z+ 20(1 + x) + 15(2 + 3) + 12(3 + x) = 240 ;
and prove the correctness of your answers.
LVI.
1. How can you ascertain, without actual division, if a
number is divisible by 3, 4, 5, 9, 1 1 respectively ?
2. Reduce -00029 and -444 to vulgar fractions.
3. Reduce 3 Ib. 4 oz. 5 dwt. to grains.
4. Multiply 7£ by 7|; and 9-16 by 19-63.
5. Add together 131-31, 14-9256, -00946, and 8-0203;
and divide -81639 by 14100.
6. Find the sura of
a3 + ab2 + ac2 - a?b - a?c - abc, ft3 + fa2 + ba? - bza - tfc - abc,
and c3 + ca? + cb2 - cza - c*b - abc.
7. Divide a?(b + c) + c2(a + b) + b2(a + c) + 2abc
by b + c.
8. If a is a whole number, what are the numbers immedi-
ately preceding, and following it ?
LVII.
1. Find the L.C.M. of 18, 54, 16, 176, 96.
2. Reduce 19 fl. 3. 3 fl. 5. to 1T]_.
3. Find the cost of 300 things at £5 8s. 4d. each.
4. Simplify
(i.) 5!f-2^r, (ii.) 100-98J§, (ill) 4025| x 360|.
5. Reduce 2| weeks to the fraction of 8 days ;
and f of 1 ac. 2 ro. 13 po. to the fraction of 6 ac. 1 ro.
12 po.
30 MATHEMATICAL EXERCISES.
6. Subtract
7xzy - 2z2 - xyz from 5z2 - 3xf + 5x^y - 2f.
7. When a = 25, b = 9, c = 4, J = 1, find the numerical value of
3 Va + 2 x/4l - x/9c + Jl6d.
8. Prove that
(x - y)s + (y- z)3 + (z - x}3 = 3(x - y)(y - z)(z ~ x).
LVIII.
1. What are the prime factors of 1232, 2112, 2268, and
2880?
2. The weight of a sovereign is about 5 dwt. 3^ grs. ; how
much will 120 sovereigns weigh ?
3. Find the cost of 5 cwt. 18 Ib. at 10s. 4d. a stone.
4. Simplify |f of |f of 5^- x 1^ of 6fi
5. In 1 year £4 of interest is gained on £100 ; how much
ought to be gained on 450 guineas ?
6. Find the value of
(x3 - Sx2 + 3x- 6)(z2 + 3x + 6).
7. Divide a3J2-2a&4 + 65 by a2J + a62-63.
8. From (4/- 3</)(3/+ 2<?) take (- 5f+g)(3f- 2g) •
and show that the square of a - b is not a? - 52.
LIX.
1. Eeduce to decimals (i.) ^ of | ; and (ii.)
2. Multiply 3 tons 4 cwt. 1 qr. 17 Ib. by 42.
3. From 121-0768 take 48*398675 ;
and multiply 8-056 by 250-635.
4. Add together 5^-, 3^-, -^, and 2£ ;
and divide IQlf by 2.
5. What is the meaning of " a fraction " ?
MATHEMATICAL EXERCISES. 31
6. Square 2b - 3c.
7. What does the identity
(x + y)(x -y) = xz-y2
become, when a + b and c are substituted for x
and y respectively ?
8. Solve the equations : —
(11 \ /70* — 7) •
11.; ax — u,
and show that your answers satisfy the equations.
LX.
1. If 5 ac. 3 ro. 4 po. are valued at £1125, what will be the
value of 44 ac. 3 ro. 1 po.?
2. Find the vulgar fraction which is equal to '78516; and
express T as a decimal.
3. How many pieces of cloth each 14 yds. 1 qr. 3 nl. long
could be cut from a piece 129 yd. 3 qr. 3 nl. long 1
4. Express 2| of £15 16s. 3d. as the fraction of £50 4s. 9d.
5. Simplify 1 '375 + 30 -2 + -07125; and find the difference
between -399 and -4.
6. Multiply a3 + a2 + a + I by a - 1. •
7. Find the factors of
3a3 - 6a2& + 9a4&2 and mx-my-xy + f:
8. If a; be a number, and it be doubled, 10 added to it,
the result halved, and x subtracted from this, show
that 5 remains.
LXI.
1. Eeduce 17|, -^£5, •£, and f to decimals.
32 MATHEMATICAL EXERCISES.
2. Add together the following quantities and divide their
sum by 7 : —
Oz. Dram. Scr.
11 7 2
432
10 5 0
9 4 1
3. If a square contain 14400 square inches, find the length
of one side of it in feet.
4 Simplify 7^of^ »
3 '14'
5. Find the value of 1 -025 of 40 guineas, in sixpences.
6. Show that
7. What are the factors of
1 6a4Z>3c + 24a25c5 - 40aW and 6z2
8. A man bought a suit of clothes for £4 7s. 6d. The
trousers cost half as much again as the waistcoat, and
the coat hah0 as much again as the trousers and waist-
coat together. What did each cost ?
LXII.
1. Keduce -375, 100-001, -9, -714185 to vulgar fractions.
2. Multiply 20 po. 2 yd. 1 ft. 6 in. by 64.
3. From Paris to Leipsig is 1225 kilometres. If 8 kilo-
metres are equal to 5 miles, find the distance in English
measure.
4. Simplify if of 3f + i£ x 9f - i of 2f - 1|.
5. Express 1-76 feet as the decimal of a mile.
6. Prove that
(a - 6)8 + (6 - c)3 + (c - a)3 s 3(a - 6)(6 - c)(c - a).
MATHEMATICAL EXERCISES. 33
7. Factorize —
x2-2x-3, z*-l, and x* + 3z3 + x + 3.
8. Simplify 2{a- (J-c)} - {a + c- 3(b + d)}.
How many times will x go into y 1
LXIII.
1. Express the product of
2i + F + in: + T^and|-F + Tr-TT!y as a decimal
2. Multiply 9 wk. 2 days H hr. by 101.
3. If I give away f of my money, then f of what remains,
then -| of what still remains, what fraction of the whole
shall I have left !
A Sirnnlifv I32_i4,35i 7 21
*• »- Hupniy TSTf TTT + 7T + 77 ~ T^'
5. How many ounces of silver are there in a piece of plate
costing £84 6s. 8d., the price of the silver being *255+ '16
of «£!, per oz.
6. If (x-2)z-(x-l)(x-3) = a,
find the value of a in its simplest form.
7. Divide (a + i)2 - (a + b)(c + d) - 6(c + <2)2
by (a + 6)-3(c + d).
8. If a be multiplied by itself (i.) 8 times, (ii.) n times,
what is the result ?
LXIV.
1. Find the cubical content of a cistern 36 ft. 8 in. long,
7 ft. 11 in. broad, and 6 ft. 5 in. high, giving the
answer in cubic yards, feet, and inches.
2- Bring ^^, f , T4^ to decimals.
3. Find by Practice the value of 731f£ things at 14s. 2d.
each.
4. Simplify 2| + 3£- 3f -2|+4J-
5. Multiply 23-002258 by -004003.
34 MATHEMATICAL EXERCISES.
6. What is the square of x - 1y + 3z 1
1. Add together
and ae - bf.
8. Prove that x = 7 satisfies the equation
LXV..
1. A besieged town of 45,500 inhabitants has provisions for
3 weeks ; how many must be sent away that they may
be able to hold out for 7 weeks 1
2. Reduce -32 1, and 61-6 to vulgar fractions, and multiply
them together.
3. The sum of £8485 was left by will to be divided equally
among a certain number of persons. The share of each
amounted to £565 13s. 4d.; find the number of persons.
4. Find the value of 6-125 of £1 13s. 4d.
5. Add together 27-347, -000219, 1714-8342, and -733114;
and subtract lOO'OOOOOOl from the result.
6. Multiply ab-ac + bc-bd by ab-ac-bc + bd.
7. Factorize tf-y8, x7 - y7, x°-l 2xy - 28y2,
and 6X3?/ - 9o%2 + 3x2yz.
8. What is the sum of n a's all added together.
LXVL
1. Reduce 9876543 inches to miles, furlongs, etc.
2. If lead is worth £9 6s. 8d. per ton, how much can be
bought for £100 1
3. Find the value of 2 tons 3 cwt. 3 qr. 9 Ib. at £3 6s. 5d.
per cwt.
4. Add together 5i, 6£, 8|, and 23.
MATHEMATICAL EXERCISES. 35
5. Find the G.C.M. of 24057 and 3645.
6. Find the factors of
x8-9a^+14 and of x2-kx + lx-kl.
7. Divide afi-b6 by a -b.
8. Multiply m3 + 2wm2 - «3 by mP + ri2;
and a;2" by of.
LXVII.
1. Kesolve 3564 into its simplest factors.
2. Multiply 17 tons 13 cwts. 2 qrs. 23 Ibs. by 56.
3. Find the value of 2 Ib. 3 oz. 15 dwt. 6 grs. of silver at
5s. per oz.
4. Add together 3||, 2|£, **, and 4| ;
and divide 8£| by l£f.
5. If 7 Ibs. can be bought for £1 17s. 4d., how much can be
bought for £106 2s. 8dJ
6. When a = 3 and b = — 6, find the value of
a-
7. Add together
a3 + b3 + 3a26 + 3a62 ; 2a2(a - b) + 262(J - a) ;
and ab(2a + b) - 3a(a* + 62) - 6(3a2 + 362).
8. Knowing that (x - y)(x + y)=x2- y2,
find, without multiplication, what
(12a-13&)x(12a+13Z>)
amounts to, showing how you get your result
Lxvni.
1. Divide 109 ac. 2 roods 15 po. 12£ yds. by 56.
2. Find by Practice the dividend on £3275 10s. at 14s. 3|d.
in the £.
36 MATHEMATICAL EXERCISES.
3. Add together If, 2f , and 3f| ;
and divide f by 1^.
4. Simplify |^
5. If 20 Ibs. of coffee cost 35s., what will 4 cwt. 2 qrs.
16 Ibs. cost 1
6. Simplify 5z-3[2z+9y-2{3z-2(z-5¥
7. Break up into factors
w6-n6, m4-w*, and m?-n3.
8. Solve the equation
6(z - 1) - 3(x - 2) + 2(a; - 3) = 0.
Verify your result.
LXIX.
1. Find the L.C.M. of 3696, 286, and 19656.
2. Eeduce 2 tons 7 cwt. 2 qrs. 11 Ibs. to oz.
3. Subtract 9if- from IQJj
and multiply together |f, 3i, 2|, and 1£|.
4. Find the value of 3 cwt. 1 qr. 10 J Ibs. at £8 a ton.
5. Eeduce 3s. 6d. to the fraction of 6s. 8d.
6. Subtract the sum of
x - 4a + b, 3x+ 26, and a-x-5b,
from the sum of
b - 6x and 2a - 3b.
7. Divide
8z* - 2ax* + 3a?x* - 2a?x + a4 by 2z2 + ax + a2.
Test the correctness of your answer by putting
a=l.
8. Show that (x- 2)2- (a- !)(»- 3) = 1 ;
and explain the meaning of (x - 2)2.
MATHEMATICAL EXERCISES. 37
LXX.
1. Find the G.C.M. of 38642 and 1390.
2. Find by Practice the value of 432| articles at 17s. 8d.
each.
3. From 8| take 64 j
and multiply 4^ by ££.
4. How many grains are there in f of 1 Ib. 5 oz. 3 dwt.
14 grs. Troy 1
5. The length of a street is 937 ft. 6 in. and the breadth
66 ft. 8 in., find the cost of paving it, at 8|d. per square
yard.
6. Multiply x2 - Sax by x + 3a.
7. Divide fr + k*P + W + ti* by k* + P.
8. Factorize
a2-2az-3o;2, a2 + 3az + 2z2, and a2 + 2ab + b* - c2.
LXXI.
1. Find the G.C.M. (or H.C.F.) of 14455 and 14896.
2. What is the cost of 8713 acres of land at £Q 14s. an
acre?
3. By what must 2| be multiplied to make it equal to 4^ ?
4. Reduce f of £1 to the fraction of 1 guinea ; and ^ of £2
to the fraction of 2 guineas.
5. Add together 7316-24871, -007, 65-83, and 517-432765.
6. Multiply together
a- 26 + 1, 2a-f6-l, 2a+36+l;
and test your result by putting a = 2, 6 = 1.
7. Divide a2 + Z>2-c2-d2 + 2o& + 2cd by a + b + c-d.
8. Find the factors of
Icx + lcy-lx-ly and ie4- 12z2+ 27.
38 MATHEMATICAL EXERCISES.
LXXII.
1. Divide £5688 15s. 3d. by 276.
2. Find by Practice the value of 90072 articles at £2 17s. 6d.
per dozen.
3. Add together 4f , f , 1^, and 3| ;
and divide 7^ by 1±.
4. Simplify l±i
5. If 7 bushels of corn will last 30 weeks, and the price be
65s. a quarter, what will be the cost for 12 weeks.
6. If x = 2 find the value of x6-7x3-8.
7. Add together
1c-±d + e, 6c+3d-5e, and 4e-12c.
8. Write down, without multiplication, the value of
(3x-8y)(3x+8y),
showing by what rule you do it
LXXIII.
1. Find the L.C.M. of 8085 and 4125.
2. Eeduce 5 mi. 7 fur. to yards.
3. Find the value of 8 cwt. 2 qr. 14 Ib. at £2 4s. 4d. per
cwt.
4. What must be added to 7 f to make it equal to 12£?
5. Reduce £5 17s. 6d. to the fraction of £3; and 15s. to
the fraction of a guinea.
6. Find the value of
(x2 - ax - a?)(xz + ax).
7. From the product of
2bx + 3cz2 - 4s3 and 2bx + 4s3
take 2b2x" + Qbcx3 - I lex* - 1 2x*.
MATHEMATICAL EXERCISES. 39
8. One factor of
- s
Find the other.
What is a factor 1
LXXIV.
1. Find the G.C.M. of 2697, 3441, and 1271.
2. Find by Practice the dividend on £6421 15s. at 13s. 9d.
in the £.
3. Subtract 7j| from 10T2T;
and multiply together 3^-, 1-||, 1^-, 2||, and 2£.
4. Eeduce 6 bushels 2 pecks to the fraction of 2 qrs.
6 bushels.
5. Find the cost of carpeting a room 34 ft. 8 in. long and
1 3 ft. 6 in. wide at 4s. 6d. per square yard.
6. Multiply 3x + 2y - z by x - 2y + 3-?,
and prove the result by division.
a2
7. Divide xt-y* by %-y, and simplify — .
d
8. Split up into factors
mV + 3mn - 28 and z4 - a4.
LXXV.
1. Divide 932 mi. 1 fur. 32 po. by 56.
2. Find by Practice the value of 2870| articles at 14s. 3d.
each.
3. Add together 2J, 3f , 1^, ^, and 3^ •
and divide 8^ by
40 MATHEMATICAL EXERCISES.
5. If a 4 Ib. loaf costs 6d. when wheat is 52s. a quarter, what
ought a sixpenny loaf to weigh when wheat is 65s. a
quarter ?
6. Simplify
(a - b + c) - (b - a + c) + (c - a + b) - (a - c + b).
7. Factorize
2 + 133+15z2, z2 - 4o*/ + 3?/2, and tf-y8.
8. Solve the equation
Show that your answer satisfies the equation.
LXXVI.
1. Resolve into prime factors 105, 112, 125, 140.
2. What is the solid content of 24 blocks of stone, the con-
tent of each being 4 cub. ft. 1232 cub. in.1?
3. Find the value of 5 cwt. 2 qr. 16 Ib. at £3 15s. per cwt.
4 Simplify i*i;f* A; 5jf; !^3-
5. A man pays £42 5s. 10d. income-tax when the tax is 7d.
in the £ ; what is his income ?
6. Add bdxt + cW + Scdxt-tdW
to the product of bx - ex + dx,
and bx + cx- 2dx.
How can you test your result ?
7. Simplify 3x-('2y + z- {4y + [5y + z] + y} + 5).
8. What is the numerical value of
a2 + (2a + i)2 - (c - d)(c + d)- 4a6 - b2,
when a=l, 6 = 2, c = 3, d = 4?
LXXVIL
1. Find the L.C.M. of 111, 37, 8, 12, 3.
2. Reduce 610247 sq. Ik. to acres.
MATHEMATICAL EXERCISES. 41
3. Find the value of 15 cwt. 3 qr. 21 Ib. at £10 Os. 8d. per cwt.
4. Simplify (i.) ^-^; (ii.) f-|; (iii.) 1341^ x 10811
5. Reduce £1 12s. 7fd. to the fraction of £4 17s.
6. Divide a3 + 3a26 + Sab* + ft3 by a + b.
7. Multiply 1 + xy + x2y2 + aPy3 by 1 - xy + x*y* -
8. Prove that (a - 6)3 + &3 - a3 = 3ab(b - a).
In what different ways can you write algebraically —
" x multiplied by y"1
LXXVIII.
1. Break up into its prime factors 2273832.
2. Multiply 584 by 899 in the shortest way you can.
3. Find by Practice the value of 15 yds. 2 ft. 8 in. of carpet
at 5s. 4d. per yd.
4. Add together 5|, 4|, and 3y|- ;
and divide 4i| by 4f^.
5. If 1 cwt. 1 qr. 21 Ibs. of copper cost £4 6s. 3d., how
much would a ton cost 1
6. When a=l, b = 2, c = 3, d = 0, x = ^, find the value of
-2ab + W + cd; and of V3 ~Q2f + X9.
x(l - 3x) + x2
7. Add together
fa2-|a& + f&2, a2-2a&+*62, and - 4a2 + |a& - 6i2.
8. If (x + y)(x-y) = x*-y*
to what is (3k + 4/)(3* - 4?) equal ?
If two quantities are placed together with no sign
between them, what do you infer about them ?
LXXIX.
1. Find the G.C.M. of 8996427 and 9078399.
2. Find by Practice the value of 3 oz. 13 dwt. 18 grs. at
£3 10s. per oz.
42 MATHEMATICAL EXERCISES.
3. Subtract 9f| from llf ;
and multiply together 1£, 1£, 2|, and 3^.
4. Keduce 13s. 4d. to the fraction of £2 6s. 8d.
5. Find the cost of carpeting a room 16 ft. 7 in. long and
13 ft. 5 in. wide with carpet at 4s. 2d. per square yard.
6. Multiply together a - b, a + b, a2 + 52.
7. Divide 1 - x - 3z2 - x5 by 1 + 2x + x2 ;
and test your result by putting x= 10.
8. Break up into factors
x2 + 2ax - 3a2, x2 + (a + b)x + ab, and a16 - y16
(5 factors of the last).
LXXX.
1. What are the prime factors of 6144 and 6561 ?
Hence write down their G.C.M.
2. How far will a railway train travel in 13 hrs. at the rate
of 1 9 mi. 3 fur. 20 po. an hour ?
3. Find the value of 6 cwt. 3 qr. 9 Ib. at £2 14s. per cwt.
4. Which are the greatest and least of the fractions —
l A a a 11
2> 5' 3' 4> 8
5. If a man walks 4£ miles in an hour and a half, how long
will he take to walk 28| miles 1
6. Divide «2+ 126c - 462- 9c2 by a+2b-3c.
Show that this can also be done by factors.
7. Simplify a2 + (a + 5)2 - (c - d)(c + d) - 2ab - b2.
8. From the square of 2x + y
take (2x - y}\
LXXXI.
1. How many packets each weighing 2 Ib. 3 oz. can be
made up out of a chest of tea weighing 3 cwt. 2 qr.
91bJ
MATHEMATICAL EXERCISES. 43
2. Find the dividend on £276 14s. 8d. at 13s. 4d. in
the £.
3. Arrange in order of magnitude : — ^, fy, ^4T, T|.
4. Simplify |+_JL_.
5. If £ of an estate be worth £1003 17s. 2d., what is the
value of J- of it ?
6. Multiply x2 + y2 — z by x2 - y2.
Prove your result by division.
7. Find the square of ax2 - ax + 4.
8. Divide a3 - 2a2c - abc - 2b2c + b3 + c3
by a2 + b2 - c2 - ab - ac - be.
LXXXII.
1. Find the G.C.M. of 149291688 and 2194920.
2. Eeduce -0218 to a vulgar fraction ;
and divide 2631-874 by 70.
3. How many square feet are there in 3 acres 1 rood
4 perches 3 yards.
4. Subtract 5| from 10 J;
and multiply 3^ by 5^.
5. Find the sum, difference, and product of -025 and -03.
6. Add together
- 2(a + c) - B(b + c),
7. Divide ic4 + (a + 5)a;3 + (a + b + ab)xz + (a2 + b2)x + ab
by x2 + ax + b.
8. Substitute x and y for k + l and k-l, respec-
tively, in the following : —
(k + l)z - 3(& + l)(k -l) + 2(£ + l) + l-k.
If k = I, what do you know about the value of y ?
44 MATHEMATICAL EXERCISES.
LXXXIII.
1. Find the rent of 8 acres 3 roods 10 poles at £1 17s. 8d.
per acre.
2. Find the value of 3421^ articles at 13s. 4d.
3. Add together 3f , 4£, 5| ;
and divide Iff by if
4. Subtract &of3f from £&
47
5. K rail way fares in Russia are § of the fares in England,
and it costs £1 12s. 6d. to travel 180 miles in England,
what will it cost to travel 1 60 miles in Russia ?
6. Simplify 3a-i{3b-7(c-d)}.
7. Break up into factors
(i.) x2 - Sx - 180 ; (ii.) (a + 6)2 - (c - <Z)2 ; (iii.) tf - f.
8. Solve 9(x-2) + 12(a; + 3) = 5(4a; + 5).
Show that your answer satisfies the equation.
LXXXIV.
1. Add together 123-478, 1'03, 12, 1-0001, and -437.
2. Find by Practice the cost of 12480 things at £3 7s. 5£d.
each.
3. Add together 3f of |, ^ of If, and 2£ of | ;
and divide 23i£ by 6^.
4. Divide 4 miles 11 poles 4 yds. 1 ft. 6 in. by 29.
5. If the interest on £100 be £5, what is the interest on
£433 6s. 8d.?
6. Simplify 2a- 3(6 -c) + (a-2(6-c)} - 2{a- 3(6 -c)}.
7. Factorize a;2-3a;-4, a^-1, x7+l.
8. Solve 4(3«-2)-2(4z-3)-3(4-a) = 0.
Explain what is meant by " satisfying an equation."
MATHEMATICAL EXERCISES. 45
LXXXV.
1. Resolve into prime factors 6655 and 7168.
2. Multiply 15 fl. 3. 6 fl. 5. 30. IT^. by 42.
3. Find the cost of 19 yards 2 feet 5 inches at 5s. 7d. a foot.
4. Divide 4} of !£, by ft of 1ft of 1 ^
5. A man reaches a town in 18 days by walking 9 hours a
day; how many days would he have taken, had he
walked 12 hours a day ?
6. Find the continued product of
x + y, x-y, x* + y2, and x2-y*.
Show that your result is correct when x = 2, y=l.
7. Simplify 3b* + 2b(b + a)-(b + a)2-W.
8. When a362 - 2ai4 + 2i5 is divided by ab - bz, what is
the remainder ?
What values of b would make this remainder an even
number ?
LXXXVI.
1. Find the L.C.M. of 508, 889, and 635.
2. Reduce
to its simplest form.
3. Find by Practice the value of 3 qrs. 1 bush. 1 gall at
£2 13s. 4d. per quarter.
4. Subtract 14| from 21£ ;
and multiply together 2 If, 4-^, |^-, and ly^r-
5. Reduce 15 cwt. 2 qr. 21 Ib. to the fraction of 2 tons
10 cwt.
6. Subtract a2 + ab + b2 from a4 + a2i2 4 J4.
7. Divide «4 + a262 + 64 by a2
46 MATHEMATICAL EXERCISES.
8. Prove that if you take any number, double it, add 10 to
it, take half, subtract the original number, that 5 will
remain.
LXXXVII.
1. Find the simplest factors of 5040.
2. Multiply 14 Ibs. 8 oz. 6 dwt. 4 grs. Troy by 841.
3. Find the value of 17 acres 3 roods 25 perches at 5
guineas per acre.
4. Add together £, ^ ^g, and 4£ ;
and divide 9^ by 3^-.
5. If a bankrupt pays 3|d. in the £, and his total payments
amount to XI 54, what was his debt 1
6. Write down the numerical value of
Vs
+ y + z)(x-y + z)
x + y-z
when a; = 4, y = 5, 2 = 6.
7. Add together
8. Show how to find the value of
(m-n+p- q)(m -n-p + q),
without multiplying out.
[I.e. from x*-yz = (x- y)(x + y).]
LXXXVIII.
1. How many days will a snail take to crawl 516 yd, 3 in.
at the rate of 2 ft. 9 in. daily ?
2. Find the price of 3 cwt. 1 qr. 20 Ib. at £1 2s. 2d.<per
quarter.
3. Add together 4012|, 639j|, 15023^, and 98i|.
MATHEMATICAL EXERCISES. 47
4. Simplify - *"*
5. Find a man's income who spends 12 guineas in 35 days
and saves £100 a year.
6. Divide 16x* - x* + IQa? - 25z2 by 4a;3 + 5a;2.
7. From the product of (d + e-f) and (e+f-d) take
(rf + *-/)2.
8. Subtract (2z + y)2 from (-Bx-y)z;
and find by how much x is less than 20.
LXXXIX.
1. Find all the factors which are common to both 3786 and
576.
2. Multiply 3 qrs. 2 bush. 1 pk. 3 qts. by 17.
3. Find by Practice the value of 7719 articles at £1 7s. 6d.
per dozen.
4. Add together if, 1^, and 5^ ;
and divide 4| by 12^.
5. In 100 half-guineas how many fourpenny pieces ?
6. If a = 4, b — \, c = I, find the value of
7. Add together
(J> - c)x + (c - a)y + (a - b)z
and (2a - b)x + (3& - c)y + (4c - a)z.
8. Find the product of
tf-Pm + m? and P-Z^-m3,
without actual multiplication, explaining how you do it.
XC.
1. Find the G.C.M. of 101080 and 138945.
48 MATHEMATICAL EXERCISES.
2. Find by Practice the dividend on £3760 15s. at 17s. 9d.
in the £.
3. Subtract 3T\ from 8| ;
and multiply together 2f- of 1^-, 2f- of -|, and 3£ of -j^.
4. Express 2| of !T5^d. as the fraction of 2s. 6d.
5. Find the expense of paving an area 125 ft. 8 in. long and
48 ft. 6 in. broad at Is. 4d. per square yard.
6. Multiply together
a + b, a2-62, a-b, and a?+2ab + b2.
7. Divide
42z2 + 216a;?/ + 30y2 by ±x-(y-x)- 3{2y- 3
8. Find the factors of
(i.) 4o:2 - 9?/2, (ii. ) a2 - ab - 6Z>2, (iii.) 3x*y - 27 f.
XCI
1. Divide 727 Ib. 10 oz. 16 dwt. 12 gr. by 92.
2. Find by Practice the value of 12 ac. 2 ro. 12. po. at
£37 10s. per acre.
3. Add together 3f , 4£, £, and £| ;
and divide 4^ by 12|.
4. Simplify || + ^.
dT LS
5. If 20 men can mow a field in 84 hours, how many fields
will 75 men mow in 84 hours ?
6. Simplify 4(x - y) + 6(x + y) + 3x- (2x + y).
7. Resolve into factors
x2 -6x +5, a^-jy, m2w2-l, F + 4&-96.
8. Solve the equations : —
(i.) 4(z-3) = 5(aj-4),
(ii.) (x - 3)(z + 4) + (x - 5)(x + 2) = (2x + 7)(x - 4).
Test your results.
MATHEMATICAL EXERCISES. 49
XCII.
1. Find the volume of water which will fill a tank 5 feet
long, 4 ft. 7 in. wide, and 2 ft. 4| in. deep.
2. Express j^7T, 5f-, and ^ of 3^-| as decimals.
3. Find the value of 642 If things at 13s. 9d. each.
4. Add together 1^, 2|, -^-, and ^-;
and subtract 12£ from 14^.
5. Find the value of (i.) -0032 x 23-45 ; (ii.) 181-3 - -00037.
6. Explain how to find the result of
(3m -
without multiplying.
7. From a2 - £&2 - (3d2 - £c2) subtract 62 - fa2 - ( fc2 -
8. Solve the equations : —
(i.) 15z-10 = 20z-35.
(ii.) 5(2x+l)-2(7-a;) = 33.
XCIII.
1. What number divided by 48762 will give 6285 for a
quotient and 26108 for remainder?
2. Multiply 15 Ib. 9 oz. 17 dwts. 18 grs. by 23.
3. Find by Practice the value of 12 cwt. 3 qr. 3| Ib. at
48s. per cwt.
4. Add together 5|, 3^, and 1-^;
and divide 6| by 2F%.
5. If £3745 put out at interest amounts to £3932 5s. in 2
years, what will £100 amount to in the same time?
6. If a= - 1, b = 2, c - - 3, d = 0, find the value of
(4a+3&- 2c + d)(-a + b + c + d).
7. Add together
3z - 3a + by, 4y-4x + 6a, 5a - 5y + 1x,
and x-3a + y;
and test your result by putting x = y = a = 1 .
50 MATHEMATICAL EXERCISES.
8. If (x-y)(x + y) = x*-y'2, what is
(2Jcl - 3mn)(21d + 3mri)
equal to ?
XCIV.
1. Find the G.C.M. of 1.57 days 7 hrs. 4 min. 7 sec. and
243 days 2 hrs. 11 min. 49 sec.
2. Find by Practice the value of 319 cwt. 3 qr. 16 Ib. at
£2 12s. 6d. per cwt.
3. Add together $ of f, ^ of f, and 1£ of £ ;
and divide 2^- by 3^.
5. Find the number which can be exactly divided by all the
numbers up to 15.
6. Simplify (x - y)2 -(x + y}(x -y)- {x(2y - 2) - (y* - 2x}} ;
and show that this expression vanishes if y = 0.
7. Find the factors of
kxz + lxy-kxy-ly2, and cc3 - 3ax* + 2a?x.
8. Solve the equation
xcv.
1. How many flagstones, each 5-76 feet by 4*15 feet, are
required to pave a cloister which encloses a rectangular
court 45'77 yards long and 41 '93 yards wide, the
cloister being 12-45 feet wide ?
2. Divide ^pr—y by JT ; an(j reduce the result to a
'
-j-g-
decimal.
MATHEMATICAL EXERCISES. 51
3. Find by Practice the dividend on £731 14s. 6d. at 14s. 2d.
in the £.
4. Add together 13£ of 2J, Hi of 5|, 1^ Of 9^, and
I off.
5. Divide the square of 1-25 by the cube of -05.
C. Find the value of
(2a - 3& + 4c)2 - (2a + 36 - 4c)2
by means of the identity
7. From a3 take 63-c3.
8. Show that a; = 3 is an answer to the equation
3+ ^TT~2
XCVI.
1. If 97 Ib. of coffee cost £14 2s. lid. what will 73 Ib. cost 1
2. Reduce -US, 8-3, 42-679 to vulgar fractions.
3. Reduce 27045 guineas to half-crowns ;
and 5 bush. 2 pks. 3 qts. to pints.
4. Add together ^5- of 2s. 6d., and ^ of a guinea, and from
the sum deduct 3T7^ of Is. 6d.
5. Add together 7'076, 19-245, and 18-203.
6. Multiply
2z3 - 3x2 + 4z - 5 by 7x + 9 ;
and test your result by putting x=l.
7. If a + 3b is a factor of
a? + 3a?b - ab* - 3bz,
find the others.
8. Prove that if you add together the squares of
x, x+1, and x2 + x,
the result will be equal to the square of
x2 + x + 1.
52 MATHEMATICAL EXERCISES.
XCVIL
1. Find the cost of carpeting a room 16 ft. 8 in. long and
13 ft. 5 in. broad, leaving the hearthstone which is 4 ft.
2 in. by 1 ft. 8 in. uncovered. The carpet is f yard
wide and costs 6s. a yard.
2. Give the result of -714825 x -013 as a vulgar fraction
in its lowest terms.
3. Find the value of 3 cwt. 16 Ib. 10 oz. of sugar at £3
16s. 4d. per cwt.
4. Reduce 7s. lO^d. to the decimal of a guinea.
5. Divide 27 -5264 by -00374.
6. Solve the equation
7. Divide x* + x2y2 + y* by
Hence find the factors of
y? + sty* + y8.
8. What is the G.C.M. of
3x2y and I8x?y5 1
What is the meaning of G.C.M. ?
XCVIIL
1. A room is 24 ft. long, 131 ft. wide, and 9£ ft. high ; the
ceiling is white-washed at 7d. the dozen square yards ;
the walls are papered at f d. the square foot (deducting
2*5 of the area of the walls for doors and windows), and
the floor is carpeted at 3s. 7d. a square yard. Find
the whole cost.
2. Reduce -3545454 ... and -00525 to vulgar fractions, and
find the value of -046875 cwt.
3. Find by Practice the value of 5 qrs. 3 bush. 1 pk. at
£1 14s. 8d. per quarter.
MATHEMATICAL EXERCISES. 53
4. Reduce £7 18s. 8^d. to the fraction of £27 15.s. 5f, and
bring the result to a decimal.
5. Divide 17'465 by 98-6 to 3 places of decimals.
6. Solve the equation
4a; - 3 + 2x - 8 = 3x+ 2 - Wx + 26.
7. Divide
±0xf - 26aty2 + 8as* - 1 6x3y + 24^ by Ix3 - Qf - 1 3a;?/2.
Test your result by putting
X+ 1 =y = 2.
8. Find the simplest expression which can be divided by
a?b, a?bc, ab?c, and abc3.
What is the name given to this expression ?
XCIX.
1. Reduce ^^5- to a decimal.
2. Multiply 4 tons 15 cwt. 2 qrs. 27 Ib. by 195.
3. If the wheel of a railway carriage is 3 yds. 1 ft. 7 in.
round, how many revolutions will it make in travelling
9£ miles 1
<•«
5. Reduce 45 of 2-16 of 2s. 6d. to the fraction of -027 of
1-18 of £5.
6. Prove that
(m + ri)3 + (m- w)3 = 2w(m2 + 3?i2).
7. Find the factors of
x*-llxrj+lQf, a? + b3, and (x-yf-z^.
8. What number is greater than its fifth part by 48 ?
C.
1. Find the cost of painting the four walls of a room 18 feet
long, 16 feet wide, and 15 feet high, at 2s. 3d. a square
yard.
54 MATHEMATICAL EXERCISES.
2. Express 2-&£g as a decimal, and 1 '07 1875 as a vulgar
fraction in its lowest terms.
3. Find by Practice the dividend on £5608 10s. at 14s. 9|d.
in the £>.
4. Add together 7|, 2^, 4, and | ;
and subtract 7^ from 10^.
5. Multiply -0263 by 2-356, and 1-6 by 3.
6. Find the value of
{8(a + 6) - 9(c + d)} {8(a + b) + 9(c + d)}
without multiplying, indicating how you arrive at the
result.
7. Find the sum and difference of
3a;2_4? + l
y y
and 4z2 + 2 *
y f
8. Solve the equation
8(3z - 2) - 4(4z - 3) - 6(4 - x) = 0.
CL
•1. What length of paper 22^ inches wide would be required
to paper the walls of a room 18 ft. 9 in. long, 13 ft.
3 in. broad, and 14 ft. 6 in. high?
2. Express ^, -|Y, and ^f-g as decimals.
3. If 24 cwt. of cheese be worth 90 yds. of cloth, how much
cloth is 1 cwt. of cheese worth, and how much cheese
is 1 yd. of cloth worth ? (Give the answers in yds.
qrs., and Ib. oz. etc., respectively.)
4. Arrange in order of size the fractions
6 24 7 2105.
2T> TT» TS"» "9 > "STT '
and find their sum.
5. Find the values of -0056 x 2-625 ; and 1'72 of -276 of 15.
MATHEMATICAL EXERCISES. 55
6. What is the result of multiplying together
and (k*
7. Take a- + 5e- from
8. Solve the equation
5(z-l)-10(2a;-9) = 2z.
Show that your answer satisfies the equation.
GIL
1. If a franc is worth 9'386d. what is the value of £1 in
francs ?
2. Reduce -1634, -591, '972 to vulgar fractions in their
lowest terms.
3. Divide 606 tons 1 cwt. 2 qrs. 27 Ib. by 145.
4. Find the value of ^ of £7 5s. + £ guinea - f of
5 shillings.
5. Simplify 100-081 - 25-789 + 701-21 - 82-6854.
6. Multiply a2-3a&-462 by 2a2 - ab + 562,
and find the value of (a3)2 and (a4)5.
7. Simplify 2xz + 4xy - 3y2 - (x - 2y)z.
8. If a = 1 , b = 2, c = - 3, find the value of
a3 + 863 + c3 - Qabc.
Find also, the factors of this expression.
GUI.
1. What would be the cost of painting the four walls of a
room whose length is 24 feet 8| in., breadth 16 ft.
3^ in., and height 11 ft. 6 in., at 4s. a square yard 1
2. Reduce 4i of £2 13s. 7fd. to the fraction of £2 14s. 8|d.
3. Find the value of 12 miles 3 fur. 16 po. of telegraph wire
at £7 a mile.
56 MATHEMATICAL EXERCISES.
4. Find the values of
(i.) -3473 of 1 ton 3 cwt. in Ib. and the decimal of a lb.,
(ii.) 21 '027 of 4 furlongs, in miles, fur., po. etc.
5. Find the value of
•64 91-165
of—-
•769230 5-097
6. Solve the equation
4 3
7. Divide
z4 + 2x3y + xy3 + *2y* - 3x% - 3y3z by x + 2y - 3z ;
and divide
a3r by ar.
8. What is the L.C.M. of
2a*b, 3a262, and 5aVl
CIV.
1. -A sack of potatoes weighs 178 lb., and if 6 such sacks
cost £2 4s. 6d., what would 44 lb. cost 1
2. Add together 51-362 + 28-793 + -1824 + -5 + 3-736.
3. A room containing 893 cub. yds. 14 cub. ft. of space has
250 people in it. How much space is that for each ?
4. What part of £6 Is. Ifd. is £ of £3 12s. 8£d. ?
5. From 1036-5 subtract 389-762;
and divide 3-42 by -45.
6. Multiply 2x - 3y - 4(z - 2y) + 5[3x - 2(x - y)]
by 4s-(y-a;)-3{2y-3(fl5 + y)}.
What is the value of (x3)3 ?
7. What are the factors of
3x*-Wxy + 3y-, 3x2-4xy + y*, and x2 - 4xy + 3«/2 ?
Hence write down the G.C.M. of the first two ex-
pressions.
MATHEMATICAL EXERCISES. 57
8. Write down the product of the sum and difference of
a and b.
CV.
1. What will be the cost of carpeting a room 9 '75 yards
long, and 4£ yards wide, with carpet 1 yd. 3 in. wide
at 4s. 6d. per yard ?
2. Eeduce -01314 and -001533 to vulgar fractions, and then
divide the first by the second.
3. When coals cost lls. 6d. a ton, what will be the annual
expense of fuel for an engine whose daily consumption
is 2£ tons. (By Practice.)
4. Reduce 16 cwt. 1 qr. 21 Ib. to the decimal of a ton.
5. Divide 201-4095 by -703 ; and find the value of a bar of
gold weighing 1-683 Ib., if an ounce of gold is worth
£4-0099.
6. Solve the equation : —
6(x + 1) + 4(z + 2) + 3(x - 5) - 168 - 0.
7. Divide
a;7 + 2187 by x + 3;
and xm+n by xm.
8. What is the greatest quantity which will exactly divide
3a462, 27b3c, and 12o6cd?
CVI.
1. What are the vulgar fractions which are equal to -5l3,
5-13, 51-313?
2. Multiply 7 qr. 5 bush. 2 pk. by 227.
3. Find the income tax on £100 17s. 6d. at Is. 4d. in the £.
71 x i of *
4- Simplify ^°fT-
58 MATHEMATICAL EXERCISES.
5. The circumference of a circle is 3 '141 59 times its
diameter. Find the circumference of a circle whose
radius is 6 '85 feet.
6. Show that
(x + y}3 - (x3 + y3} = 3xy(x + y) ;
and that
(i.) a8-ra3 = as~3,
(ii.) ar-ra'=ar-'.
7. What are the factors of
ac + bc-ad-bd, 3a% - 6a2J4c + 9a2Jc4,
and x*-7x + lQ1
8. What do you mean by solving an equation ? How can
you tell if your answer is right ?
CVII.
1. Multiply 3 tons 15 cwt. 1 qr. 16 Ib. by 154.
2. Find the volume of a room 14 ft. 7 in. long, 12 ft. 5 in.
wide, and 11 ft. 3 in. high.
3. Find the value of $ of a cwt. + 8f Ib. + 3^ oz.
4. If 1 Ib. of gold be coined into .£46 14s. 6d. what is the
weight of a sovereign ?
5. Multiply -007302 by -6025 ;
and divide '532 by 323 to 4 places of decimals.
6. Divide a4 - 1 by a - 1 ;
and multiply a2 + — „ + 1 by a - -.
a2 a
7. Subtract ^-^j + \\xf-f
from - ix3 - f z2y + 2zy2 - l^/3.
8. Solve the equations : —
(i.) 15z- 25 = 10^+10.
,.. , a-2x_ 1 _ 1
(IL) abx P ab'
Prove your results.
MATHEMATICAL EXERCISES. 59
CVIII.
1. Find the cost of 59476 bales of goods at £15 9s. lO^d.
per bale.
2. Add together ^ of 12|, ^ of 12|, 2| of 1^,
and IJ ;
and divide 135r by 17^-.
3. Find the decimal fraction which is equal to
4. Express '0425 of a mile in yards.
5. Simplify
mn(tf> +l) + (nz + m2)^ + x) -
arranging the answer in descending powers of x.
Show that x + 1 is a factor of the expression.
6. Resolve into factors : —
(i)*-l.
(ii.) a;6 + 703-8.
Prove your results.
7. Prove that in an equation any quantity can be trans-
ferred fromi one side to the other by changing the
sign.
8. Find the L.C.M. of
I8z*yz, Wtffz*, and
CIX.
1. Divide 18917 ac. 2 ro. 7 po. 7 yd. 3 ft. 108 in. by 54.
2 The area of a rectangular piece of ground is 28 ac. 22 per.
Find its length if its breadth is 625 links.
3. Simplify I of 1£ - 1 V?.
3f
4. If 2 horses can plough 7 acres of ground in a day, how
many horses will be required to plough 161 acres in
ll^days?
60 MATHEMATICAL EXERCISES.
5. Multiply -401296 by 8-05 ; and 1-001 by -000001.
6. If a = 8, b = 4, c = 5, d = 7, find the value of
a - b _ 4(c - d)
a+ b c + d
7. Solve the equations : —
(i.) 88(3.r - 9) - 56(z + 1) = 77(3z - 1 4).
/•• \ tt — b . a — b 7
(n.) - % + ——a; = a + b.
a b
Prove your results.
b
ac
to a common denominator, and add them together.
8. Bring * -L, and
3b2 9bd
CX.
1. Find the L.C.M. of 1836, 1482, 1938, and 8398;
and add together ^g, -^^ and jfa.
2. Subtract 3^ from 4^-;
and multiply 14^ by 2^.
3. Reduce 5 Ib. 10 oz. 11 dwt. to the decimal of 14 Ib. 2 oz.
Troy.
i. Multiply together 5'49, -0318, -047610, and 500;
and divide the result by 21.
5. Prove that
6. Find the G.C.M. of
z5 - Sz4 -i- 4s3 + 26z2 - 92o; + 55 and z2
7. If any number be taken, trebled, and added to its cube ;
then 1 added to three times its square; and the second
result subtracted from the first; prove that we have
the cube of the number next below the original
number.
MATHEMATICAL EXERCISES. 61
8. Multiply together
o
and divide the result by
CXI.
1. Reduce 4 tons 3 cwt. 1 qr. 9 Ib. to ounces.
2. Find the cost of paving a court at 3s. 2d. a square foot,
whose length is 27 ft. 10 in. and breadth 14 ft. 9 in.
3. Reduce 13s. 4d. to the fraction of £'2 6s. 8d.
4. If 28 Ib. of tea at 3s. 6d. a Ib. be mixed with 56 Ib. which
cost £5 12s., how much a Ib. is the mixture worth ?
5. Add together 3281-6013, 42-307, 8-42, and -05869.
6. Divide
x2 - (a + b)x + ab by x - a,
and test your result by putting
x = 3, a = 2, 6=1.
7. Show how to write down the value of
without actual multiplication.
8. Solve the equations : —
o 4
\ 24a/ — 8y = 40.
Prove your results.
CXII.
1. Find the value of 5 cwt. 3 qr. 21 Ib. at £1 5s. 8d. per
cwt.
62 MATHEMATICAL EXERCISES.
2. Add together 1|, 2^-, £, and 3™ ;
and divide 8|- by -3^-.
3. Bring -S57 142 to a vulgar fraction, ajid i
decimal.
4. Subtract 3375 of a minute from T95 of an hour.
5. Simplify
(x + y)(x -y}- x(x + y) + y(x + y),
without multiplying out, if you can.
6. Factorize (i.) pq-gr+pr-r2,
(ii.) 3x%-9^2,
(iii.) p*q8-r™.
7. Of a certain number of boys in a class, one-fourth are
away at examination, 2 are seger, and the class is in
consequence only f of its usual size. Find the number
in it usually.
8. Find the L.C.M. of
a + x, a-x, and a2 -a2.
CXIII.
1. In a million inches, how many miles, furlongs, etc.?
2. How many cubes, each of 3 inches edge, can be cut out
of a cube of 12 inches edge ?
3. Simplify ^of If + 1^of6|+2ix2| + i±i.
^2
4. Find the income tax on £356 10s., at 5d. in the £.
5. Subtract 17-2398 from 27-06;
and divide 92-3784 by -623 to 3 places of decimals.
6. Add together
I , m n I m n A I m n
-- + —+- _ + _-_, and --_ + -,
x y z x y z x y z
and find the value of the result when
MATHEMATICAL EXERCISES. 63
7. Solve the equations : —
,. ^ x-a _x + a_ 2ax
(ii \ 3a:~2 _ 4s -3 _ 4-a;
8. Add together
3, and tey(*±y)^
7
x-y' xP-y^ ' (x2 + xy + y2)(x - y)
CXIV.
1. Reduce -If-fr to its lowest terms.
2. Subtract 19^ from 45FV;
and multiply together 6j^, 1-^, 16|, and |.
3. Express 6 Ib. as the decimal of 1 cwt.
4. Find the value of -33$ x 2f£ -=- 6-48.
5. Find the factors of —
(i.) a2x2 - 2a?x + a4,
(ii.) a2-a6-1262,
(iii.) 2^ - 2m« - ^2 - 12 + m2 + n2.
6. Find the G.C.M. of
3z3 - 3x - 72, and 2x2 - 2x - 1 2.
7. If from a number its fourth part be taken, and then its
third part be added to it, show that the result is larger
than the original number.
8. Simplify
xy
What is the meaning of - a ?
cxv.
1. Divide 261814 acres 1 rood 32 perches 1 foot 3 inches
by 72.
64 MATHEMATICAL EXERCISES.
2. Find the cost of the lead lining for a rectangular cistern,
3 ft. 2 in. long, 2 ft. 8 in. wide, and 2 ft. 6 in. deep ; at
Is. 9d. a square foot.
3. Express 12 oz. 8 drs. as the fraction of 2 cwt. 26 Ibs.
4. The construction of a railway 12 miles 6 furlongs 20 poles
in length cost £328000. What was the average cost
per mile ?
5. Multiply 10-101 by -001 ;
and divide 126-04745 by 3-925.
6. Divide
xt-(a + b + c + d)x3 + (db + be + ac + ad + bd + cd)x2
- (abc + acd + abd + bcd)x + abed
by 22 - (a + V)x + db.
7. Subtract 3x-y + z
from the sum of
3x-5y-6z, 2z + 4« + 3y, and 3y+3?-4a;.
Test the result by putting
x = y = z—\.
8. Solve the equations : —
2-1.2-2
CXVI.
1. Find by Practice the dividend on £2560 5s. at 12s. 8d.
in the £.
2. Subtract 3£ from 10£;
and divide 3f by l^.
3. Express ^^ as a decimal fraction.
4. Find the value of 1-25 of £1 13s. 4d.
MATHEMATICAL EXERCISES. li.'.
.">. Simplify
(x-yy-1y(x-yy + y*(x-y}.
If a be multiplied by b 20 times, what is the result '/
6. Find the factors of —
(i.) x(x -y)+ k(x -y) + l(x - y).
(ii.) x7-y7.
(iii.) 2z2 - 5xy + 2y*.
7. A certain number of two digits when inverted is equal to
£ of the original number ; one of the digits is 4, find
the other.
8. Find the L.C.M. of
24 (x3 + xzy + xy2 + f) and 1 6(a* - x*y + xf - y3).
CXVII.
1. Multiply 5 tons 16 cwt. 2 qr. 11 Ib. by 84.
2. The walls of a room 21 ft. long, 15 ft. 9 in. wide, and 11
ft. 8 in. high, are painted at an expense of £17 17s. 3|d.
Find the cost per square foot.
3. Simplify SfxZl^l*
4. If 6 yards of silk cost £1 7s. 9d., what would 177 yards
cost1?
5. Divide 2054-95 by -0563;
and add together 1'375 gallons and 3'5 quarts.
6. If a=l, ft = 2, c = £, d = Q, find the value of
a - 6 - c _ ad + bc _ MS ^3
a-b + c bd-ac
What is the difference between nbz and (ab)3 1
7. Simplify
<i-b a-b a2 + &2
a b nl
E
66 MATHEMATICAL KXERClsKfS.
S. Solve the equations : —
_
(ii.) (a - b)x + ab = bz.
Show that your answers satisfy the equations.
CXVIII.
1. Simplify the fraction fyfyf.
2. If a piece of cloth is 4| yds. long, how much will be left
after 11-f feet have been cut off1?
3. Reduce -35 of 2 days 13 hours to minutes.
4. Divide 91-663 by 87-56.
5. Prove that
x\y + z) + y-(z + x) + z\x + y) + 2xyz = (x + y)(y + z)(z + x).
6. Find the G.C.M. of
tf + ab, a4 + 3a?b + 3a262 + ab3, and
7. Simplify
a4 + a?b -abs-b* a-b
8. Having given any two whole numbers, show that
if from the product of the two numbers next above
their squares,
be subtracted
the square of the number next below their product,
the remainder will be
the square of the sum of the two given numbers.
CXIX.
1. Divide 10 miles 1 fur. 6 poles by 22.
2. Find the cost of painting the ceiling of a room 21 ft.
long, 15 ft. 9 in. wide, and 11 ft. 8 in. high, at 5d. a
square foot.
MATHEMATICAL EXERCISES. 07
3. Add together 21T8T lb., 8^BT oz., and 18| dwts., and ex-
press the answer in Ibs.
4. If 14 grains of gold cost 2s. 2£d., what is the price per
ounce ?
5. Subtract 56-293104 from 400-0738;
and divide 17-6 by 275.
6. Multiply
7. Subtract (a - b)x* + (a2 - ab)xz - (a% + ab*)x
from (a + b)x* - (ab - b*)x* + (ab* - b*)x.
8. Solve the equations : —
(i.) 14(o;-2)-2(a; + 2)-28 = 0.
(ii.) a?x
cxx.
1. Find, by Practice, the value of 3 oz. 15 dwt. 12 gr. of gold
at £3 15s. per oz.
2. Add together 2f , ^ 5£, and & ;
and divide 14£ by 9^.
3. Reduce i£|£ to its lowest terms, and then bring it to a
decimal.
4. Add together 2-15 oz. and -0075 of a cwt., and give the
answer in ounces and drams.
5. Simplify
(a + y)(x + 2y) (x - 2y)(x - y).
If k be multiplied by itself 10 times, what is the
result ?
6 Write down the expressions whose factors are respec-
tively— (i.) x-3y, and y + 3x.
(ii.) W-lm, and k2 + lm,
(in.) a, b, b, and c.
68 MATHEMATICAL EXERCISES.
7. Find the number, of which the half is greater than the
third by 5.
8. What is the L.C.M. of
z4?/2, y*z, and zV 1
Explain the meaning of L.C.M., and thence prove that
your answer is correct.
CXXI.
1. In 156704 square inches, how many square poles, square
yards, and square feet are there 1
2. If it takes 40 yards of carpet to cover a room 15 feet
wide by 18 feet long, find the width of the carpet.
3. Simplify Ig-.
1 +
3 + |
4. If a clock is 2 minutes and a half fast at noon on
January 1, and gains 3£ minutes a week, what time
will it indicate at noon on January 311
5. Multiply -42881 by 5-63 ;
and divide 242 by -55.
6. Solve the equations : —
,. N 3z-4 4rc-3 2, R.
<L)-2T- -3~=7(X-^
... ,. 2x- b 2x-a
(11.) = — j —
a b
What is meant by the expression " — satisfies the
equation " ?
7. If a = b + c, find the value of
a2 + 62 - 2ab - c2.
By how much is x greater than y ?
8. Add together
4(«-^ 3(0+^ and i3a
MATHEMATICAL EXERCISES. 69
CXXII.
1. Explain what is meant by Greatest Common Measure,
and Least Common Multiple.
2. Simplify (i.) 19^ -12ft
(ii.) 34x6^x^x21.
3. Express '545 of a guinea in pence and the decimal of a
penny.
4. Multiply 8 7 '5 6 by 1 '049073, giving the answer as a
decimal.
5. Which of the expressions
x2 -y2, x3- y3, a? - y*, etc.,
can be exactly divided by x + yl
Which by x-yl
6. Find the G.C.M. of
x* + x2 + l and a?+l.
7. Prove that the difference of the squares of two consecutive
whole numbers is equal to the sum of the numbers.
8. Simplify
ax3y abxz _._ ax3
byz2 b2yz2 ' bz*
When may you " cancel out," and why ?
CXXIII.
1. Eeduce 424768 oz. Avoirdupois to tons, cwt., etc.
2. A room is 27f ft. long, 21| feet wide, and 4 yards high.
Find the cost of papering the walls at 6d. a square yard.
3. Reduce 4 sq. poles 2 yds. 6 ft. 108 in. to the fraction of
4 roods 20 po.
4. If £39 19s. will buy 47 tons, how much may be bought
for 11s. 8d. ?
70 MATHEMATICAL EXERCISES.
5. Multiply 169-35 (i.) by -00265, (ii.) by 2-6500, (iii.) by
2650.
6. Multiply
7. Subtract
x2-xy + y2 from x* - x2y2 + y*.
Prove that
a - (b - c) = a - b + c.
8. Solve the equations : —
(ii. ) ax + bx = a2 - b2.
CXXIV.
1. Find the value of 2 bush. 1 pk. 1 gall. 2 qts. at £3 16s.
per bushel.
2. Simplify (i.) 12f + li + ^+2if
(ii.) 191|-7|.
3. Reduce '429 and -0375 to vulgar fractions in their lowest
terms.
4. Reduce -275 of £1 13s. 4d. to pence.
5. Simplify 2 - [2- 2{ -(2 - 2 . 2^~2)}].
6. If x-2y is one of the factors of
a;3 - 2x2y - 4a^2 + Sy3,
find the others.
What is a factor ?
7. A sum of £3 is divided among 30 women and two-thirds
that number of men ; each man has half as much again
as each woman. What do they each receive ?
8. Find the L.C.M. of
(x - a)(x + a), and (z2 - a2)(o;2 + a2).
MATHEMATICAL EXERCISES. 71
CXXV.
1. Reduce 27 po. 5 sq. ft. to sq. ft.; and | acre to sq. links.
2. A room is 16 ft. long, 8 ft. 4 in. broad, and 7 ft. 6 in.
high. Find the expense of painting the four walls at
4|d. per sq. ft.
3. Find the value of (§ of £ + £ of ££) of 3s. 4d.
4. What quantity of silver can be bought for £36, when
14 Ib. 6 oz. 15 dwts. costs £45?
5. Subtract 4-354 from 5'8693;
and divide -067206 by 24-35.
6. Multiply
a2 + J2 + c2 + d2 - a(l + c + d) - b(c + d) - cd
by a + b + c + d.
Show that your answer is correct when
7. Subtract az + 2a?b -
from 3a26 - £a&2 - 26s.
8. Solve (i.) KJB - 2) - }(x + 2) = 1.
. .
x-l x-3
Test your results.
CXXVI.
1. Find, by Practice, the value of 1 25 yds. 3 qrs. 2 nls. of
cloth at 4s. 6d. per quarter.
2. Divide | + | + | + f by |+ 1-^ + 3^ + f£.
"3? "052
3. Reduce — — x — — to a vulgar fraction.
4. Find the value of -0735 of a guinea, in pence.
«•>
(ii.)
72 MATHEMATICAL EXERCISES.
G. Find the factors of
(i.) (a2 + ab + 52)2 - (a- - b*)-,
(ii.) 2x2-8,
(iii.) 2x(x-2)-x(x-'2)-x + 2.
1. What are the odd numbers nearest to 12 ? If 2n be a
whole number, what are the nearest odd numbers to it?
8. Find the L.C.M. of
b3c, aV, and
CXXVIL
1. What is the weight of a dozen silver tea-spoons, if each
weighs 14 dwt. 13 gr.?
2. Find the volume of a vessel 8 ft. 7£ in. long, 5 ft. 7^ in.
broad, and 3 ft. 7 in. high.
3. Simplify
5 V i * - 7
« x 8 9" THB"
4. A garrison of 4500 men is provisioned for 15 weeks at
the rate of 1 3 oz. per diem per man ; how many men
must leave, that the same provision may last those who
remain, 27 weeks at 10 oz. per diem ?
5. Find the vulgar fraction which is equal to — -
3-42552
6. If a = 2, b = 3, x = 5, y = 1, find the value of
(a + y)(a-y) (b-y)(x + y)'
7. Solve (i.)
"•'
8. Simplify
,
_ 3z-4y) _
7 /
6~ 7 / 1 7 8 J'
Test your result by putting x = y=l.
MATHEMATICAL EXERCISES. 73
CXXVIII.
1. Make out a bill for : — 13| yds. ribbon at Is. 4|d. per yd.,
47 yds. calico at 8|d. per yd., 2£ doz. pairs of gloves at
3s. 8d. per pair, 5 table-cloths at a guinea and a half
each, 25 reels of cotton at f d. each.
2. Subtract 7|f from ll/^;
and multiply together 6|, ^-, ^, and 1|.
3. Find the value of -5275 of £1 5s. + -245 of half a crown
- -095 of half a guinea.
4. Divide -7 by -636.
5. Prove that
(x - yf + (y - zf + (z- x)s = 3(x - y)(y - z)(z - x).
What is the meaning of the sign = ?
6. Find the G.C.M. of
82? +27 and I6x* + 36z2 + 81.
7. To the number x add 4, double the result, subtract
3 from this, add 3 times the original number, and
divide the result by 5.
*. Simplify <i.) ^|; <ii.) l+l + l.
CXXIX.
1. Divide 17 sq. mi. by 88, giving the answer in acres, sq.
chains, and sq. links.
2. How many square feet of glass are required to glaze
5 windows, each containing 14 panes of glass, measuring
18 in. by 6 in. ?
3. Keduce 4 Ib. to the fraction of a cwt.
4. If the duty on hops at l|d. a Ib. amounts to
£26,357 9s. 10R, on what quantity is it paid?
5. Divide -0068 by 340; and 314 by -0005.
74 MATHEMATICAL EXERCISES.
6. Divide a3 + 863 + <? - Qabc by a+2b + c.
7. Find the product of a + b-c and a — b-c, without
multiplication, and explain your process.
8. Solve 11
(ii.) 3 (a? - 1) - 4(a; - 2) = 2(3 - x).
cxxx.
1. Find, by Practice, the value of 2741J things at 12s. 7d.
each.
2. Divide 4|of1Ar by 2£ + 5f.
3. Reduce f , f, If to decimals, without division.
4. Subtract 18 '5 2 of a shilling from 21-43 of a crown.
5. Simplify
6. Factorize : — (i.) ot? + y?y + xy1 + y3.
(ii.) cy2-llc%+2802.
(iii.) a2 - 2ab + V* - c2.
7. If four times a certain number be added to 7 the result
is 31. What is the number?
8. Find the L.C.M. of
4a3x, 8a?xy, 24a?x*z, and 9a^.
CXXXL
1. In 3706529 inches how many miles, furlongs, poles, etc.,
are there ?
2. Find the cost of papering a room 10 ft. long, 17 ft. wide,
and 10 ft. high, with paper 15 in. broad, at 3|d. the
yard.
3. Simplify Si-J-f-A-A-^r-
MATHEMATICAL EXERCISES. 75
4. 500 boys are distributed in 3 houses. The largest house
contains y£ of the smallest, and the smallest contains
^ of the whole. Find the number in each.
5. Multiply 526-32 by -0573, and divide the product by
•0072.
6. If a = 3, b = 3|, c -• 0, find the value of
7. Solve (i.)
(»•:
- c2) - (2a -
2x -
-4= -1.
8. Simplify 08a;2 0 y— + x .
«2 _ 4/J.2 y + 2x 2x — y
Test your result.
CXXXII.
1. The sum of £399 2s. 6d. was distributed as prize-money to
a number of soldiers ; each received £3 1 7s. 6d. How
many were there ?
2. From 5* of 4£ deduct 3J of 3i.
3. Find the number of square yards in -3 of 1 ac. 3 roods.
4. Divide -0089 by the recurring decimal -267267
5. Prove the identity : —
(9x2-4
6. Find the G.C.M. of
z2-3a;+2, and z2-f3a:-10.
7. By how much is a+ 5 greater than b + 10 ?
8. Simplify
a2 - x2
76 MATHEMATICAL EXERCISES.
CXXXIII.
1. The solid content of some of the stones at Baalbec is
8640 cub. ft.; find the weight of one of these stones,
supposing a cubic foot to weigh 1 cwt. 2 qr. 12 Ib.
2. A box is 6 ft. long, 4£ ft. wide, and 3 ft. deep. Find
the difference in cost of covering it with sheet lead at
8d. and 8id. a square foot.
3. What fraction of 300 yards is I fur. 2 po. 6 yds. 2 ft.?
4. A man owns ^ of a mine and sells -47 of his share.
How much has he left ?
5. Divide -7036 by -73 to 4 places of decimals.
6. Subtract a2 -06 + ft2 from a* + ab + b*.
By how much is x greater than y ?
Explain the result when
x = 4 and y = 6.
7. Divide ie8 + ofy4 + y8 by x2 - xy + if.
8. Solve (i.) z-*-**±™*
o 45
(ii.) a(x - b) = b(x - a).
Test your results.
CXXX1V.
1. Find, by Practice, the cost of 453| things at £1 14s. 3d.
each.
2. Add together 1TV, 2|, 1 ^, and 2£; and divide 3^ by
3. Divide -0043007450 by '5326, and express the result as
a vulgar fraction in its lowest terms.
4. Express 7s. 9|d. as the decimal of £8.
5. Simplify (i.) (l + jWl - 1) * (a - &)«,
(ii.) 4(3z - 2) - 2(4z - 3) - 3(4 - x).
MATHEMATICAL EXERCISES. 77
6. Find the factors of: — •
(i.) 2z2 - I3xy - 45y2,
(ii.) x7 + y7,
(Hi.) (a - b)x - y(b - a).
1. Find the number which added to its seventh part makes 16.
8. Find the L.C.M. of
and
cxxxv.
1. If 1438041 be divided by 165, by successive division by
the prime factors of 165, show how to find the
remainder.
2. Find the cost of painting the 4 walls of a room at 8d. a
sq. yd., if the room is 18 ft. 6f in. long, 16 ft. 5£ in.
broad, and 12 ft. high.
48* 7_3_
3. Simplify _.JI— 4.- L and reduce the result to a
1085^ 174f\
decimal.
4. If 200 men in 12 days, of 8 hours each, could dig a trench
160 yds. long, 6 wide, and 4 deep, in how many days
of 10 hours each will 90 men dig a trench 450 yds.
long, 4 wide, and 3 deep ?
5. Add together
20134-01992, 2-0028803, -00001, and 30-00001 ;
and subtract -00099901 from 4321.
6. If a= 1, b = - 1, c = 2, d = 0, find the value of
qg-62 tf-cd c3-^
a* + b*+2b2 + cd+ Zabc '
7. Solve the equation
5(z-4)-3(a;-5)-£(a--6) = 0.
«. r- 2a2-2a+l a 1
8. Simplify --- T+-,
a2 - a a - 1 a
and multiply a" by cf.
7S MATHEMATICAL EXERCISES.
CXXXVI.
1. If the 6dy. loaf weighs 4 Ib. when wheat is 39s. a quarter,
what ought it to weigh when wheat is 52s. a quarter 1
2. Subtract 3|-f from 100^; and multiply ^V by ^
3. Express 2-425 of a Ib. Troy in grains.
4. Multiply -43§ by l'65&i, and express the answer as a
decimal.
a»-I
a3 1
5. Prove that r- = a2 + 1 +. — , by division.
a-- a
6. Find the G.C.M. of
x2 - xy - 1 2yz and xz 4 5xy + 6y2.
7. Divide 18 into 2 parts so that one may be twice as large
as the other.
8. Divide
(x + y)2 + 3(x + y)z+2z2 by x + y + z,
without removing the brackets.
CXXXVII.
1. Eeduce 4 cub. yds. 12 cub. in. to cubic inches.
2 Find the number of rods in a piece of ground 79 ft. 1\ in.
long, and 69f in. broad.
3. Express 10 hrs. 30 min. as the fraction of a week.
4. If 15 cwt. 3 qr. 24| Ib. cost £9 11s. 7|d. what would
1 ton cost ?
5. Multiply 8-0327 by -00698, and divide 24-495 by -0426.
6. Multiply
l+4z-10o;2 by l-Qx + 3x2,
and test your result by putting x = 1.
'7. Subtract
3(as _ j) + 9(as _ J2) from 8(a2 - b) - 4(a3 - 62) ;
and divide ce* by x*.
MATHEMATICAL EXERCISES. 79
8. Solve (i.) *_i!_-H.
,..,( 2Qx+3Qy = 430.
'
CXXXVIII.
1. Find, by Practice, the dividend on £3447 15s. at 13s. 7d.
in the £.
9ft 5
2. Subtract H of 3^ fr0m ^ft.
^^5
3. Distinguish between a decimal and a vulgar fraction.
4. Express £3 6s. 6d. as a decimal of 7 half-crowns.
5. Simplify (i.) (x + y)(x - y) - x(x + y) + y(x + y).
a c
(ii.) Ixi
c a
~& b
6. Factorize (i.) kl - Ix + ky - xy,
(ii.) z2-81y2,
(iii.) (a - ft)2 - (a + b)2.
7. Write down the square of
2 + 3x - 4z2
without multiplying, and explain how you do it.
8. Find the L.C.M. of
z2 - 1/2, (x + yY and (x - y)2.
Prove that aV is the L.C.M of
a?x and ax3,
explaining what is meant by L.C.M.
CXXXIX.
1. Reduce H Ib. Troy to grains.
2. Find the cubical content of a block 2 ft. 9 in. long,
1 ft. 8 in. wide, and 1 ft. 4 in. deep.
SO MATHEMATICAL EXERCISES.
3. Find the value of
13 ' 5 '
•.: 7 TT
4. Divide £857 5s. 8Jd. among A, B, C, and D, so that A
may have twice as much as B, B three times as much
as C, and (7 four times as much as D.
5. Divide -01461201 by 357; and -75973 by -00205.
6. Find the value of
(a - b)z (b-cY (oL^c)2
vat
7. Solve
a + b b + c a + c
when a = 5, 6 = 3, c = l.
\ 2a; a;- 1 11
. _ _..__
(ii.) a(2x - a) - &(3a - ft) - 3a2 + 862 = 0.
(ii.) a6 x a2 -f a3,
(iii.) axa2-ra3.
CXL.
1. Make out a bill for 10 Ib. of raisins at 10|d. a Ib. ; 4| Ib.
nuts at 2s. 4d. a Ib. ; 15 Ib. figs, at 3s. 6|d. a Ib. ;
8 Ib. figs at lid. a Ib. ; 5f Ib. currants at 8d. a Ib. ;
120 oranges at 7|d. per half-dozen.
2. Subtract l&fa from 20^; and multiply together 3TV,
6 11 Qrifl 2
7> X¥> an<1 T-
3. Subtract 2-63 dwts. from 12-13 oz. Troy.
4. How can you find out whether a vulgar fraction will pro-
duce a terminating, or recurring decimal ?
5. Find the factors of
(i.) 3.7^ - 6x5y + 9x*f ;
(ii.) -<3-l;
(iii. ) w2 - 2pq -pz- q*.
Prove your results by multiplication.
MATHEMATICAL EXERCISES. 81
6. Find the G.C.M. of
24a63, and
Show that your answer is correct.
7. Divide 24 into two parts, so that one may be 7 times the
other.
8. Multiply together
a2 _ 3.2 1
azx2 -TT— = ;;, and
CXLI.
1. Add together
Lb. Oz. Dram. Scr. Gr.
5 10 7 2 19
18 5 2 0 12
25 10 4 1 8
20 9 3 2 15
2. Find the number of yards in the side of a square park
containing 109 ac. 3 ro. 8 po. 9 sq. yd.
3. Reduce £ of a Ib. Avoirdupois to the fraction of a cwt.
4. If I get 1 Ib. of bread for 3d. when wheat is 6s. a bushel,
what should be the price of wheat per qr. when I get
12 oz. for IdJ
5. Add together 34-25, 235-632, -00685, and 1400-06;
and multiply 652-394 by -0476.
6. Divide yzz - yz2 + z2x -zx2 by x - y ;
and a2"1"1"" by am.
7. Express the product of
a - 2b - 3c and a + 2b - 3c
without actual multiplication, and explain how you
do it.
8. Solve (i.) 3z-2
(ii.) (x-(i}
Prove that your answers satisfy the equations.
s-j MATHEMATICAL EXERCISES.
CXLII.
1. Find the cost of 48 cwt. 3 qr. 3| Ib. at 24s. per cwt.
2. Add together 3^, ii, lif, and"^ ;
and divide 13^ by 2|f
3. Reduce 5^-, and f-f to decimals. Do the first without
using division.
4. Express -315 of 13s. 4d. in pence.
5. Simplify
(i.) {&2
<rj v '
r 6. If a man has £a invested in business, loses a quarter of
it during the first year, and then gains ^ as much as
what he has left, during the second, with how much
does he begin the third year 1
7. Find the factors of
(i. ) &2 - 2 E + Z2 - w2 + 2mn - n2,
(in.) m2-12mw-85w2.
8. Find the L.C.M. of
• x-y, xz — y2, and z3-?/3.
Show that your answer is correct.
cxLin.
1. Multiply 32856 by 121711 in 3 lines of multiplication
only.
2. A clock loses 5 sees, in every 24 mins. At 10 P.M. on
Sunday it is 19 mins. fast. When will it be right ?
3. Simplify ^t¥:3i7 of Itff
MATHEMATICAL EXERCISES. 83
4. If 17 men can build a wall 22 yds. long, 5 ft. high, and
1 foot wide in 9 days, working 1 1 hours a day, in how
many days can 51 men build a wall, 60 yds. long, 7 ft.
high, and 2 ft. wide, working 1 2 hours a day ?
5. Express 6-105 Ib. Troy in grains.
Prove your answer by bringing it back to pounds.
6. Find the value of
3z2-7a;+2
when a; = 3.
7. Add together
a + b j a-b
- and --
a+c a-c
Add k to I.
8. Solve (i.) (a + c)x-bx = a-
(ii.) 9
CXLIV.
1. If a regiment march 2 paces a second, and the length of
a pace be 28 inches, how long will they take to march
35 miles 1
2. Take 6| from 11^; and multiply 8^ by 1^2T.
3. Find the value of 2-604" of £4 2s. 6d.
4. Divide (-01)3 by (-05)4.
5. Prove that
(a2 + c2)(62 + d2) = (a6 + cd)2
if ad = be.
6. Find the G.C.M. of
8^ + 27 and 16^ + 36z2 + 81.
7. Take any whole number, the one next to it, and a third
equal to the product of the other two. Show that the
sum of their squares is equal to the square of the
number next greater than the third.
84 MATHEMATICAL EXERCISES.
S. Simplify
«
211
(ii.) a'xrt'-r-a*.
CXLV.
1. Reduce 5 ac. 3 ro. 4 po. 4 yd. to sq. inches.
2. How many yards of carpet f yd. wide will cover a room
15 feet by 18 feet ?
3. Reduce f of £1 to the fraction of 1£ of £3 5s.
4. If the rate of income tax is 5d. in the £, what is the
income on which £59 Is. 3d. is paid ?
5. Reduce TVT to a decimal, and divide the difference be-
tween this and 3-236 by the sum of -002 and -0005.
6. Multiply
1 - a3 + a2 - a by 1 + 2a + a2 ;
and a* by a".
1. Subtract
from a3
, So,ve (i.)
(ii.) (a -b)(x-c)-(b- c)(x -a)-(c- a)(x - b) = 0.
Prove your results.
CXLVI.
1. Find the value of 45 cwt. 3 qr. 21 Ib. at £1 6s. 8£d.
per cwt.
2. Add together, 3Jj-, |, £, and ^ ;
divide 3£ by 8.
3. What is a recurring decimal ? Write in full the decimal
fraction 1-1031, and find its equivalent vulgar fraction.
MATHEMATICAL EXERCISES. 85
4. Reduce -49325 of £4 12s. 5d. to pence and the decimal
of a penny.
5. Simplify
(i.) (2s-l)(2a* + iX2*+l),
(ii.) a?(b + c) + b2(c + a) + c2(a + b)-(a + b)(b + c)(c + a).
6. Resolve into factors : —
(i.) as + bs + c*-3abc,
(ii.) m'' + n3,
(m.) ax2 - 2a?o? + 3a5x.
7. Show that x = 2 is a solution of the equation
8. Find the L.C.M. of
3x2y, Zxy3, 5xyz, and
CXLVII.
1. In 25 plots of ground each 9 sq. yds. 2 sq. ft. 105 sq. in.,
how much area is there ?
2. A stack of 125,000 bricks is 41 ft. 8 in. long, 16 ft. 8 in.
wide, and 14 ft. 7 in. high ; if each brick is 10 in. long,
and 3| in. thick, find its breadth.
Prove your answer.
1 34
3. Simplify 3| of 19ft- ^.
*TZ
4. If railway fares in Prussia are |j- the fares in England,
and it costs £1 2s. 6d. to travel 180 miles in England,
what will it cost to travel 160 miles in Prussia ?
5. Subtract 594-1307 from 601-05.
6. If a = 4, b = 3, c=2, find the value of
_ Jc2 + db
// c
86 MATHEMATICAL EXERCISES.
7. Solve the equations : —
(i.) 21(z-2) + 28(2:c-6)=14(z-6),
8. Simplify :9-^_9 + .
Divide the excess of the square of a above 9, by the
sum of a and 3.
CXLVIII.
1. Find the L.C.M. of 1547, 1729, and 4199, and add
together ^T, T^ir, and T}£ v.
2. A postman has to walk 10 miles in 3 hours. He walks
the first 4 miles at 3| miles an hour, the next two at
3^ miles an hour, and the next mile at 3 miles an
hour. At what rate must he walk the rest, to finish
at the proper time ?
3. Reduce -0162 of 4s. 6d. to farthings and the decimal of
a farthing.
4. Divide 8'06 by -1809523, giving the answer as a decimal.
ii. Find the factors of : —
(i.) (2x + 3y)«-(3s-2y)»,
(ii.) a4
(iii.)
6. Find the G.C.M. of
84zyt26, eOz3^9, and
7. What sum is as much greater than £1, as its half is less
than £1 ?
So- T£ x x — 3 x x + 3
. Simplify j-— — x
x-3 x x+3 x
Test your answer by putting x — 6.
MATHEMATICAL EXERCISES. 87
CXLIX.
1. Find the value of 2 ro. 19 po. 19 sq. yds. at £40 6s. 8d.
an acre.
2. Express 4 Ib. 3 oz. 12 dwts. 6 grs. Troy as the decimal
of a Ib. Avoirdupois.
3. If 8000 metres be equal to 5 miles, and a cubic fathom of
water weigh 6 tons, and a cubic m£tre of water 1000
kilogrammes, find the weight of a kilogramme in Ibs.
4. Find the vulgar fraction in its lowest terms that is equal
to 10-56240, and show that its square root is 3-249.
5. Find the Simple Interest on £2600 for 4£ years at
3 1 per cent.
6. Multiply a - b by c - d in one line.
What do you mean by homogeneous 1 Are these
quantities homogeneous? If the multiplier and the
multiplicand are both homogeneous, how can you test
your working ? ^»
— £*• 1 •/• o*^ *• & *T~ A ^^" ^> — "*~"i^ I
'. Simphfy 3,m-2*-=T "^
8rt i /• \ *C** *£ "i A *C— "T* »C *T" J. .—
. Solve (i.) — + — - = 2z.
x-l x+l
Cx+l y-l 5
(ii.) \ y y ~y'
a: -y-l.
Prove the results of these equations correct.
CL.
1. Three taps A, B, and C can fill a cistern ; A in 24 mins.,
B in 10 mins., and C in 27 mins.; they are all turned
on for 4| mins., and then B and C are turned off.
How long will A take to complete the filling of the
cistern ?
88 MATHEMATICAL EXERCISES.
2. Express £ + 1- + '1% + -161 as a recurring decimal.
3. Find the value of
3-2? of 4s. 6d. + 7-454 of 16s. 6d. + -357142$ of 3s. 6d.
4. In how many years will the Simple Interest on £3260
amount to £326 at 4 per cent. 1
5. Find the factors of
tf-f, x8 f, at-y5.
6. Explain how you can get the product of
a+b-c-d and a-b-c + d
without actual multiplication.
7. Find the L.C.M. of
9z2-4, 27^-8, and 27^ + 36a;- + 24£ + 8.
8. Find two numbers whose sum is 51, and difference 15.
DLL
1. Find the square root of 8264-446281.
2. Express f of 1 2s. 6d. as the decimal of £40.
3. If 20 men dig 30 yards of earth in 4 days, how many
yards can be dug by 40 men in 6 days ?
4. A train starts with its full complement of passengers ;
at the first station it drops ^ of these, and takes 96
more ; at the next it drops \ of the new total and takes
in 12 more. There are now 248 in the train. How
many were there at first 1
5. Simplify 8|of^ + ^x5| + ||of§ +
6. Find the value of
when x = -g-, y = — ^>
7. Solve (i.) l_£r*JLl|_l.
6 6 12
(n.) - +1= r •
x - a n b
MATHEMATICAL EXERCISES. 80
8. Simplify
3 + 2 x-2 ]
__
x-2 2(z+2)~4'
Test your answer by putting x=3.
CLII.
1. The area of a square is 87 ac. 1 ro. 2 po.; find the length
of its side to the nearest foot.
2. Express 1 cwt. 1 qr. 21 Ib. as the decimal of four tons
and a half.
3. Add together 3T%, 2i, 1T"T, and £| ;
and divide 5T3^ by 29.
4. Find the amount of £3750 in 4 years at 4£ per cent.
Simple Interest.
5. Prove that
«2(& + c)+ i2(c + a) + c2(a + b) + 2abc = (a + b)(b + c)(c + a).
6. Find the G.C.M. of
4a3 - Stfb - 6ab* and 8
7. Simplify
a + x a3 + y?
a3 -a? ' a?-x~'
8. Find the divisor,
when (4a2 + 7ab + 5&2)2 is the dividend, ,
8(a + 26) the quotient,
and 62(9a+115)2 the remainder.
Do this, if you can, by the use of factors.
CLIII.
1. If 15 cwt. 1 qr. 14 Ib. of coal cost £1 5s. 7|d., what is
the price of a ton 1
•2. Add together 4-302 Ib. and 17-532 dwts., and give the
answer in grains and the decimal of a srrain.
90 MATHEMATICAL EXERCISES.
3. In 34512876 sq. inches how many acres, roods, etc.
4. Multiply -0263 by 2-36, and divide -4336218 by 7362.
5. If either 5 oxen or 7 horses eat up the grass of a field in
87 days, in what time will 4 oxen and 6 horses eat up
the same ?
6. Divide
9a3 + 21a26 + 33aZ>2 + '28bs by b + fa.
7. Subtract (4»-ll)(2a;+l)
from (3z-8)(3:e+2).
8. Solve the equation
b(x + a) + (x + a) = (a + b)(x -a) + (x-a)
and prove your result.
CLIV.
1. If a brick is 9 in. long, 4| in. wide, and 3 in. thick,
find the number of bricks in a solid stack, 30 ft. by
16 ft. 6 in. by 9 ft. 3 in.
2. Subtract £& from 5f of 1T7T.
^1-5
3. Find the value of 3-6 of -954 of -428571 of Is. 6d.
4. In 1861 the population of 3 towns was 22368, 43415, and
10632. In 1871 the two former had increased 6
per cent, and 8 per cent, respectively, and the latter
decreased 10 per cent. Find the average population
in 1871.
-(y-x)(-x + y).
- a2 - 62 + 2mn - 2ab,
MATHEMATICAL EXERCISES. 91
7. Find the L.C.M. of
9(z2-t/2), 8(*-y)2, and
Prove that your answer is correct, by division.
8. A and B have 8s. between them, A and C have 10s.,
B and C have 1 2s. ; what have they each ?
CLV.
1. A man has £642 6s. 8d. per annum, after paying income
tax at 4d. in the £. What is his income ?
2. On the 15th of May, 1872, a man had lived 30,000 days.
Find the year and day of the month of his birth.
3. Find the square root of 516§£f'
4. Find the average of 13£f, 21, 7|, -0023, 3£, 0, 106|, and
57¥V. Express the answer decimally.
5. If a piece of work can be done by 5 men in 15 days of
12 hours each, how many hours a day must 7 men
work to do f of it in 9 days 1
6. Find the value of
1 _ 2(fl - b)
when a = - 3, 6 = 4, c = 0.
7. Solve
8x+l
13 IT 9
f /» _i_ O»i 1
(ii-) I
(i.) ^V- +
(
8. Simplify
(i) 1 l I 1
2(«-l) a5-2^2(»-3)'
Test your answers by putting x— - 1.
02 MATHEMATICAL EXERCISES.
CLVI.
1. Express as a fraction of an acre the ground taken up by
a path 3 feet wide round -a house which is 57 feet long
and 37 feet wide.
•2. At what time between 12 and 1 o'clock are the hands of
a clock exactly opposite to each other ]
3. The Simple Interest on £956 for 2£ years is £119 10s.
Find the rate of interest.
Prove your answer.
4. Reduce -20735 of £1 3s. 3d. to pence and the decimal of
a penny.
5. Prove that
(Xs - ax2 - a2x + a3)(a^ + ax2 - a2x - a3)
6. Find the G.C.M. of
x4 - y4 and x5 - if.
1. Simplify 1
z+-^2
x
8. A is twice as old as B, but ten years ago he was three
times as old. Find their present ages.
CLVII.
1. Two clocks are right at noon on Monday. One gains
2 min. in 24 hours, and the other loses 5 min. in 48
hours. Find the time indicated by the slower when
the other is at 3 p.m. on Friday.
2. The distance between two telegraph stations is 18 mi.
1 fur. 44 yds. How many posts will be required
between them, if they are 8 rods apart ?
MATHEMATICAL EXERCISES. !tt
3. Find the cost of 2 tons 21 Ib. at £12 10s. per cwt.
4. Express 2 cwts. 1 qr. 27 Ib. as the decimal of 10 cwt. 5 Ib.
5. Divide -238096 by -3428571, and find the square root of
the quotient.
6. Multiply x*-a(x-2a) by z2
7. Subtract \tf-\x- 2£ from |z* + \xz + 3x - 1 f .
8. Solve (i.)
- _
x-l 4x_2(2x-5)
~
Prove the truth of your answers.
CLVIII.
1. Find the length of the side of a square whose area is
equal to that of a rectangle, whose sides are 94-28 yards
and 6720 yards.
2. Simplify -?2l.
3. Reduce yly of half a rood to the decimal of if of an
acre.
4. Divide 110 tons 11 cwt. 2 qr. 16 Ib. 8 oz. by 79.
5. Simplify (i.) {y2- (1 -y)}x- (x* + x^l .y} ;
("•) tfl&p.
6. Find the factors of
(i.) &-f-*l*-z\
(ii.) x3 - ax2 - 4a?x + 4a3,
(iii.) y«- 12008 + 3600.
7. If A does a piece of work in 10 days, which A and B
can do together in 7 days, in what time would B do it
alone ?
94 MATHEMATICAL KXERClSKs.
8. Find the L.C.M. of
4(a2-Z>2), 6(a-6)2, and 8(a*-64).
What is the meaning of the Least Common Multiple of
two or more quantities ?
CLIX.
1. Find the dividend on £9648 8s. at 16s. 0|d. in the £.
2. Of a regiment of soldiers ^ are killed and wounded in
the first battle, •£§ of the remainder in the second, and
^j- of the remainder in the third ; 5 1 2 are left. Find
the strength of the regiment.
3. Find the Simple Interest on £658 10s. at 4^ per cent.
for 10 years.
4. Divide £700 between A, S, and C, so that for every £1
that A has, B may have 8s., and C, 7s.
5. Find the continued product of -05, -01557, and 20-007.
6. If x = find the value of
What is the square of ,«/£?
7. Solve the equations : —
(i\ 2x g~3_i/K
5x + 4y = 44 - 5x -
8. Reduce to their lowest terms
/• v a2 + 2a& +
(ii.) a2 x a4 x a3 x a x a .
MATHEMATICAL KXK1MMSKS. 95
CLX.
1. If a postage stamp be 1 inch long and -f inch broad, how
many will be required to paper a room 18 ft. 10 in.
long, 16 ft. 9 in. broad, and 9 ft. 6 in. high ?
2. Express 2-465 of a ton in ounces.
Prove your result by working it back.
3. Find the amount of £4850 in 3 years at 4 per cent.
Compound Interest, to the nearest penny.
4. The rates of a railway train and a coach are as 7 to 2 ;
the former can go 42 miles an hour ; how long will the
latter take to go 342 miles 1
5. Find the factors of
(i.) 3x*-4xy-6x; (ii.) kl* - ly2 - ql2 + p*q ;
A'ii ^ r2 4- Rr — 9ft
^111.; x + ox — ^u.
6. Prove that
xz - 3x - a, and x2 - 4z - 5
are both divisible by the same quantity if a = 10.
7. A square floor would contain 17 square yards more area,
if each side were 1 yard longer ; find its area.
8. Simplify
2ab - 2b2
9 19 1. ' O '
Q" — 0 ff, — Q «)
CLXI.
1. The price of silver being 5s. 6d. an ounce, what is the
value of 8 oz. 6 dwt. 16 gr ?
2. Subtract -986 of a pint from 1-032 of a bushel.
3. If 172 cwt. 2 qr. 18 Ib. cost £87 6s. 3d., how much can
be bought for £3 19s. 4£d. t
4. Multiply 375-61 by 68-02 ; and divide 38-5866 by 66-3.
5. Find the Simple Interest on £2333 15s. for 3 years, at
1| per cent.
96 MATHEMATICAL EXERCISES.
/X'Simplify
f (i.)
(ii.) !-
7. Write down the result of
(a;2 - 2xy + f
without actual multiplication, and explain how you
do it.
8. Solve the equations : —
(i.) 2(z-3) + 6(z-2) = 54.
(ii.) ~-l-— + 3ab = 0.
a c
/ Show that your answers satisfy the equations.
CLXII.
1 . How many yards of carpet f- yd. wide will cover a room
10 feet wide, and 36 feet long ?
2. Find the smallest whole number which exactly contains
IT§T> 22i|, and 1^ ; and extract its square root.
3. Express -45895 Ib. Troy in oz. dwts. and grs.
4. Find the Simple Interest on 200 guineas for 4 years 235
days at 4| per cent.
5. Multiply «* + &* by a* -6*.
6. Find the factors of (i.) a4 + «262 + 64.
(ii.) 8«2-26a;+15.
(iii.) x*-x2y2.
7. Two trains travelling at 26, and 30 miles an hour
respectively, start from two termini 120 miles apart.
In what time will they meet ?
Prove your result correct.
8. Find the L.C.M. of
xy, x-y, and y3 - x2y.
MATHEMATICAL EXERCISES. 97
CLXIII.
1. At what time are the hands of a clock exactly opposite,
between 4 and 5 o'clock 1
2. Find the square root of 30789ff|.
3. Divide 361-25 by -049 to 4 places of decimals.
4. If 50 workmen can do a piece of work in 12 days, how
many more will be required to complete it in 5 days ?
c o 710 Q s is
5. Multiply together ff, iiJL, £jJ? and ±gr
1TT 5T -S-5
6. Find the value of
3a* a
when a =1, 5 = 3, c = 5.
7. Solve the equations : —
(i.) 3(z
(ii.) ^+ =
' bx ax
Prove your results.
8. Simplify
'^ x(x-y)(x-z) y(y-x)(y-z) + z(z-x)(z-y)'
(ii.) a multiplied by itself 6 times, and by x 20 times.
CLXIV.
1. Find the cost of papering the walls of a room, 27 ft. 8 in.
long, 21 ft. 4 in. broad, and 12 ft. high ; the price of a
piece of paper 12 yards long and f yard broad being
4s. 6d.
2. Subtract -036 of a mile from 6 furlongs, and give the
answer in yards, and decimal of a yard.
3. Find the amount of £7200 in 3 years at 4 per cent.,
Compound Interest.
98 MATHEMATICAL EXERCISES.
4. Find, by Practice, the dividend on £1942 5s. at 12s. 7d.
in the £.
What is the creditor's loss ?
5. Prove that
x(a? -ax- a2) + ax(x - 26 - 3c) - a(ax - 2bx - 3cx) = x3.
6. Find the G.C.M. of
o;5+x2 + 2o;+2 and x* + x2 + l.
7. Find three numbers A, B, C, such that A with half B, B
with a third of C, and C with a fourth of A may each
be 1000.
,8. Simplify
x
abx^y ' fe5
1
V"'/
1--
a
Test your results by putting a = b = x = y = 2.
CLXV.
1. Find the value of 1 ton 5 cwt. 2 qrs. 7 Ib. at £3 a ton.
2. Find the value of -g- of £1 - f of a guinea, and express the
answer as the fraction of half a guinea.
3. How many yards of cloth at 10s. a yard are worth 189
yards at 6s. 8d. a yard 1
4. Divide 1 by 3'1415 to 4 places of decimals.
5. Find the difference, in yards and fractions of a yard,
between 30 chains 15 links, and 3 fur. 6 rods.
6. Find the continued product of
2x2 + ax + a?, 2x-a, and x + a.
7. Add together bx2 + (a2 - ab)x + a?,
and (a-b)x* + (a? + ab)x-(a* + b3)
without removing the brackets.
MATHEMATICAL EXERCISES. 99
8. Solve the equations : —
'' 2x - 2 2x + 2
(ii.) 5-3(4-z) + 4(3-2o;) = 0.
Show that your answers are correct.
CLXVI.
1. Find the volume of a tank 13 ft. 7 in. long, 9 ft. 3 in.
broad, and 2 ft. 5 in. deep.
2. Define Prime Factors, and Least Common Multiple.
Add together ^-^, 278l7, -rrre-j an(* multiply the sum
by 46189, giving the answer as a decimal.
3. Express 2 cwt. 3 qr. 14 Ib. as the decimal of 1£ tons.
4. What sum invested at 3| per cent, for 4 years (Simple
Interest) will be increased by £533, through the
accumulation of interest ?
Prove your answer is right, by finding the interest on
it.
5. Simplify
(x - y){(x + 2y)3 + (2x + y)3} *(z + y).
6. Factorize (i.) (z4-?/4)2,
(ii.) tfy-f,
(iii. ) xa + y3 + i^- 3xyz.
7. If two numbers differ by a unit, show that the difference
of their squares is equal to the sum of the numbers.
8. Find the L.C.M. of
3x*, 6x?y, 5xyz, and
CLXVII.
1. If 16 tons 5 cwt. of provisions will enable a garrison of
2100 men to hold out for 13 days, what weight of food
will enable 4340 men to hold out for 6 weeks ?
100 MATHEMATICAL EXERCISES.
2. Simplify
3. Multiply 2-7956 by -08003 ; and divide 200-1 by -3553 ;
each to 4 places of decimals.
4. Find the square roots of 64009, and of 17£f.
5. A cistern has 2 taps which will fill it in 1^ and 2 hours
respectively ; and a waste pipe which will empty it in
1 hour. If all are open together, in what time will it
be filled or emptied ?
6. When a = l, b= -2, c = 3, d= -4,
show that {d - (c - b + a)} { (d + c) - (b + a)}
= d2- (c2 + ft2) + a2 + 2(6c - ad).
7. Solve the equations : —
(ii.) 2(z-3) + 8(2z-6) = 2
8. Add together * A, JL
b ax axa
CLXVIII.
1. Find the difference in weight between a Ib. of gold and a
Ib. of feathers.
2. The area of a circle is obtained by multiplying the square
of the radius by 31416. Find the area of a circle whose
diameter is 20 yards.
3. Add together -05735 of a mile, and 46-25 yards, and
reduce the result to the decimal of 7 furlongs.
4. In what time will the Simple Interest on £1150 amount
to £215 12s. 6d. at 2£ per cent.?
Prove your answer by direct working.
MATHEMATICAL EXERCISES. 101
5. Resolve into their simplest factors : —
(i.)
(ii.)
(iii.) mz-nz
6. Find the G.C.M. of
60? - y? - Sx*
and
How can you try if your answer is right ?
7. Divide a into two parts, so that the greater shall be
5 times the less.
8. Simplify
( i *2 V d i a2 V '
V a2 + aV • V a2 - a2/C_ —
CLXIX.
1. If 3 qrs. 27 Ib. 8 oz. cost £17 8s. 9fd. how much can be
bought for £8737 15s. Ofd.?
2. Add together ^ of a square mile, y7^ of an acre, and
f of a rood, giving the result in acres, roods, and
perches.
3. If 200 men can make an embankment 5 miles long in
25 days, how much overtime must 60 men work to
finish an embankment 2 miles long in 32 days ; 1 2 hours
being a day's work ?
c- re K a AAK-7 4-2325 -39424
4. Simplify '0057 + .__ + __.
5. Find the times between 4 and 5 o'clock when the hands
of a watch are at right angles.
6. If x is the multiplier, and the product is twice the
multiplicand, find x.
102 MATHEMATICAL EXERCISES.
7. Find the value of
(ft3 — o?b + (i62)(d3 4- a?b — aW)
by means of the formula
8. Solve
.• ^ x x — 4t -_ 2 4- 4#
' ' 13 7~ 91 '
(ii.) 2ax + 6 = 3oe + 4a.
Prove your results.
CLXX.
1. A roller 4-12 feet in circumference, and 2-45 feet in
width, makes 35 revolutions in passing from one end
of a lawn to the other. Find the area rolled after the
roller has passed over 42 times the length of the lawn.
2. If 13 solid inches of copper balance 17 of iron, and 15
of iron balance 1 6 of tin, and 1 9 of tin balance 1 2 of
zinc, how many solid inches of zinc will balance 1235
of copper t
3. Divide £420 between A, E, and C, in the proportion
15, 52, 33 respectively.
4. Find the Simple Interest on £5781 for 10 years at
4J per cent.
5. Simplify
6. Factorize
(i.) 9a2d - b(cd
(ii.) a2 + 6a;-7,
(iii.) aW-6463.
7. Show that the difference of the squares of two numbers
is always divisible by the sum of the numbers.
8. Find the L.C.M. of
axy, a(xy-y2), and x(x + y).
MATHEMATICAL EXERCISES. 103
CLXXL
1. The circumference of a carriage wheel is 2 yds. 2 ft. 2 in.;
how many times will it revolve in going 27 miles
554 yards 1 ft. 4 in.1
2. Multiply 5-75 by 3-16, and divide -1040864 by 3-52.
3. At what rate per cent, will £425 amount to £573 15s. in
5 years at Simple Interest ?
4. Find the average of 6f , ^-, 3£, ^.
5. When a= -3, 6 = - 4, c = - 1, find the value of
a(a + 3c) - 52 + c(c - a).
6. Solve
r , 2x x-l
(ii.) =.
a+b-c+x
7. Simplify
(i.) «-* + _*
a; a + x
8. The sum of two numbers which are in the ratio of 2 to 3
is 60 ; find them
CLXXIL
1. Find the cost of painting the walls of a room 20 ft. 10|
in. long, 15 ft. 1£ in. wide, and 10 ft. 3 in. high, at
9£d. a square yard.
2. Express 1-59685 cwt. as cwt. qr. Ib. and oz.
3. Find the amount of £3650 in 3 years at 5| per cent.,
Compound Interest (neglecting fractions of a penny).
104 MATHEMATICAL EXERCISES.
4. Three men and 6 boys working together can do a piece
of work in 1 1 days. In how many days could 4 men
do it, the work of 2 men being equal to that of 3 boys ?
5. Prove that
(p + q + r)2- (p2 + q2 + r2) =p(q + r) + q(p + r) + r(p + q);
and without multiplying out, if you can.
6. Find the G.C.M. of
9x2 - 25, and 9x2 + 3z - 20.
7. Prove that if a number divide any two quantities, it also
will divide their difference.
8. Simplify
8a2& c2^ 4aJ bed - cd2
Test the truth of your answer, by assuming
a = b = c = 2, d = l.
CLXXIIL
1. Find, by Practice, the value of 5 cwt. 3 qr. 21 Ib. at £15
a ton.
2. Find the value in pence of
6f of f^A - (2* of £) of a guinea.
"TTF ol T
3. An estate of 625 acres 1 rood 10 poles is bought for
£7878 18s. 9d.; find the price per acre.
4. What number multiplied by 65-29 will give 917-3245 as
a product ?
5. What is the cube of
2z + By - z ?
6. Simplify
(x2-!)2'
12
MATHEMATICAL EXERCISES. 105
7. Solve the equations : —
-b z-a_ax-b*
' ~T = a
8. Produce a straight line so that it may be of given
length.
CLXXIV.
1. How many yards of carpet, 25 inches wide, will be
required to cover a floor 19 ft. 7 in. long, by 18 ft. 9 in.
wide?
2. Simplify
100 - ( 3A of 4f + 2A x 4A + 4 of
3. Express 1 oz. 5 grs. as the decimal fraction of 3 oz.
4 dwts. 16 grs.
4. Find the Simple Interest on £4950 for 12 years at 3f
per cent.
5. Find the factors of : —
(i.) a?-W,
(ii.) a2 + 62 - c2 - d* - 2ab - 2cd.
6. Find the L.C.M. of
(a - b)(a - c), (b - a)(b - c), (c - a)(c - b).
7. Find without actual multiplication the product of
a;2 + xy + y2 and x2 - xy + y2 ;
and multiply x* by x*.
8. Find that number, to which if its third part be added,
the sum will be less by 8 than 4 times the original
number.
106 MATHEMATICAL EXERCISES.
CLXXV.
1. A and B together can do a piece of work in 5 days;
A alone does it in 7 days ; in how many days would
B alone do it ?
2. Subtract 48-978654 from 279-018; and multiply 20-6754
by -203.
3. Find the square root of 930677049.
4. A bankrupt's debts amount to £30,732 12s. 6d. and his
assets to £10,756 8s. 4|d. ; how much can he pay in
the£?
5. If x = 4, y = 5, z = - 6, evaluate
' - z)(x -y-z)
\~
6. Simplify
3a - {2a - (3a - &)2} + 3a!'.
and show, without multiplying out, that
7. Solve the equations :—
(i.) (2 - x)(a - 5) = 4 + Sax,
(•• \ O*C Zijj n o
T-y = 3A-
8. Prove that the diagonals of an oblong are equal.
CLXXVI.
1. Find the cost of carpeting a room 23 ft. 9 in. long, and
16 ft. 3 in. wide, at 2s. 9d. per sq. yard.
2. Find the value of
•75 of 6s. 8d - 1-84375 of 4s. + 3-9796 of 2s. 6d.
3. A person completes a journey of 1056 miles in 12 days
travelling 1 1 hours a day ; in how many days would he
complete 480 miles going 6 hours a day at the same rate?
MATHEMATICAL EXERCISES. 107
4. At what rate per cent., Simple Interest, will £2630
amount to £3090 5s. in 3 J years ?
5. Resolve into their simplest factors : —
(i.)
(ii.)
(iii.)
6. Find the G.C.M. of
49a3 - 98a2z and 7a4 -
7. Simplify
, . 2a2 2m2
' m*n - mn2 ' a?n - an2
(ii.) a8"* a*.
8. Divide 64 into two parts which shall be in the ratio of
7 to 9.
CLXXVII.
1. Find, by Practice, the dividend on £5476 10s. at 13s. 6d.
in the £.
2. Find the value of 32£ of £3 6s. 8d. in pence.
3. At what time are the hands of a clock together between
9 and 10 1
4. Simplify -00123 + 21-4 + 3605 + 1-021379 + 400-006 ;
and divide 17-637 by 84-2 to four places of decimals.
5. Write down the square of
and explain how you do it.
Simplify (a*)*.
6. Reduce
tf + gjy?- 9a2z2 + 1 Ia3x - 4o*
to its lowest terms.
108 MATHEMATICAL EXERCISES.
7. Solve the equations : —
*
33
14 10 ~140
(ii.)™ + ^=l, %»=l.
X y z y
8. ABODE is an equilateral and equiangular pentagon;
prove that the triangle ACD is isosceles.
CLXXVIII.
1. Find what length of matting, f yd. wide, will be required
to cover a room 39 ft. 6 in. long by 25 ft. 6 in. wide,
and its cost at 4s. 6d. a yard.
2. Simplify t<*tt-2i<*tt + faf&
°F 6t
3. Find the value of
•285714 of £30 + £6 -$57142 + 5-8 of £2-4114583.
4. Find the Simple Interest on £2620 for 5| years at 4£
per cent.
5. Find the factors of: —
(i.) #-ft
(ii.)
(iii.)
6. Find the L.C.M. of
b2 + bxy and bxy - x^y2.
1. Simplify
(i.) (a2 - ab + J2)2 - (a2 + oh + &2)2,
(ii.) (ar + a')(ar-a')
by means of the formula
(x-y)(x + y) = x*-y2.
8. Nine years ago A was three times as old as B, but now
he is only twice as old : find their present ages.
MATHEMATICAL EXERCISES. 109
CLXXIX.
1. A cistern can be filled by one pipe in 20 minutes, and
by another in 30 minutes ; in what time would both
together fill it 1
2. Multiply together 9-965, -098, and -00253,
and divide the result by -0001.
3. Find length of the side of a square which contains
1703025 sq. yards.
4. A metre being 39-371 English inches, find the length in
metres of 4 miles 6 furlongs 1 1 poles.
Give the answer also in kilometres, hectometres, etc.
5. Prove that the equation
y a? y xy x xy
is satisfied by x = 4, y = 3.
6. Prove the identity
(a; + a)(x + b)(x + c) = a? + (a + b + c)x2 + (ab + be + ca)x + abc.
7. Subtract 3x - ^ - (2x + 3x2)
o
8. Prove that the three exterior angles of a triangle are
together equal to four right angles.
CLXXX.
1. Find the cost of papering a square room 22 ft. 6 in. long,
and 1 2 ft. 4 in. high, with Lincrusta Walton 2 ft. 4 in.
wide, at 9d. a yard.
2. Eeduce 5s. 4|d. to the decimal of £1 7s. 6d.
3. The population of six parishes is 1236, 452, 364, 516,
3430, and 1541 respectively. Find the average
population.
110 MATHEMATICAL EXERCISES.
4. In what time will £2640 amount to £3425 8s. at 4J per
cent., Simple Interest ?
Prove that your answer is right, by finding the Interest
on it.
5. What are the simplest factors of : —
(i.) 105a2W»,
(ii.) a2 + f,
(in.) #»-P.
6. Find the G.C.M. of
2a56 + 2a264 and 4a5 + 4a3Z>2 + 4ai4.
7. What is meant by a coefficient, a power, an index, a root ?
8. Express in the form of three simple fractions in their
lowest terms
a?xd+b*d-ab
abxd
CLXXXI.
1. Find the cost of 16 cwt 1 qr. 19 Ib. at 35s. per cwt.
2. Find the value of 1^$^ of a guinea.
3. A watch loses 2 hrs. 13 min. a week. If it is right at
6 a.m. on Monday, what time will it show at 6 p.m. on
Tuesday ?
4. Find the difference between 7'999 and 8 ; and between
7-9 and 8.
What is the meaning of " decimal " ?
.,,,.., 2a 3b , 4a2 ab 962
5. Multiply --- by -9- + ^ + ^.
a a?-db ab + b* a2-&2
MATHEMATICAL EXERCISES. Ill
7. Solve the equations
(i.) 35(x - 1) + 1 2(6« - 1) = 28(2z + 7) + 20(z + 8),
(ii.) 5x- 3y = 3x + 5y = 51.
Prove your answers.
8. Construct a quadrilateral figure equal in all respects to
a given quadrilateral.
CLXXXII.
1. A piece of carpet 72 yards long, and 27 inches wide,
costing £12 12s., is used to cover a room. Find the
cost per square yard.
2. Simplify ^-25j|.
T5T
3. Express £2 3s. 7£d. as the decimal of £5.
4. Find the Simple Interest on £3121 6s. for 5 years at
3| per cent, per annum.
5. Find the factors of
(i.) 21z2-2a;-3,
(ii.) (x* + xy + y*)*-(x*-f)*,
(in.) (a + '2b)* + (2a + b)3.
6. Find the L.C.M. of
5x(x* + 2x+l), 10a;2(a;2-l), and I5(x+ l)(x2- 2x+ 1).
7. Find the product of
a + 26-3c and a — 25 + 3c.
8. A person bought some cloth for £12 ; if he had bought
one yard less for the same money, each yard would
have cost him ^ as much again as it did. How many
yards did he buy 1
CLXXXIII.
1. Find, by Practice, the cost of 7326516 things at £1 3s. 6d.
a dozen.
112 MATHEMATICAL EXERCISES.
2. Multiply -1809523 by 22-2&9473684210526315?, and
give the answer as a decimal.
3. Find the length of the side of a square park containing
439 acres 33 perches and 5| sq. yards.
4. Express the length of a seconds pendulum (39-1393 in.)
as the decimal of a metre, correct to 4 places.
(1 metre = 39-371 inches.)
5. Multiply |x2-a; + f by
6. Simplify
209
(a - b)(a -c) (b- a)(b -c) (c- a)(c - b)'
7. Solve the equations : —
(L) 26(2z-l)-9(3z-2) = 9.13,
{x + y = a + b.
x + a_b
y + b~a'
8. If one angle of a triangle is equal to the sum of the
other two, of what kind is the triangle ?
CLXXXIV.
1. After a certain number of men had been employed on a
piece of work 24 days, and had half-finished it, 16 more
were set on, and the remaining half was finished in
16 days. How many men were employed at first ?
2. Find the value of 1-3 tons + -001 cwt. + -93 qr. + -25 lb.,
giving the answer in Ibs.
3. The value of unwrought silver being 4s. lid. per oz.,
and the workmanship equal to | of the value of the
metal, find the cost of a salver weighing 45 ounces.
MATHEMATICAL EXERCISES. 113
4. What sum will amount to £486 14s. 4£d. in 2| years at
3£ per cent., Simple Interest ?
5. Find the factors of
(i.) x*p-y°9; (ii.) tf-y8.
6. Find the L.C.M. of
3? + 2ax* + a?x + 2a3 and x3 - 2az2 + a%c - 2a3.
7. Show that
b 6a*b + WJa + b)*
a-b a-b
8. Out of a cask of wine of which % had leaked away,
21 gallons were drawn ; it was then half full. What
did it hold 1
CLXXXV.
1 . Find, by Practice, how much a man can save out of an
income of £200 a year, if his daily expenses are 9s. 6|d.
2. Add together 13^ of 2s. 8d. and 5/T of £1 3s. 2d.
3. When are the hands of a watch exactly opposite to each
other between 1 2 o'clock and 2 o'clock ?
4. Subtract the sum of 7-076 and 19-246 from 44-52494221.
y? — v2
5. Find the value of -= — -„
z* + y2
when x-a + b, y = a-b.
6. Simplify
1 _ (i _ i _ 2)4.20;- (3 - 5x) + 2 - ( - 4 + 5x).
7. Solve the equations : —
(i. ) \(±x - 21 ) -f %(lx - 28) = x + 3| - |(9 - 7z) +
If two triangles have three parts of the one equal to three
parts of the other, each to each, are they equal in all
respects 1
114 MATHEMATICAL EXERCISES.
CLXXXVI.
1. A rectangular cistern 13| feet long, and 6 feet broad,
contains 294^ cub. ft. of water. Find the depth of the
water, and its weight. (1 cubic inch of water weighs
252-4 grains.)
2. What fraction multiplied by 3| of i produces 2| of -^ ?
3. Express 2s. 6d. as the decimal of £5.
4. Find the Simple Interest on £2987 10s. 6d. for 2| years
at 3J per cent.
N.B. — This may be done in 2 lines of division. How 1
5. What are the factors of
(i.) xpf-xps?,
(ii.) a2 + o2 - c2 + 2a6,
(iii.) a2-62-c2+2Z»c?
6. Find the G.C.M. of
(3x - 5)(4a;2 - 6x + 2) and (6z2 - 8* + 4)(3z - 5).
7. Explain the method for addition and subtraction of
fractions. What do you mean by cancelling ? When
may you cancel 1
Find the sum, difference, product, and quotient, of
o, i &
^- and - .
0 a
8. A and B have each an income of £400 ; A spends £40
a year more than B, and at the end of 4 years their
joint savings amount to one year's income of either.
What do they each spend a year ?
CLXXXVII.
1. Find the price of 3 qr. 2 bush. 1 gal. at £2 13s. 4d. per qr.
2. Divide 7-026 by -8242, and give the answer as a decimal.
3. Find, within an inch, (i.) the side, (ii.) the diagonal, of
a square field containing 3|- acres.
MATHEMATICAL EXERCISES. 115
4. If a train which travels at the rate of 35 J miles an hour,
performs a certain journey in 11 hrs. 45 mins , in what
time would a train do it travelling 42| miles an hour ?
5. Divide
23oft3 - 1 2 J4 - 9a262 + a4 - 3a3b by a2 - 362 + 2ab ;
•, 5,2
and x3 by a*.
6. Simplify (i.)
a
a — b +
a + b
I 1/1 _2
a + 2\b + 3c
a 3\b 2c;
Test your answers by putting a = b = c = 1.
7. Find three numbers in the proportions 1, 2, 3, the sum
of whose squares is 56.
8. Solve the equations : —
= 2z + — ,
(iL) [3x-y = ^(2y + x)-3.
CLxxxvm.
1. A cistern is filled by 2 pipes in 20 and 24 minutes
respectively ; half of it can be emptied by a tap in
15 minutes; what part will be filled in 15 minutes
when all three are open ?
2. Express '45 of a perch as the decimal of an acre.
3. If pasturage for 28 oxen for 1 1 days can be obtained for
£7 10s., what should be paid for a herd of 33 feeding
for 63 days 1
116 MATHEMATICAL EXERCISES.
4. A sum of £3024 lent at Simple Interest produces £378
in five years ; what is the rate ?
5. Resolve into their simplest factors : —
(i.) 3z2 + 4z - 4,
(ii.) (a + Z>)2 - (a - If,
(iii.) a3 + 125ft6.
6. Find the L.C.M. of
a + b, b-c, c + a, a2-&2, &2 + c2, «2-c2.
7. Find the value of
(a + b + c + d + e)z - (a - b + c - d - e)2
8. Prove that a square is a parallelogram.
CLXXXIX.
1. The population of 3 towns in the year 1841 were 21326,
42324, and 6706 ; and in the year 1851 it was found
that the first two had increased 12 and 10 per cent.
respectively, and the last decreased IS percent. Find
the average population in 1851.
2. What decimal of a Ib. Troy is f of a pennyweight ?
3. Find the Compound Interest on £5000 for 3 years at
4 per cent.
4. Divide 2-63 by 3-954, giving the answer in the form of
a decimal.
5. Prove that
Xs - y3 + £ + 3xyz
x-y + z
is numerically equal to
when «=-!, y = 2, z = 2.
6. Simplify
(i.) 2(x - y) - 2/3(0; - y) - (to - 3y - ^=-4
v »
(ii.) a* x a" x a*.
MATHEMATICAL EXERCISES. 117
7. Solve (i.) i?~^f "fi = 1-
(ii.) x + y = 5(x-y)= 10.
8. Find the size of each angle of an equilateral triangle.
cxc.
1. A and B run a mile race. A takes 3-3 steps of 5 feet
each, per second, all through the race. B takes 3 steps
of 6 feet each, per second, for f mile ; but in the last
\ mile he only goes 5 -5 feet at each step, and takes
2-5 steps per second. Which won, and in what time 1
2. Simplify ^J3 x III x ^ x lo.
3. Find the value of
•54 of 8s. 3d. + -027 of £2 15s. + -3125 of £2 2s.
4. Find the Simple Interest on £1760 14s. 6d. for 9 years
at 3f per cent.
5. By what are each of the following exactly divisible ?
(i.) *<>-
(ii.)
(iii.)
6. Find the G.C.M. of
1 2<i2 + 1 Bab + 3b2 and 6a2 + 2 J
7. Simplify
3a
2(1 + a)/'
(ii.) (a2 - 7a6 - 3062) x («2 - 2ab - 1 562) -r (a2 - 15aft + 5062).
8. A man has 1200 books; twice as many duodecimos as
octavos, and 5 times as many octavos as quartos. How
manv has he of each kind ?
L18 MATHEMATICAL EXERCISES.
CXCI.
1. Find the cost of 9 tons 4 cwt. 3 qr. 21 Ib. at £14 15s. 9d.
per cwt.
2. Keduce 16 '08 pennyweights to the fraction of a pound
Troy.
3. Find the amount of £6845 in 3 years at 4 per cent.,
Compound Interest.
4. Subtract 48-978654 from 279-018;
and divide 40-303662 by -00542.
5. Divide x5 - I — p(tf - x) + q(x* - x2) by x-l.
6- sim
7. Solve the equations : —
(i.) 3z-i(a;-ll)
f OT + 3v — 22
(ii.) ^+^-^»
8. Prove that the straight line which bisects the vertical
angle of an isosceles triangle, is perpendicular to the
base.
CXCII.
1. How many planks 15£ ft. long, and 1£ ft. broad, will be
required to floor a room 60 feet by 30£ feet 1
2. Simplify ^1— + f|— DL - (1 - i||).
7 , 7 TT "Tff
' +l+i
3. Add -16875 of £1 to -375 of 3s. 4d., and from the result
subtract of 5s.
MATHEMATICAL EXERCISES. 119
4. Find the Simple Interest on £4775 for 3£ years at 3£
per cent.
5. Find the factors of
(i.) a? + bs + c*-3abc,
(ii.) a6-?/6,
(iii.) xz - (a - b)xy - ohy2-.
6. Find the L.C.M. of
2z3 + a;2+2z-12 and 2a^- 7z2 + Ux- 12.
How many times are each of these quantities contained
in the L.C.M.
7. Find the product of
y4 + mPy2 + m* and y4 - m2^2 + m4.
8. A and B have £4900 between them : A invests £ of his
share, and B \ of his, and then each has the same
amount left. Find their shares.
CXCIII.
1. What should be paid for 1397 yards of fencing, if
£360 19s. l|d. be the cost of 1 mile 3 furlongs
1 6 poles 3 yards ?
2. Divide 3-Sl by 2-227, and express the answer as a
decimal.
3. Find the Present Worth of £2674 6s., due in 3 years, at
4f per cent.
4. Each page of a book of 560 pages contains 40 lines, and
each line contains on an average 41 letters. How
many pages will there be in another edition of the
book consisting of pages of 60 lines, each line one
quarter longer, and printed in type occupying more
space on the line in the proportion of 6 to 5 ?
120 MATHEMATICAL EXERCISES.
5. If a = 0, b = -I, c = 1, d = 3, find the value of
/• v abc + bed ab (b + c)d
W + + ~~~
(ii.)
b+-L
(ii.) {3a-
7. Solve the equations : —
C \ x 3z 5x_qi (a; -4- 15)
* ' 2 + T~T ^ <T~
(ii.) a2 =16.
8. State the converse of Euc. I. 4. Where is it proved ?
CXCIV.
1. Find the cost of turfing a rectangular grass plot 36 yds.
long and 24 yds. broad, at 4d. a square yard.
2. Express 6 cwt. 1-125 qrs. as the decimal of a ton.
3. If 30 men working 9 hours a day can build a wall in
16 days, how many youths must be employed to do
half as much in 20 days at 8 hours a day, the work of
4 youths being equal to that of 3 men.
4. At what rate per cent, will £1427 amount to
£1658 17s. 9d. in 5 years?
5. Find the factors of
(i.)
(ii.)
(in.) 2
6 Find the G.C.M. of
a?s?-, abx*- and
MATHEMATICAL EXERCISES. 121
(IT — f)^r -I* c
1. Break up into 3 simple fractions.
xy
8. A man distributed p shillings among n persons, giving
9d. to some and 15d. to the rest. How. many re-
cipients were there of each ?
(ii.) If you cannot do this, substitute 22 instead of p, and
20 instead of n, and then work out the problem.
cxcv.
1. Find the value of 4 acres 3 roods 30 perches at £5 13s. 4d.
per acre.
2. Subtract /, of 2f of £3 6s. 6d. from -0475 of £100.
3. Find the amount of £89 10s. in 3 years at 5 per cent.,
Compound Interest.
4. Find the continued product of 30-75, 29-417, and -00123.
5. Multiply «4__ + _l by
y f y y*
6. Simplify
/. v a + 3 5 - (6 - 3a) 2a-(
~T~ ~T~
di.) - l
i_
How can you test your answers ?
7. Solve the equations : —
1-*
/. \ £-& 2 _x x-l
~2T' ~|~~4+ 3 '
(ii.) a?(x - 1) -ab(x- 2) = //-'.
8. If the angles at the base of an isosceles triangle ABC be
bisected by the straight lines BD and CD, show that
the triangle DBC is isosceles.
122 MATHEMATICAL EXERCISES.
CXCVI.
1. Find the cube root of 30-664297.
2. What number is that, from which if ^ - ^ be deducted,
_4
and to the remainder -5- added, the result will be 3|| ?
3. Express 9 hrs. 1 9 min. 3 sees, as the decimal of 9 days.
Prove your answer.
4. Find the Simple Interest on £533 6s. 8d. for 146 days,
at 3 per cent.
5. Bracket like powers of x together in the following ex-
pression : —
as? — bx* + cxz- dx3 + ex- fxz + dx*,
and arrange the result in descending powers of x.
6. Find the L.C.M. of
6xf(x + y), 3x*(x-y)2, and 4(z2-y2).
7. Use the formula
(x-y)(x + y) = x2-y2
to find the value of
(a + b + c)(a + b - c}(a -b + c)(c + b — a).
8. A garrison of 1000 men was victualled for 30 days;
after 10 days it was reinforced, and the provisions were
exhausted in 5 days. Find the number in the rein-
forcement.
CXCVII.
1 . Two clocks point to 8 o'clock at the same instant on the
morning of New Year's Day 1880 ; one loses 6 seconds,
and the other gains 10 seconds in 24 hours. When
will one be half an hour before the other, and what
time will each then show ?
2. Find the length of the side of a square field containing
3952 ac. 3 ro. 18 po. 21-| sq. yds.
MATHEMATICAL EXERCISES. 123
3. Divide 3-714285 by 5-0571428" and express the answer
as a decimal correct to 4 places.
4. What is the discount on £2775 19s. due 6 years and
4 months hence, at 7| per cent. 1
How can you test your answer ?
5. When a = b + c, find the value of
a3 - ft3 - c3 - 3abc.
6. Simplify (i.) a(a - b)x2 + b(a + b)x - 2b2,
(ii.) x2^+af+a2.
7. Solve the equations : —
(i.) z-2-(2a; + 3) = ^±i,
Z
(ii.) 3x - 8y + 4 = 0, x=2y.
Prove that your answers satisfy the equations.
8. Prove that the bisectors of adjacent angles are at right
angles.
CXCVIII.
1. A rectangular garden contains 1200 square yards, and
the length is J as much again as the breadth ; find its
dimensions, and the cost of fencing it at 3s. 6d. per
yard.
2. Add together -032 of £1, 2-35 of a shilling, and 5-42 of
a penny, and subtract '35 of half a crown from their
sum. Give the answer in pence.
3. A can do as much work in 3 hours as B in 4, and B can
do as much in 5 hours as C in 6. A gets through a
certain work in 15 hours, how long would it have
taken C 1
4. At what rate per cent, will £550 amount to £1060 9s. 4id.
in 11£ years, Simple Interest?
l->4 MATHEMATICAL EXERCISES.
5. Find the factors of
6. Find the G.C.M. of
and
7. Stopiify (i.) — •
(it)
13z+15a;2"
Test the result of (ii.) by putting x=l.
8. I have five times as many shillings as half-crowns, and
my money altogether amounts to £3. How many of
the coins are half-crowns ?
CXCIX.
1. Find, by Practice, the wages of a man for 3 weeks,
2 days, 4 hours at £3 a week ; reckoning 6 days to a
week, and 1 2 hours to a day.
2. Add ^ of ^V of 1 cwt. 2 qr. to & of £4$ of 3 qr. 12 lb.,
and give the answer in ounces.
3. Find the amount in 3 years of £2425 at 4 per cent.,
Compound Interest, neglecting fractions of a penny.
4. Add together 4-00607, -617634, and 376-473,
and subtract the sum from 400.
5. Divide
I a^-l^x+l^a^ + ^ax^-x4 by fa-fcc.
6. Write down two expressions, each containing powers of
x up to the third, which have 3x+ 2 as a G.C.M.
7. Break up 3^+2x<2~x + l jnto simple fractions.
xy
8. Prove that every triangle must have two acute angles.
MATHEMATICAL KXKI5C1SKS. 1-Jf,
CO.
1. Find the cube root of 34 correct to 3 places.
9 7
2. A person who owned ^ of an estate, sold ^3J of ^ of
^TTT
his share for £96. What would be the value of
2JL
s-j of ^7 of the whole estate ?
<%
3. How many lengths of -0375 of a foot are contained in
31-7296875 feet, and what decimal of a foot would be
left?
4. Find the amount of £625 16s. 8d. in 5| years at 2f per
cent., Simple Interest.
5. Find the factors of
(i.) a^-Ga^+iiz-e,
where (x~l) is one factor,
(ii.) GaP-ixty-Zxyt + top.
6. Find the simplest expression which is exactly divisible by
2o:2 - 8, 3z2 - 9z + 6, 6z2 + I8x + 12.
7. If 4a^ + 12x3y + oa;V+6a^3 + yl
is the square of
2xz + 3xy + kf,
find a and Tc.
8. A labourer was engaged for 36 days on condition that
for every day he worked he should receive 2s. 6d., and
for every day he was idle he should forfeit Is. 6d. ;
he received £2 18s. How many days did he work ?
CCI.
1. A cistern which can be filled by one tap in 28 hours is
emptied by four others, in 10 hours, 12 hours, 15 hours,
and 21 hours respectively. If the cistern be full and
all the taps open, how soon will it be emptied ?
126 MATHEMATICAL EXERCISES.
2. Find the value of 3(1-1139434 + -4771213) + 6 x -30103.
3. What ready money will discharge a debt of £1056
lls. lOd. due 8 months hence, at 4| per cent. ?
4. Divide £32818 among 4 persons in the proportions §, f,
4 5
T» tf"
5. Prove that the difference between a number, and half of
(10 more than the number), is 5.
6. Simplify
(i.) (2<i - 6)2 + 2b(a + b)- 3a2 - (a - 5)2 + (a + b)(a - b),
/ii) a(x2 + c*)-2acx h
(U° 6(x2+C2)-26Ca;' V
7. Solve the equation
a b o to
^- + — = a2-62.
to ox
8. Prove that the diagonal of a square is greater than its
side.
COIL
1. If the cost of painting the 4 walls of a room whose
length is 24 ft. 8£ in., breadth, 16 ft. 3| in. at 4s. a
sq. ft., be £188 12s., find its height.
2. Find the value of 2-749" of -254 of -190476 of £2 5s.
3. By selling 10 acres of land for £4699 8s. 3d. a man
gained T2^ of what the land cost him. What was the
original price per acre 1
4. What sum will amount to £4694 lls. in 7 years, at 4-J-
per cent., Simple Interest ?
5. Resolve into factors
(i.) 4z2-9y2, (ii.) 8x3-27^,
(iii.) a2-a6-652.
ATHEMATICAL EXERCISES. 127
6. Simplify
(i.) |(2o; - 3y) - ^(3x + 2y) 4- — ,!» >
12
,..v
xyz \xyzj
7. Find an expression which will exactly divide
x3 + 2ax2 + a^x + 2a3 and a? - 2ax* 4 a?x - 2a3.
8. Find the number whose third part added to its seventh
part makes 20.
CCIII.
1. Find the value of 7 qrs. 3 bush. 3 pks. of wheat, at
£5 4s. per qr.
2. Simplify 2^ of 6f + 4T^ of f + 6J x T3T+ ?|.
3. Find the amount of £2540 in 3 years at 4 per cent.,
Compound Interest, neglecting fractions of a penny.
4. Multiply 1-231056 by -81231056 correct to 7 places of
decimals.
6. Simplify (i.) __ + —__,
(iL) (^HT..H
7. Form two equations whose roots shall be 4, and a
respectively.
8. Straight lines which are respectively perpendicular to
two parallel straight lines, are themselves parallel.
1-28 MATHEMATICAL EXERCISES.
CCIV.
1. Find the cost of painting the four sides and bottom of a
tank 24- yards long, 4 feet wide, and 4J feet deep, at
4d. per square foot.
2. Simplify 4^ mile + ^y furlong - ll^f poles.
3. A can copy 150 pages in 30 hours; A and B together
can copy 280 pages in 35 hours. In what time would
B copy 200 pages ?
4. Find the Simple Interest on £1368 15s. for 106 days at
4f per cent.
5. Find the factors of
(\ ^ c$b^ — b^c^ — c^ct^ *f Qctbc^
6. Divide the product of
(a - b + c), (a + b - c), and (b + c-a)
by a2 - &2 - c2 + 2bc.
1. Simplify —^
ff j.
»v T -i
8. A horse is sold for £42 at a loss ; had it been sold for
£57 the gain would have been four times its present
loss. What did it cost ?
ccv.
1. If 4 cwt. 1 qr. 26 Ib. of tobacco can be bought for
£56 9s. 6d., how much can be bought for £4 19s. 9d. ?
2. Simplify without reducing to vulgar fractions :—
(•5 + -75) x (2-5 - -4) -r (-125 + -L\
3. Find the Discount on £528 15s. due 4 years hence, at
5| per cent.
MATHEMATICAL EXERCISES. 129
4. A watch, set on Friday at 9 p.m., gains 45 seconds in
12 hours ; what time does it show on the next Sunday
at 3 p.m.?
5. If x = 5, a — 8, find the value of
(Ltjx*1 — 3a + x^/x2 + 3a.
6. Add together
1 - [1 - (1 - »)], 2x-(3-5x) and 2-(-4 + 5z),
and from the result subtract
a - x - (x - 2a) + 2a - x - {a - 2x - (2a - x) + (x - 2a)}.
7. Find the cube of ax-by + cz.
What is the cube of %]x ?
8. Given a triangle construct a right-angled triangle of the
same area.
CCVI.
1. A rectangular courtyard is 100 ft. long by 60 ft. broad.
Two paths cross it at right angles, parallel to the sides,
each 5 feet wide. Find the cost of turfing the remain-
ing area at 6d. a sq. yard.
2. Find the value of 03249 of £1.
3. A cistern can be emptied by 3 pipes in 3 hours ; one
pipe alone would empty it in 6 hours, another in 9
hours ; how long would the third pipe take ?
4. (i.) In what time will £1300 amount to £1493 7s. 6d.
at 3£ per cent, per annum, Simple Interest ?
(ii.) State this as a " Present Worth" sum.
5. Prove that an - b" is divisible by a-b.
When is it also divisible by a + b 1
6. Find the G.C.M. of
2ooy, 4a&2»y, and (Sasxf - tabx^y).
7. The product of two factors is (3z + 2y)a - (2ar + 3y)2,
and one of the factors is x— y ; find the other.
130 MATHEMATICAL EXERCISES.
8. A is twice as old as E ; twenty -two years ago he was
four times as old. Find their ages.
CCVII.
1. Find the cost of 3 oz. 10 dwts. 15 grs. of gold at £3 10s.
per oz.
2. What fraction of a cwt. is 1 qr. 22 Ib. 14 oz. 8T8T dr. ?
3. Find the difference between the Simple and Compound
Interest on £933 6s. 8d. in 2| years at 2| per cent.
4. Find the value of 7 '653478 x -03576 to 6 places of
decimals.
5. Divide 9a4- I2a3b + 34a262- 20o&3 + 2564
by 562 + 3a2 - 2ab.
6. Simplify (i.) Tf~AX"V
*C ~~ ~~s — Jj T~ A
.
v '' (x + a)(x - a) ' (x2 - a?)(x + of
7. Solve the equations : —
7x-W 16z+15 2f
-- -— +'
-5z= -5.
4x-5y + 3z=lO.
5x + 3y-4z= 3.
8. Any three sides of a quadrilateral are greater than the
fourth side.
CCVIII.
1. Find the length of the edge of a cube which contains
18-Q62 cubic yards. What is the area of its face ?
MATHEMATICAL EXERCISES. 131
2. Simplify
3. What decimal of £6 5s. is 3s. 6dJ
4. In what time will £4550 amount to £5573 15s. at 3 per
cent., Simple Interest 1
5. Resolve into their elementary factors : —
(i.) a3 - a?x —
(ii.)
(iii.)
6. Write down the square of a-b + c-d, explaining the
process.
7. If " « show that ^±-t=^±
o a c+d c—d
8. Find the G.C.M. of
Sx3 - 3x*y + xy* - y3 and 4^ - xy - 3y* ;
and the L.C.M. of
9s3 + 532? - 9z -18 and xz+llx + 30.
CCIX.
1. Find the square root of '0003418801, and the cube root
of -25 to 3 places of decimals.
2. Divide -000324 by -018, -0000018, and 1800.
3. What is the Present Worth of a bill for £694 10s., due
3| years hence, at 4| per cent. ?
4. A, B, and C start in business; A puts in £5000, S,
£6000, and C £9000. At the end of three months
C leaves, at the end of 7 more B leaves, both taking
out their capital ; but the profits, which amount to
£816 13s. 4d. are not divided till the end of the year.
How much does each get ?
132 MATHEMATICAL EXERCISES.
5. If a = 0, J = l, c = 2, d = 3, find the value of
c-d- — + ad
3 2c - bd + 2ac - b
6. Simplify
/.. x 4a4 2a2 a a
^2 2
a + x a-x
Test your results.
7. Solve the equations : —
r \ a b b a
(i.) - + -=p, - + -=q.
x y ' x y
/.. v 2a- 1 a; + 4 _ 5x - 1
( ' ~3~ ~9~ "T2~-
8. ^ is twice as old as B, and 4 years older than C ; the
sum of the age of A, B, and C is 96 years : find J5's
age.
OCX.
1. A room is 20 ft. long, 18 ft. 6 in. broad, and 11 ft. high;
it has a door 4 ft. by 7 ft., and a window 5 ft. by 8 ft.
Find the cost of papering it at 9d. a square yard.
2. Reduce 13s. 6f d. to the decimal of £4.
3. If 12 men can build a wall 6 feet high and 3 feet thick
and 20 yards long in 9 days, how many men would
build one of the same length, 5 ft. high and 4 feet
thick in 24 days ?
4. In what time will £4550 amount to £13,195 at 3f per
cent., Simple Interest ?
MATHEMATICAL EXERCISES. 133
5. Find the factors of
(i.) a>"-y",
(ii.) 5z2 + Sxy - '21 f,
(iii.) x5 + 3x* + 3x+l.
6. Find the G.C.M. of
a2-4z2+12a;-9 and a2 + 2a - 4z2 + Sx - 3.
7. Simplify (i.) 4{(2a&V)4-2a&2c(2a&2c5)3},
(ii.) (a7-r(a«)'.
8. BAC, DAF, are two straight lines bisecting each other
at A. From B and C perpendiculars BK, CL, are
drawn to DF. Prove that DK is equal to -Z/.F.
CCXI.
1. Find the dividend on £4146 12s. 6d. at 11s. 8d. in the £.
2. Find the value of
^ of £1 + f of Is. + /¥ of a guinea + f of £1 6s. 8d.
3. Find the Compound Interest on £133 6s. 8d. for 3 years
at 2£ per cent.
4. Simplify
£{1-1139434 + £ x -C9897 - £ x -845098}.
5. Divide (i.) 3Qxf- \§x*y + 8x*- 30x2y2
and (ii.) a$x^ by a?x*.
6. Simplify _J ^.^J^^.
1 - 2x 1 + 2x 4x* - 1
7. Solve the equations : —
Ix- 11^-3 = 0.
5y - 6x + 7 = 0.
8. Construct a rectangle double of a given triangle.
134 MATHEMATICAL EXERCISES.
CCXII.
1. Find the quotient of 4 divided by l^y, and the quotient
of this result divided by ^.
2. A man saves ^ of his income, gives away -^ of it, and
spends the remainder, £520. What is his income ?
3. Find the value of
•65 of 4-il of 3 of 2432 of 13s. 6d.
1 0
4. What is the difference between the simple interest on
£266 13s. 4d. for three months at 4| per cent., and
the discount on £83 due 15 months hence at 3 per
cent.?
, 4z2-5z-6 6a;2+22a;+20
2 l
X- 1 X X+l
6. Find the L.C.M. of
y*)(xz + f),
7. Find the value of
(ax + by- cz)z - (ax -by + cz)2.
8. There is a certain fraction which becomes \ when 1 is
added to its numerator, and ^ when 1 is added to its
denominator. Find it.
CCXIII.
An engine while driving machinery burns coal at the
rate of 1 ton 12 cwt. 2 qrs. in 8 hrs. 40 min. When
the machinery is not in motion the consumption of
coal is only -^ this rate. How much coal will it burn
in 1584 hours, working •§ of the time?
MATHEMATICAL EXERCISES. 135
2. Divide by -000039.
o »
3. Find the Present Worth of £2457 12s. due 9 months
hence, at 3| per cent., Simple Interest.
4. When are the hands of a watch together between 7 and
8 o'clock ?
5. Find the value of
a
6. Simplify
(i.) |(y + z) + f (* + «) + f (z + a;) + f (x + y)
(ii.)
x-l
X
~T
X-
iC+1
7. Solve:—
Test your answers.
8. Find the square root of
CCXIV.
1. A rectangular field is 214-35 yards long, and 123-26
yards wide. Give its area in acres, roods, and poles ;
and find, to the nearest inch, the side of a square field
of equal area.
136 MATHEMATICAL EXERCISES.
2. Add together -6375 gallon, -01283 pk., and -0041 2 bush.;
subtract the result from -93 gallon, and give the answer
in pints and decimal of a pint.
3. If £1 is equivalent to 25 francs, 36 ducats to 375 francs,
and 19 ducats to 40 roubles, how many roubles are
equivalent to £95 1
4. At what rate per cent., Simple Interest, will £1885 15s.
amount to £2569 6s. 8^d. in 7 years and 3 months t
5. Find the G.C.M. of
4s3 - 6x + 2, and 9Z3 - 3z2 - 6,
and the L.C.M. of
z4 - a?x, ax3 + a4, z4 + a2«2 + a4.
6. Simplify
1 1 1
a(a - b)(a - c) b(b - c)(b - a) c(c - a)(c - b)'
7. If (i.)
(ii.) a?
(iii.) am + bm,
are divided by a + b, what are the remainders ?
8. Prove that
(b - c)(l + db)(l + ac) + (c- a)(l + 6c)(l + ba)
+ (a-b)(l + ca)(l + cb) = (b- c)(c - a)(a - b).
1. Simplify
ccxv.
Qf 2-i
__
ill of 2±i
3-1 2-^
2. Multiply 3-85704 by -36479 correct to 3 places of
decimals.
3. Find the square root of 2, to 6 places of decimals.
4. Find the value of 5354f articles at £4 4s 8d. each.
MATHEMATICAL EXERCISES. 137
5. Simplify
a2 - a + I
2g(a-l)2 2a2(a2-!)2
a4 + a2 + 1 a8 + a4 + 1 '
a2 + a + 1
6. Find the G.C.M. of
a4 -
and
7. ^4, who travels 3| miles per hour, starts 2J hours before
.#, who is going at 4£ miles an hour in the same direc-
tion. When will B overtake A ?
8. ABD is an isosceles triangle having the angles at B and
D each double of the angle A. If DC bisects the angle
at D, and meets AB in C, prove that AC is equal to
CD.
CCXVI.
1. Find, by Practice, the dividend on £1726 4s. at 4s. 5|d.
in the £.
2. Multiply 43-133 by 2-34S to 3 places of decimals.
3. Find the cost of carpeting a room 10 yds. 2 ft. long, and
7 yds. 1 ft. broad, with carpet f yd. wide at 4s. 6d.
a yard.
... _ -004-^-0005
mpllfy 2-423 + 3-576 + 2-0001911'
5. If a= 1, b - 2, c = 3, d = 4, find the value of
6. Find the G.C.M. and L.C.M. of
x?-xl + x?-x* + x-I and x5 - 1 .
7. A, after doing f of a work in 30 days, calls in B, and
they finish it in 10 days : in how many days could
each do it alone ?
138 MATHEMATICAL EXERCISES.
8. Show that in converting to a decimal any fraction whose
denominator is 7, the same digits always occur after
the decimal point, but in different order, and point out
how they are connected one with other.
CCXVII.
1. Reduce 82976432 inches to miles, and 7 miles 14 poles
3 1 yards to inches.
2. Add together £15-125, 17-3125 shillings, and 975 pjnce,
and reduce the result to a decimal of £25.
3. Find the cube root of 3 to 4 places of decimals.
4. Make out an invoice of the following : — 5 prs. of blankets
at £1 4s. a pair ; 12| yds. of merino at 2s. lid. a yard;
15f yds. of cloth at 9s. 6d. a yard; 5| yds. of flannel
at Is. 9d. a yard; 2 counterpanes at £1 3s. 9d. each;
25^ yds. of calico at 9d. a yard.
5. Solve the equation : —
x y - y
6. If a = 2, 6 = 3, c = 6, d = 5, find the value of
d)(5d - 4c)} + ^c-a.d-b.
7. A, B, and C are employed on a piece of work. After
3 days A is discharged, ^ of the work being done.
After 4 days more B is discharged, ^ more being done.
C then finishes in 5 days. How long would each take
separately ?
8. Construct an isosceles triangle such that each of the base
angles is three-tenths of the vertical angle.
CCXVIII.
1. How many times does the 29th day of the month occur
in 400 consecutive years ?
MATHEMATICAL EXERCISES. 139
2. A tank is 8 ft. long, 5 ft. 4 in. wide, 4 ft. 6 in. deep ;
find the number of gallons it contains.
(1 cub. ft. of water weighs 1000 oz., and 1 pint of water
weighs 1£ Ibs.)
3. Find, by Practice, the rent of 134 acres 3 roods 6 poles
at £3 13s. 4d. per acre.
4. Simplify
x
1 +
,
1 +x +
l-x
5. Find the square root of
37aty2 - 30z3y + 9s4 - 2
6. Solve the equations : —
,. v f2(3-3)-i(y-3)
''l3(y-5) + $(a5-2)
~l ,4-2(2a;-5
~
3
7. Multiply together
x + a - by and x - a + b ;
and show that the product is zero, if x is equal to
the difference of a and b.
8. ABCD is a parallelogram : from any point P in BD, the
straight lines PA, PC are drawn. Show that the
triangles PAS, PCS are equal in area.
CCXIX.
1. If silver plate weighing 3 Ib. 4 oz. 7 dwt. 8 gr. is valued
at £20 3s. 8d., what is the price per ounce 1
2. Multiply f+l^-J by 4f-3T2T + 1^,
and express the result as a decimal.
140 MATHEMATICAL EXERCISES.
3. Merchandize weighs 4 tons 4 cwt. 1 Ib. after being
reduced in weight 3 per cent. What was the original
weight ?
4. Simplify
(7x2 - 5xy + 6y2) - (2«2 + Sxy - y2) - (3x2 - 2xy) + (±xy - 5y2).
5. Divide
6. Solve the equations : —
,.. Qx-l 1 / 1 - x\ 1
_?
1. Find the G.C.M. of
a3 - 3a2 + a + 1 and a2 - 3a + 2.
8. Prove Euc. I. 12, without using I. 8.
ccxx.
1. How much tin and copper does a bell weighing 150 Ib.
contain, bell metal being composed of 75 per cent.
copper, and 25 per cent, tin 1
2. Find, by Practice, the cost of 3 cwt. 3 qr. 17 Ib. at
£1 7s. 9d. per cwt.
3. A, B, and C rent a meadow for £43. A puts in 10
horses for a month, .#,12 oxen for 2 months, and C
20 sheep for 3 months. The quantities eaten by a
horse, ox and sheep are as 4 : 3 : 1. What should
each pay ?
4. Solve the equations : —
(ii.) 3^ + -10 =
MATHEMATICAL EXERCISES. 141
5. Simplify
' x-y
6. A man bought a certain number of sheep for £35 ; after
losing 2 he sold the rest at a profit of 10s. a head, and
so gained £] on the transaction. How many did he
buy?
7. Find the L.C.M. of
I6a2b(a-x), 24&2(a2 + ax), and 9a&(a2-a;2).
8. Divide a triangle into four equal parts, by straight lines
through the vertex.
CCXXI.
1. Divide £7 6s. 6|d. between 2 men, so that for every 3d.
received by the one, the other may have 4d.
2. Find the value of 19672 things at 3s. 7£d. each.
3. Express f of f of 2£ of a mile in kilometres.
(32 metres = 35 yards. 1000 metres = 1 kilometre.)
4. Solve the equations : —
(i.) 6a;-12y=l,
,.. x 3-x 4-a;
(1L)
5. Find the value of
_
yz + zx-xy 2(y + z-x)'
when x = 3, y= - I, z = 2.
6. A and B do a piece of work in 20 days ; A and C in 12
days; B and C in 15 days. How long would each
take separately ?
142 MATHEMATICAL EXERCISES.
7. Show that (x - 2)2 is a common factor of
&7-6a*J + 13z6-123* + 4a£ and z7 - 5z6 + 8x5 - 4Z4 j
and find the G.C.M. of the two expressions.
8. Through a given point in a circle draw a chord which
shall be the farthest possible from the centre.
CCXXII.
2. Find the cube root of 74991286313.
3. Divide £31 17s. 6d. into 2 sums of money, one of which
shall contain as many sixpences as the other contains
fourpences.
4. Find the G.C.M. of
a?y? - a?bx2y + atfxy2 - b3y3 and
5. Solve the equation
/x - a\3 _ x - 2a - b
\x+bj
6. The product of 2 numbers is 750, and the quotient when
one is divided by the other is 3^. Find the numbers.
7. Find the square root of
!-4a+10a2-12a3 + 9a4.
8. If the straight lines bisecting the angles at the base of an
isosceles triangle be produced to meet, they will con-
tain an angle equal to an exterior angle of the triangle.
CCXXIII.
1. If 15 pumps working 8 hours a day, can raise 1260 tons
of water in 7 days, in how many days will 30 pumps,
working 12 hours a day, raise 7560 tons ?
2. Find the cost of paving a floor whose length is 33 ft. 2 in.
and breadth 18 ft., at 6s. a sq. yd.
MATHEMATICAL EXERCISES. 143
3. A contractor agrees to supply sugar at 9d. a lb., and
buys the sugar at 70s. per cwt. What will he gain per
cwt., and per cent. ?
4. Solve
, - -_
~9~ "IT". ,
(ii.) (3z
5. What is the continued product of
x - y, x + y, xz + y2, x* + y*, x8 + y81
6. Divide Xs + tfy2 + x*y* + x2y^ + y8,
by a4 + x*y + 2%2 + xy3 + y*,
and a4 — - by a ---
a4 a
7. A sum of money is divided among 3 persons. The first
receives £-10 more than a third of the whole; the
second £15 more than a half of what remains ; and the
third receives what is over, which is £70. What did
each receive ?
8. If the diagonals of a quadrilateral bisect each other, it is
a parallelogram.
CCXXIV.
1. Reduce 1 ton 8 cwt. 3 qr. 17 lb. to the fraction of 3 tons.
Q. r. 6-60416 2-772
2. Simplify — -j- — — .
33-02083 11 -6&
3. What sum will amount to £463 Is. in 3 years at 5 per
cent., Compound Interest ?
4. If a reservoir of water be 6 ft. 4 in. wide, and 25 ft. 4 in.
long, how many tons of water must be drawn off, that
the surface may sink 7 ft. 6 in.
(1 cub. ft. of water weighs 1000 02.)
144 MATHEMATICAL EXERCISES.
5. Solve the equations : —
/. v x + 3^x-3_x-5_
(ii.) 10a;-4y=ll, 3«+2y = 14|.
6. Find the square root of
4?/2 cc2 2y x **
7. Simplify
,. x z+l x+2 z2-4
\ v —
x 2
' 9" 3^a;-6 3(»-6)
8. In a division, a motion was carried by a majority equal
to -^ of the votes of the winning side ; had 8 votes
been transferred to the losing side it would have
been lost by a majority of 1. Find the number of
votes.
ccxxv.
1. Reduce to their simplest forms : —
19f
2. Find the value of 1-657142 of a guinea.
3. Divide 81 into 2 parts which shall be in the ratio 4 : 5.
4. Divide (a;2 - y2)3 - & by x^-y^-z2.
5. If a : b : : c : d, then
(a + b + c + d)(a + d-b-c) = (a + c-b- d)(a + b-c-d).
6. If a + b men and b - a boys make a2 - J2 tables in a2 + b2
days, how many men can make a2 - ab tables in the
same time ; given a boys = b men ?
MATHEMATICAL EXERCISES. 145
7. Solve the equations : —
,n x+1 x+2_ x+3
( ' ^T+^2~2^T3'
(ii.)
8. Simplify
g2 &s C2
(a - b)(a -c) (b- c)(b -a) (c- a)(c - b) '
9. Find 2 numbers such that £ the first and f of the second
together may be equal to the excess of 3 times the
first over the second, and this excess equal to 11.
CCXXVI.
1. If I give away f of my money, then f of what remains,
then ^ of the remainder, what fraction of the whole
have I left?
2. Simplify
•2845 gal. + -0494 qt. + 3-32 pt. - -21185 gal.
3. Given the principal £375, Simple Interest, £3 2s. 6d.,
and rate per cent. 2|, find the time.
4. Find the values of a and b, which will make xz + c3
exactly divisible by
x2 + ax + b.
5. Simplify
x y x y
-+-+2 -+--2
x y x
x
/.. x
(ii)
,. .
(i.) ^ - + •
x + y
(x-2a)2 x*-5ax + 6a2 x-3a
(iii.) (x% x
146 MATHEMATICAL EXERCISES.
6. Solve the equations : —
(H.) j j*_£"jz-
7. A certain fraction becomes ^ when 1 is added to its
numerator, and | when 1 is added to its denominator.
Find it.
8. Of all straight lines drawn through the intersection of
the diagonals of a rectangle, and terminated by the
sides, the diagonals are the greatest.
CCXXVII.
1. The driving wheel of a locomotive 5 feet in diameter
turned 5000 times in going 12 miles. If the circum-
ference of a circle = 3*141 6 times its diameter, what
distance, in miles, was lost by the slipping of the wheel
on the rail ?
2. Find the value of (12-5)2 -5- (-005)3.
3. Find the cost of 1 ton. 17 Ib. 5 oz. at £22 8s. per cwt.
4. If a = 2, find the value of
5. Solve
(i.) (
(ii.)
6. Divide
kzl3 - m(k2 + m)l + km2 by kl-m;
and
without removing the bracket, or bringing the fractions
to a common denominator.
MATHEMATICAL EXERCISES. 147
7. Simplify
(i.) J* ^.-^JL E±i,
x-^2 x + 2 ' x-2 x-2
4 +x-3 3 + x-l
8. Show that the sum of any two consecutive whole num-
bers is equal to the difference of their squares.
ccxxvm.
1. Reduce -1215625 of a mile to inches.
2. If gold is worth £3 17s. 10|d. per ounce, and silver
5s. 2£d. per ounce, how many sovereigns will weigh as
much as 5000 shillings ?
3. Find the True, and Mercantile, Discount on £2000 for
If years at 3 per cent.
4. A man sells an article at 5 per cent, profit. If he had
bought it at 5 per cent, less, and sold it for Is. less, he
would have gained 10 per cent, profit. What was the
cost price ?
5. Find the square root of
6. A man starts from A to walk to £, a distance of 16
miles, at 4£ miles per hour, at the same time that
another leaves B for A at the rate of 3^ miles per hour.
Where will they meet ?
7. Solve
148 MATHEMATICAL EXERCISES.
8. If four straight lines meet at a point so as to make the
vertically opposite angles equal, they are two and two
in the same straight line.
CCXXIX.
1. Simplify -65 of 4-ii of ff of 2432 of 13s. 6d.
13
2. A bankrupt can pay 13s. 4d. in the £ ; if his assets were
£1000 more he could pay 14s. Find his debts and
assets.
3. A man buys 4200 articles at the rate of 7 for Is., and
sells them at the rate of 12 for 2s. 6d. What is his
whole gain and his gain per cent.?
4. Keduce to its lowest terms
5. Solve
' 1 9z-5y = 69.
(iii.) a2 -123z+ 1332 = 0.
6. A bag contains half-crowns, shillings, and sixpences, and
the three sums of the several coins are of equal value.
If there are in all 51 coins, find the number of each
sort.
7. If a = 4, x = 2, find the value of
8. The straight line bisecting the exterior angle at the
vertex of an isosceles triangle is parallel to the base.
MATHEMATICAL EXERCISES. 149
ccxxx.
1. Extract the square root of 15356-28125 of £f£f
2. In how many years will a sum of money double itself at
2£ per cent., Simple Interest ?
3. Simplify
4. Simplify
5. Solve
r, 3x-l3 _
~~ ~~~ '
(ii.) x =
5-
5-
5-a;
6. A man walks from A to B and back, a distance of 10
miles, taking 2 hrs. 51£ min. to go, and 2 hrs. 55 min.
to return. He goes 3 miles per hour up hill, 3£ on the
level, and 4 miles an hour down hill. Find the length
of level road between A and B.
7. Write down 2 expressions each containing powers of x
up to the 4th, which have x2 + x + 1 as a common
factor.
8. Describe a circle with given centre, and touching a given
circle.
150 MATHEMATICAL EXERCISES.
CCXXXI.
1. Simplify
1 1
1+— ^- 5+— 1
2 + --*- 4+ l
1 1
ft J_ O i
O T ~i — -T O T ,?i
2. A metre is 39-371 inches. Find the length of a mile in
metres.
3. By what must 23520 be multiplied, to make it a perfect
square ?
4. Factorize
5. Find the G.C.M. and L.C.M. of
a5-a4 + a3-a2 + a- 1, and a3-!.
6. Solve the equations : —
({} 2x x-Q _3a_u
^ ' I5+ 12 20 ~ *f
(ii.) Jx- Ja+ *Jxta-b= Jb.
7. The sum of £255 is to be divided among 20 men, 30
women, and 45 children, so that a man and a child
may receive as much as 2 women, and the women
may together receive £90. What do they respectively
receive ?
8. In the parallelogram A£CD, BL, DN are drawn per-
pendicular to the diagonal AC, and AK, CM. to the
diagonal ED.
Show that (i.) the triangles BLC, AND are equal;
(ii.) that KLMN is a parallelogram.
CCXXXII.
1. Find the value of s/3- ^3 to 3 places of decimals.
MATHEMATICAL EXERCISES. 151
2. Express 4s. 7d. as the decimal of £2 5s. lOd.
3. If 8 men can reap 76 acres in 8 days, how many men can
reap 114 acres in 6 days?
4. Simplify
^7 v I 1 , 41 Q 9 Q2
' -
»• ' 2+-
°"ff- ' ~S~ ^°¥TF + ¥ *T
5. Find the L.C.M. of
^-27^, x2-l5xy + 3Qf, and z3 - 3x2y - 2xy* + 6«/3.
6. Divide x into two parts so that the first part divided
by a and the second part divided by b may
together equal y.
7. Simplify
lx + 2y x\ /x+2y x \
\ z + y y) \ y x + y)
ind test your result by putting x = y = l.
8. If a + b-c = 0,
ehow that a3 + abc + a?b = act.
CCXXXIIL
1. Find "he value of
•003 of £1 5s. + -069 of £5 - •& of 2s. 3d.
2. Find Ihe cost of carpeting a room 16 ft. 7 in. long and
13 ft 5 in. wide, with carpet f yd. wide, at 4s. 2d.
a yarl.
3. A farmer bought some wheat for £25 10s. At what
price nust he sell it to gain 10 per cent.?
4. Simplify
— in
: X
n m
152
MATHEMATICAL EXERCISES.
5. Solve
(i.)
n {
6. Show that
(x2 - yz)y + (y2 - zx)s + (z2 - xyf - 3(x2 - yz) (y2 - zx)(zz - ty)
is an exact square.
7. Find the G.C.M. of
ic4 - 2x3 + x2 - 1 and a;4 - 3z2 + 1,
and the L.C.M. of
1 5a3x(a + x)3, 20ax3(a - x)3, 36aV(a2 - a;2)2.
8. If from a point in one side of an equilateral triangle,
parallels be drawn to the other sides, the parallelogram
thus formed has its perimeter always the same length,
wherever the point be taken.
CCXXXIV.
1. Find the square roots of 1-014049 and 175ff.
2. A invests £552 in the 3| per cents, at 92; .# invests
£679 in the 3 per cents, at 97. Find the dfference
in their incomes.
3. What must be the least number of soldiers in a regiment
that will allow it to be drawn up 3, 5, or 6 leep, and
also to be formed into a solid square ?
4. Find the L.C.M. of
/>*4 1 ("W2 4. Q />»4 i 1 0*7*3 j. 90/>*2 1 Oo* _ 91
*t/ A \JtHj i «/5 *(/ T^ X \JtJU T & VtO — L V»v £i Jf
___ J ..A . -I _.Q f»n..O A .. . C\~t
5. Solve
a; + 6 x-3 x+2 x+l
(ii.) _=
MATHEMATICAL EXERCISES. 153
6. Multiply
1 by a*+l.
7. Factorize
a8 -of, 4ft4-12a3a:+13a2a;2-6a:c3 + z4, a4 + a2 + l.
8. Prove that if a perpendicular be drawn from the vertex
of an equilateral triangle, to the base, the square on it
is equal to three times the square on half the base.
ccxxxv.
1. Find the square root of 1015-6969,
and cube root of 4^^£|-.
2. What is the income corresponding to an income-tax of
£10 at lOd. in the £ 1
3. Find the cost of painting the 4 walls of a room at 1 Od. a
square yard, the length of the room being 18 ft. 10£ in.,
breadth, 17 ft. If in., and height, 12 ft. 3 in.
4. Simplify
X*
a +
&
a + V
5. Extract the square root of
6. Solve
7. Find the time between 2 and 3 o'clock, when the hands
of a watch are together.
8. Show that no parallelogram except a rectangle can be
inscribed in a circle.
164 MATHEMATICAL EXERCISES.
CCXXXVI.
1. If I invest £3591 in 3 per cents, at 85|, what income
shall I get, and what interest per cent, on the invest-
ment?
2. Find the square root of -07 to 5 places of decimals.
3. A legacy remains unclaimed for 4 years. At the end of
that time it amounts to £14,586 Is. 6d. at 5 per cent.
Compound Interest. What was the original sum 1
4. Simplify
^ of a ton + £ of 2 cwt. + if of a qr. + 98 Ibs.
- 3£ of 2^5- of a cwt.
5. Simplify
2(3z-2y) 2(3x+2y)
_a ut
(ii.)
a3 -
2
2 -
(iii.) (a
6. Solve
.. v /, x- 12 £-4 z-
!.,- 6— _=_+_
(ii.)
7. Two sums of money together equal £54 12s., and there
are as many £ in one as shillings in the other. Find
them.
8. If 3 sides of one triangle be perpendicular to the 3 sides
of another, the triangles are equiangular.
MATHEMATICAL EXERCISES. 156
CCXXXVII.
1. By selling a horse for ,£82 5s. a man lost 6| per cent.
How much per cent, would he have lost or gained, had
he sold it for £90 10s. ?
2. Find the dividend on £2738 16s. 8d. at 9s. 4|d. in the £.
3. What sum of money must be lent at 2£ per cent.,
Simple Interest, for 5^ years, so that at the end of the
time it may amount to £814 10s.?
4. Divide a^ + Sor3 by x+2x~\
5. Find the L.C.M. of
2(a26 - aft2), 6(a26 - 63), 9 (a3 - a26 + ab2)
and 15(a?
6. Solve
,. v x 11 3x + 5 3x-S
22=-T-
(ii.) (a; + 4)(y+5) = (x
7. Find 2 consecutive numbers the difference of whose
cubes is 91.
8. If A, B, are two points on a straight line, find a point
C on BA produced, such that the square on A C may
be equal to the rectangle contained by BA and BC.
(SeeEuc. II. 11.)
CCXXXVIII.
1. Find the number of cubic chains and links in a rect-
angular solid, whose edges are 189 chains, 2 chains
10 links, and 63 links.
2. Divide 4'03 by -1 $09523. (Answer in recurring
decimals.)
3. Find all the prime factors of 111540, 296352, 404352,
and thence write down all the numbers which will
exactly divide all three.
156 MATHEMATICAL EXERCISES.
4. Divide
a? + a?$ - a?b* -ab + atb* + b* by a? + b*.
5. Prove that any common factor of two quantities is a
factor of their sum, or difference.
6. Find the square root of
9 7 ab lax b2 a2
,2-te--+_ + T+-.
7. Solve the equations : —
/. s a; — a x-b a A
x-a — c x-b + c a + c b-c'
8. If D be the middle point of the base BC of a triangle
ABC, the squares on AB, AC, are together double of
the squares on AD, DC.
CCXXXIX.
1. The map of a country is drawn on a scale of ^ of an
inch to a mile. What area on the map will represent
a lake 4000 acres in extent ?
2. Find the square roots of '4 and '04, correct to 4 places
of decimals.
3. A merchant sells tea to a tradesman at a profit of 60 per
cent., but the tradesman becoming bankrupt pays only
2s. 6d. in the £. How much per cent, does the mer-
chant lose or gain ?
4. Find the G.C.M. of
4z* - 1 3a?y + 9y*, and 1 Oz4 + 23aty - 28xf -
5. Solve the equation
x 5 xz 8a -1
~\iC *T" o) «C *™ v ^r(*C ^ O)
MATHEMATICAL EXERCISES. 157
6. Multiply
and divide 3'2x + y? by
7. Find the value of
a;3 + 2y3 + 2z? + Qxyz
when £=y+2= 4/4.
8. If an eight sided figure be inscribed in a circle, show that
the sum of either four alternate angles is equal to 6
right angles.
CCXL.
1. Simplify
gj + Sjof I.
H °f 3* -if
2. A merchant buys 6 doz. of wine at 72s. per doz., and
20 doz. at 54s. What is his gain per cent, if he sells
the mixture at 63s. per dozJ
3. Find the value of
•72 of -2016 of 3 mi. 4 fur. 24 po.
4. Find the G.C.M. of
Sx4 + x-l and 4z3-3z+l.
5. Solve
/•N 2x-3 y-8_y + B x-1 . 4y+l_0
~f~ ~5~ 4 ~3~ ~TT
f" \ x~ ^ X_X~I g-2 a;-3
~2~~3~~3~ ~4~ ^~
6. ^4 and 5 begin business with equal sums. At the end
of a year A has gained £50 and B lost ^ of his capital.
A has then twice as much as B. How much had each
at first 1
7. What is a Geometrical Progression ? Give examples.
158 MATHEMATICAL EXERCISES.
8. Through a point K within a parallelogram ABCD, straight
lines are drawn parallel to the sides. Show that the
difference of the parallelograms of which KA and KC
are diagonals = twice the triangle BKD.
CCXLI.
1. If the weight of a cubic foot of water be 62-35 Ibs.
Avoirdupois find the error in calculating the weight of
1000 cub. ft. on the assumption that 1 cub. ft. weighs
1000 oz.
2. Find the value of -2671875 of £6 in shillings and pence,
and decimal of Id.
3. Find the number of gallons of water which pass in ten
minutes under a bridge 17 ft. 8 in. wide, the stream
being 10 ft. 11 in. deep, and its velocity 8 miles per
hour. (1 gall. = 277-72 cub. in.)
4. Find the square root of
5. Prove that in an equation any quantity may be trans-
ferred from one side to the other, by changing its sign.
6. If a:b :: c:d
prove that ^ = i
cx + d a
7. Find the L.C.M. of
8. Draw a tangent to a given circle which shall be parallel
to a given straight line.
CCXLII.
1. Supply the term wanting in
( ): 286 :: 581:182.
MATHEMATICAL EXERCISES. 159
2. If 11 horses eat 30 bushels of corn in 13 days, how many
horses can be fed at the same rate on 67£ qrs. for
78 days ?
3. Simplify
(ii.) _•
14
4. Construct the equation whose roots are 3, and - J.
5. Find 2 numbers whose product is 3, and such that their
sum divided by their difference is 4 times their differ-
ence divided by their sum.
6. If a = l, 6 = 0, c= -£, find the value of
a2 - c - 3ac(6 - 2c)
7. Solve the equations : —
(ii.) 1 y + 4:
10
8. Describe a circle to pass through a given point, and
touch a given straight line at a given point.
CCXLIII.
1. Multiply 2-81809781 by -511329 correct to 4 places of
decimals.
2. Simplify
£2-94375 + -625s. + -1825 of £5.
3. What sum must be insured at £3 10s per cent., on pro-
perty worth £1351, that in case of loss, the owner may
receive the value of property and premium 1
160 MATHEMATICAL EXERCISES.
4. Simplify
x x + y
\ y 2x + y\x + y x
*+2 *+3
5. Find the G.C.M. of
a? - Qx - 4 and Sx3 - 8x + 8
and the L.C.M. of
(3a2-3a&)2, 18(a3Z>2-o64), and
6. The sum of two numbers is 35, and their difference
exceeds i of the smaller by 2. Find them.
7. Find the value of
(i.) when a = £, 6 = £ = c,
(ii.) when c = a + b.
8. Find the shortest distance between 2 circles which do
not meet.
CCXLIV.
1. What sum must be paid to discharge a debt of £550 6s. 8d.
due 14 months hence, interest being at 5 per cent.?
2. Find the square root of 253009, and of -Ol26 to 4 places
of decimals.
3. A bankrupt who can only pay 7s. Id. in the £, owes a
creditor £563 10s. What will he receive ?
4. Divide
a + x by a-x to 4 terms.
5. Multiply
x + ^y-2 by lx + 3y,
without reducing to a common denominator.
MATHEMATICAL EXERCISES. 161
6. Resolve into factors
(i.) (a + *)«-(«
(ii.) z2-z-20,
(iii.) l + 27a3.
7. Simplify
8. From the ends of the base of an isosceles triangle, straight
lines are drawn perpendicular to the sides ; show that
the angles which they make with the base are each
equal to half the vertical angle.
CCXLV.
1. Reduce £ of 1 oz. 13 dvvt. to the decimal of If of 5 dwt.
15 grs.
2. A person sells £530 Three per Cent. Stock at 86, and
invests the proceeds at 3| per cent. Find alteration
in his income.
3. If 6 men and 2 boys can reap 13 acres in 2 days, and
7 men and 5 boys can reap 33 acres in 4 days, how
long will it take 2 men and 2 boys to reap 10 acres ?
4. If 3, x, 1083 are in continued proportion, find x.
5. Resolve into factors
a;2 -880; +1612; and 36^ + 27<c2- 20x- 15.
6. Find the cube root of
Sx6 + 48CZ5
162 MATHEMATICAL EXERCISES.
7. State and explain the rule for transferring a quantity
from one side of an equation to the other side.
Solve the equation
4x-7 6x+ 11 .
showing how it exemplifies your rule.
8. The 3 straight lines drawn from a point within a triangle
to its vertices, are together less than the perimeter
of the triangle, but greater than its half.
CCXLVI.
1. If a man sells 22 articles for the same money as he paid
for 36, what does he gain per cent.?
2. A man owns i of a ship worth £3484, which is insured
for 9 If per cent, of its real value ; what would he lose
in case of the ship being lost 1
3. Find the length of the edge of a cube which contains
1520 cub. ft. 1512 cub in.
4. Find the H.C.F. of
2x3-x2-a;-3 and x5 - .-c3 - 4z2 - 3z - 2.
5. Simplify
/i) (JL 5 W'-3 2
V ' \x-2+x-8j \3x-S x
(ii.)
6. Prove that
(x
x-y
1. Find the L.C.M. of
4z2 - fyz - (9y2 + z2), Qf + 4xz- (4.T8
and
MATHEMATICAL EXERCISES. 163
8. If A, B, C, D are 4 points in order on a straight line,
prove that the rectangle AC. ED is equal to the
rectangles BC . AD and AB . CD.
(Prove this also algebraically.)
CCXLVII.
1. Find the cube root of 12471027759.
2. The discount on £566 10s. for 9 months is £16 10s.
Find the rate of Interest.
3. I have £8 4s. in florins, shillings, and sixpences, and the
numbers of coins of the three kinds are respectively as
the numbers 7, 5, 3. How many of each kind have I ?
4. Find the square root of
tf x3 y2 0 xz
T + - + 2 - xy - 2 + -* •
4 y x* y2
5. Solve
6. Find the G.C.M. of
7x* - I0x3y
and 8z* - 1 3afy
7. A roll of cloth was bought at 5s. 6d. a yard ; and another
roll, 25 yards longer, at 5s. a yard ; the two together
cost £100 15s. How many yards were there in each
roll?
8. Trisect a right angle.
CCXLVTII.
1. What is the cost of carpet for a room 17| ft. long by
9| ft. broad at 4s. 9|d. per square yard ?
164 MATHEMATICAL EXERCISES.
2. The difference between the discount and interest on a
debt due 1£ year hence at 4 per cent, is 5s. AVhat is
the debt ?
3. A tradesman's prices are 12 per cent, above cost price.
If he allows £1 6s. 3d. discount on a bill of £21, what
profit does he make 1
4. Show that if x - k will divide ax2 + bx + c without
remainder, then k is a root of the equation
ax2 + bx + c = 0.
(Perform the division.)
5. Simplify
b-c c- a a-b
a?-(b- c)2 + W^~(c^a)* + c*-(a-bf
6. Solve the equations : —
(i.) *-
(ii.) 3z=10-
Sf Aj
(iii.) . - = —^—, ax + by = (x + w)2.
b + y a + x
1. Employ factors to find the result of dividing
a4 + J2C2 _ a2c2 _ a2£2 ty a2 + M - be - db.
8. Construct an isosceles triangle having the vertical angle
f of each of the base angles.
CCXLIX.
1. A prize of £2770 was divided among 3 persons in the
proportion of |, i|, and f. What were their shares 1
2. A circular pond has an area of 346| square yards. Find
the cost of fencing it round at 4s. 6d. per yard. (?r = 3y.)
3. Find the cube root of 1277289-^V
MATHEMATICAL EXERCISES. 165
4. Find the L.C.M. of
32xsy3(x - y)2, 1 8x2y(x* - f), 9xf(x* + xy + y2),
and the G.C.M. of
3z3 + 6a;2 + 2a;-ll and 4z3 + 6z2 + 2x- 12.
5. Simplify
/;\ {(a
+ a;2?/2 +
6. Solve
(i.) 2z(2a; - a - b) = (2a - b)(a - 2&),
(ii.) (a;-3)(a;-4)(.T-14) = (x-2)(a;-5)(a;-18).
7. If b is a mean proportional between a and c,
then a2+262:a :: &2+2c2:c.
8. Prove that the opposite sides of any equiangular hexagon
are parallel, and any two adjacent sides are together
equal to the two to which they are parallel.
CCL.
1. What must 5400 be multiplied by, to make it a perfect
square ?
2. If 5 men can do as much as 8 boys, how many days will
it take 32 boys to finish a piece of work, of which
15 men did a quarter in 16 days 1
3. Find the cube root of 136352311523.
4. Solve'
' x+3 ' x-2 x-7
5. Tf 25 = a + b f c, show that
.3
1 = s - a +s-b + 3(s - a)(s - b)c.
166 MATHEMATICAL EXERCISES.
6. A "man bought an equal number of bottles of two kinds
of wine, at 3s. and 4s. a bottle respectively. If he had
spent his money equally he would have had two bottles
more than he did. How many did he buy 1
1. Find the L.C.M. of
Sa^xy, 5abxzy5, and Gb3xy* - Sab^y2.
8. Prove that the sum of the squares on the sides of a
triangle is equal to half the square on the base, together
with twice the square on the straight line from the
vertex to the middle point of the base.
CCLI.
1. Three pipes empty a vessel in 6, 4, and 2 hours re-
spectively. How long will they take, all together 1
2. What is the least sum of money which contains an exact
number of sixpences, florins, half-crowns, and half-
guineas ?
3. Divide 7-143562 by 1-28143 to 4 places of decimals, by
contracted division.
4. Simplify
5. Find the square root of
6. Solve the equations : —
(i.) 3i-5£ 2a_-|
0 ^
,.. x x-a x-b
(n.) , + = 2,
' x-b x-c
(iii.)
-4° =1
MATHEMATICAL EXERCISES. 167
7. If a:b :: c:d, prove that
a -c : b - d :: JaF+~<? : JtP+~d*.
8. The straight lines joining the ends of two equal arcs of
a circle toward the same parts, are parallel.
CCLII.
1. Two pipes can fill a cistern in 25 and 30 minutes re-
spectively. If both are opened, when must the first be
turned off, so that the cistern may be just filled in
15 min.?
2. A, B, and C run a 300 yards race, and A beats B by
20 yards, and C by 34 yards. By how many yards
would B beat C in the same distance 1
3. The English parliamentary fare is Id. a mile ; in France
the fare is 6 centimes a kilometre. Compare the two
rates.
(8 kilometres ^ 5 miles. 25*2 francs = XI.)
4. Solve
x- 2 v- 4
(ii.)
x+2 y + 5
. x+ 1 y + 3
5. Find the cube root of
8r9 — 1 ^r8 4- fii-7 ^7r6 4- 3fir5 — Qr^ -4- 54-r3 — 97r2 _ 97
UAr X *Ji!j ^T >J«C/ ^^ O I <fj n t>V*C */»O T V^tv ^JlA* Al.
6. If ab + bc + ca = 1, show that
a 6 c 4aic
1 - a2 1 - 62 1 - c2 (1 — a2)(l - fc2)(l - c2)
7. Divide XI 120 between A and B, so that for every half-
crown that A gets, B may get a shilling.
168 MATHEMATICAL EXERCISES.
8. Prove from Euc. Book I., that the square on the side
subtending an acute angle of a triangle, is less than
the squares on the other two sides.
CCLIII.
1. A block of stone contains 5 cub. ft. 24 cub. in.; its
breadth and thickness are each 2 feet ; find its length.
2. A man buys 20 Railway Shares at 65, and sells them
when they have risen £7 each, buying 3J per cent.
Stock at 90. The shares paid £3 5s. dividend. Find
the change in his income.
3. If the births in a population of 30 millions are 1 in 20,
and the deaths 1 in 30 annually, find the number of
population after 5 years.
4. Simplify
1 1 2x 4s3 8z7
1 - X 1 + X ~ 1 +X? 1 + £4 1 + X3'
5. Solve
(i.) ^— + -4-= 2,
' x+a-c x+b-c
(ii.) 3* + = 42, + 4y = 27.
6. Divide 59 into three parts so that the 3 products of the
numbers taken two at a time may be in the proportion
5:6:8.
7. Factorize
x5+27f; z2-13z + 40; z2 + Ix - mx - Im.
8. Bisect a parallelogram by a straight line drawn through
a given point within it.
MATHEMATICAL EXERCISES. 169
CCL1V.
1. Multiply 98-41 by 9-68452 correct to 2'places of decimals.
2. Find the value of 5 cwt. 3 qr. 16 Ib. 12 oz. at 5s. 6d. a
quarter.
3. If 5 horses are worth 8 cows, and 10 cows cost £50, what
are 9 horses worth ?
4. A does £ of a piece of work in 8 days, and then B
comes and works with him for 2 days. Then A
finishes the work in 4 days. How long would B
have taken to do the whole ?
5. Solve
x y x z y z
6. Find the G.C.M. of
2zs-7o;2-15z+27 and 2z2- 13z+ 18.
7. If a = — U h = ^— c^-L-.t d = ^—
find the relation between a and e.
8. Find the point in the base of a triangle from which equal
perpendiculars may be drawn to the sides.
CCLV.
1. Which is the better investment, 6 per cents, at 113| or
5 per cents, at 98 ?
2. Simplify
170 MATHEMATICAL EXERCISES.
3. Find the cost of papering a room 14| feet long, 12| ft.
broad, and 8\ feet high, at 2|d. a square foot.
4. Solve
(i.) 7z + 32=ll, 3«-4y+2^« -12,
5. Find the G.C.M. of
a5 + 6a* -i- lla3 + 5a2 - 3a - 2
and ft6 + 3a5 - 6a3 - 5a2 - 3a - 2.
6. If 2 cwt. 1 qr. 3 Ib. of an article is bought for 15s. and
17 Ib. sold for Is. 3d; what profit is made per cent.?
7. If x be a positive quantity show that
} 0
X+l
is also positive.
8. Trisect a given straight line.
CCLVI.
1. Find all the prime factors of 2697, 3441, and 1271, and
hence write down the number that will divide all
without remainder.
2. One clock strikes 4 whilst another strikes 3. They both
begin striking a certain hour together, and the last
stroke of one is simultaneous with the last but 2 of the
other. What o'clock is it ?
3. Find the value of 3 cwt. 16 Ib. 10 oz. at £3 16s. 4d. per
cwt.
4. Find the G.C.M. and L.C.M. of
z10-l and z12-!.
5. Solve
(i.)
(ii.)
MATHEMATICAL EXERCISES. 171
6. Simplify
a2 - Sab + 2&2 a2 + ab - 2Qb* a? - 4ab + 4&2
a3 - 1 3«62 - 1 263 aA - 21 aft2 + 2063 ' a3 + 2a26 - 5ab2 - 6ft3'
7. In the year 1875 a man on his birthday was just as
many months old as half the number denoting the year
in which he was born. Find in what year he was born.
8. Prove that the three straight lines bisecting the angles
of a triangle meet in a point.
CCLVIL
1. Simplify
129 2 , 3 _ 4
-- ~
2. Divide 18596-508 by 98760 and -8l by -027.
3. A starts at 4 A.M. from X to walk to Y, a distance of
50 miles. B starts from X at 5 A.M. and passing A at
the 20th milestone reaches Y at 5 P.M. When will
A arrive ?
4. Find the weight of a cubic foot of water in oz. Avoir-
dupois; given that a gallon weighs 146'17 oz. Troy;
and a pint contains 34'66 cub. in.
5. Simplify
d*(a - b)(b -c) + V(a - d)(c - d) .
c2(a - b)(a -d) + a?(b - c)(c - d) '
and test your result by putting
a = 4, ft = 3, c = 2, d = l.
6. Find the square root of
7. After paying income tax at 8d. in the £, a man has
£406 9s. 8d. left. What is his income ?
172 MATHEMATICAL EXERCISES.
8. What value of I will make ' equal to —
I - c ^ I- a
CCLVIII.
1. Bring 8s. 4|d. to the decimal of Is. 3d.
2. A tap A will fill a cistern in 1 J hours ; 2 others, B and
C, will empty it in 2 hrs. and 3 hrs. respectively.
(i.) How soon will it be filled if A and B are both
open? (ii.) How soon will it be emptied if A, B, and
C are all open ?
3. Find the side of a square field whose area is 1 5 acres.
4. Simplify
2x + y x-y 2x
~
a2 - f ~ 2x2 + 3xy + y*
5. Find the cube root of
6. At what time between 9.30 and 10 o'clock is the angle
between the two hands of a clock a right angle ?
7. Solve
/. , 3x + 5 6x + 5 _ 2
-
-9
z-4
8. Two triangles A CD, BCD are on the same base and the
same side of it, and AD, BC bisect each other. Prove
that AB = CD. What do you know about the figure
ABCD1
CCLIX.
1. A pays £9 3s. 4d. more rates than B, whose income is
the same; they are rated at 2s. and Is. 4d. in the <£.
respectively. Find their income.
MATHEMATICAL EXEECISES. 173
2. A, B, C and D enter into partnership. A and B together
contribute £1390; B and (7, £1590; C and D, £1810;
A and.D, £1610 ; A and (7, £1500 ; how must a profit
of £1152 be shared?
3. Find the cube root of 40, correct to 4 places.
4. Simplify
(x - y)(x - z} + (y - z}(y - x) + (2-x)(z-yY
5. Show that if x + y + z - 0, then
_
2x2 + yz 2y2 + xz 2z2 + yx
6. How many pounds of tea at Is. 6d. and 2s. 6d. a Ib.
must be taken to make a box of 200 Ibs. worth
altogether £18?
7. Find the H.C.F. and L.C.M. of
and 9z* - 3xsy - l&f + 3xy3 - y4.
8. If a quadrilateral is bisected by one diagonal, the other
diagonal is bisected by the first.
CCLX.
1. Find the square of the number whose cube is 469097433.
2. Express "015 of a cubic yard in cubic inches and the
decimal of an inch.
3. A man buys goods at £35 a ton ; at what price per Ib.
must he sell them to gain 10 per cent, on his outlay?
4. Solve
(i.) 2z-£(a;H-27) = 16,
V — 2
(•• \ o •*/•»•
u.) z = § + __|.
174 MATHEMATICAL EXERCISES.
5. Divide a yard into two parts, so that half of one part with
22 inches may be double the other part.
6. Find the square root of
7 Tf ai _ a2 _ as
i. ~ JT ~ T" '
?1 »2 *3
prove that each of these fractions is equal to
8. Divide a circle into 2 parts so that the angle contained
in one segment, shall be twice that contained in the
other.
ANSWERS.
I.
1. 37. 2. £424514 15s. lOd.
3. 100 ac. 2 ro. 37 po. 28f sq. yds. ; or, 100 ac. 2 ro. 37 po.
28 sq. yds. 6 sq. ft. 108 sq. in. 4. 13080 grains.
5. 3^. 6. I 7. -a. 8. a + 3b-d.
II.
1. 19152. 2. £17 17s. 10d.— and 22 farthings over.
3. 6178 tons 17 cwt. 2 qr. 24 Ibs. 4. £2 5s.
5. 7||. 6. -4. 7.
8.
III.
1. 87. 2. £15851. 3. 3 ro. 5 po. 27yds. 4ft. 7 2 in.
4. 23 mi. 1 fur. 18 po. 4 yds. 2 ft. 3 in. 5. 27.
6. (i.)l; (ii.) -31. 7. 3x + Sa. 8. llz+10*.
IV.
1. 7560. 2. £368 12s. 9£d.
3. 620 tons 18 cwt. 1 qr. 21 Ibs. 4. (i.) £f ; (ii.)
5.2^-5. 6. (i.) 6 ; (ii.) 2. 7. a3-
8. z5 + s4 + 4z + 4.
175
176 MATHEMATICAL EXERCISES.
V.
1. 37. 2. £648467 16s. lid. 3. 2 qrs.
4. 4738 qrs. 2 bush. 2 pks. 1 gall. 3 qts. 1 pt. 5. 6f|£.
6. 4. 7. 0. 8. -2a+26-2c+2cZ; -4a-4c.
VI.
1. 22176. 2. £133s. ll^d.
3. 20 acres 1 ro. 1 po. 23 sq. yds. 108 sq. in. 4. 65 men.
5. _s_«_ 6. -16. 7. -4a3+13a26-6oi3-2R
8. tf-tf-k+l.
VII.
1. 493. 2. £482780 5s. 6|d.
3. 5 qrs. 3 bush. 2 pks. 4. 212096 drams.
5. 11^. 6. -1. 7. 7a-35&.
8. 56 + 2c; 4a-7c.
VIII.
1. 1260. 2. £250 13s. 5d.
3. 63 Ib. 8 oz. 2 dwt. 18 grs. 4. V/; 5T5f- 5. 22.
6. -^. 7.
8.
IX.
1. 43. 2. £87380 4s. 9|d.
3. 1 Ib. 5 oz. 17 dwt. 9 grs. Troy weight.
4. 25 tons 2 qrs. 27 Ibs. 6 oz. 5. 5||. 6. 2 ; 1|.
7. 6x. 8.
X.
1. 125. 2. £6694 8s. Hd.
3. 256,324,441,2,4,6,11. 4. 63283 sq. yds.
5. 3|. 6. 92. 7. 7a-6. 8.
ANSWERS. 177
XI.
1. 2640. 2. £300 16s.
3. 17 Ibs. 5 oz. 5 drs. 0 scr. 5 grs. 4. £6 Is. 4d.
5. 3^. 6. 0. 7.
8.
XII.
1. 677. 2. £34180 5s.
3. 3 tons 15 cwt. 1 qr. 16 Ibs.
4. 27 tons 4 cwt. 2 qrs. 16 Ibs. 5. 1. 6. 0.
7. 3m. 8. -
XIII.
1. 63072. 2. 80 times.
3. 8 tons 9 cwt. 2 qrs. 24 Ibs.; 1 ton 13 cwt. 3 qrs. 21f Ibs.
A 5)99 . AK 9 K 1_JL_ ft 07
* ~22" > *J'2T' °« *T5^' O. 27.
7. « + 6 + c + ^. 8. a3 - a3Z> + 2a2 + a2i - a&3 - 2a&.
XIV.
1. 379. 2. £467405 16s. 4d.
3. £21 19s. 3fd. 4. 82703| sq. yds. 5.
6. 48. 7. «. 8. x
XV.
1. 2520. 2. £20 7s. IJd. 3. 7. 4. £90.
5. 10^. 6. 223. 7. i?-v*x-5vxy.
8. a2x2
XVI.
1. 881. 2. £16293 6s. 8d. 3. 9090.
1 4 Ibs. 1 oz. 2 dwts. 5. 41J. 6. 2272.
7. 0. 8. 2z2 - 20z + 20.
178 MATHEMATICAL EXERCISES.
XVII.
1. 40. 2. £16235 2s. 3. 261 hhd. 7 galls.
4. 112112 oz. 5. 5#£. 6. 24 = 24.
7. -4fl-26 + e. 8. 2c2 + 3c-6/.
xvni.
1. 2788. 2. £2 17s. 6d.
3. 243 acres 3 ro. 24 po. 27| sq. yds. 4. 120 weeks.
5. 7 sax. e. is. 7. -^ + 4«V + 3a;y3-3xy2-y4.
8. a3 + 63; 6 -a.
XIX.
1. 1849. 2. £22240 19s. 4|d.
3. 1 dr. 0 scr. 1 2|£ grs. 4. 1 3 1 Ibs. 7 oz. 1 8 dwts. 20 grs.
5. ±. 6. 6. 7.
8. 1 + 2z + 3xz + 4s3 + 5s*.
XX.
1. 52948. 2. £13 19s.
3. 112 tons 10 cwt. 3 qrs. 21 Ibs.
5. 1¥%V 6. 189.
4a-79&-a;+2y. 8. 3o*-.
XXI.
1. 571. 2. £3791 11s. 3|d.
3. 14 days 7 hrs. ll.min. 17 sees. 4. 431700 oz.
5. 11. 6. 33. 7.
8. 3i
XXII.
1. 1008. 2. 5125 mi. 3 fur.
3. 1 ac. 1 ro. 29 po. 3 sq. yds. 5 sq. ft. 65T439T sq. in.
lVfcV#> Rltfitt 6.51.
7. »-y + y*j 5 -a. 8.
ANSWERS. 179
XXIII.
1. $ ; f 2. £445 9s. 4d.
3. 4 st. 3 Ibs. 3 oz. 3| drs. ; w 4 st. 3i Ibs.
4. 5 sq. yds. 7 sq. ft. 105 sq. in.
5. M, If, |, f; 31'303228-2A- 6. 32a2&-3a&+15&2.
7. 480 = 480. 8. 6 + 2c + 3A
XXIV.
1. 77. 2. £268711s. 3. 188 days; 184 days.
4. 762 bush. 3 pks. 5. 36 ; 2f .
6. - Zx"2 + 4xy - 6y2 ; x + 4. 7.8. 8. 26 + 2c.
XXV.
1. 2571. 2. £4251 10s. & 35 times.
4. 15 qrs. 3 bush. 3 pks. 5. 2££; •&,
6. 3a2 - ab + Qb'2. Take the sum of the last 2 from the sum
of the first 3. 7. 7a2 + Qxy - 2yz + 14.
8. -7.
XXVI.
1. 15120. 2. 2 Ibs. 9 oz. 15 dwts. 3. 40 Ibs..
4. 3 fur. 36 po. 1 ft. 5. 11^-; |f-
6. 4o;4-h5az3-2rt2x2 + 2a3a; + 20. 7. x.
8. 340 = 340.
XXVII.
1. 57. 2. £112 9s. 10|d. 3. 100 times.
4. 2551443 sees. 5. ^. 6. 2z2 - 3f - 9z. Take
the sum of the last two from the sum of the first two.
7. -2 + 7x*y+Sxy*. 8. 74779.
180 MATHEMATICAL EXERCISES.
XXVIII.
1. 421. 2. 1464 hrs. 3. 56 lea. 0 mi. 5 fur. 25 po.
4. f; ^ K If >ft and £1 >T£-
6. 3z2 + 2|rty+2Jy2. 7.
8. lOcz-f 4<fo-5e.
XXIX.
1. 19. 2. £1299 19s. 3|d. 3. 6 sq.ft. 132 sq. in.
4. 81 tons 12 cwt. 3 qrs. 4 Ibs. 5 oz. 6 drs.;
81 tons 5 cwt. 0 qrs. 12 Ibs. 15 oz. 7 drs. 5. yf.
6. bx - 2cx. 7. b2c - 3d2e +/2 ; a;2 - x.
8. 54.
XXX.
1. 780. 2. 2 qrs. 24 Ibs.; 2 cwt. 2 qrs.
3. 4d. 4. 1555 mi. 4 fur. 10 po. 5. 2£ ; ^.
6. Ila3x + 14a«3 + 40. 7. 2ibx - 6cx + Sdx - 2e + 2f.
8.
XXXI.
1.1350. 2. £17251 13s. 4d. 3. 91668793 miles.
4. 3 ac. 7 sq. ch. 5250 Iks.; 7 ac. 2 sq. ch. 5. 633J.
6. -2-4z. 7. 2z3-9a2z-a3 + 4.
8. - 252 = - 252.
XXXII.
1. 12600. 2. £254 9s. 4^d.
3. 3257 cub. yds. 6 cub, ft. 1597 cub. in. 4. |; ^.
5. 6; 1TV 6. 28a2 + 3ai
7. Sum = 5. 8. -4?nw2-27i3.
ANSWERS. 181
XXXIII.
1. 72. 2. £799 19s. 2d. 3. 8 qrs. 4 bush. 3 pks.
4. 1 mi. 21 po. 4| yds. ; 1 mi. 1 fur. 11 po. 2 ft.
5. 1£; 9f. 6. 4ax + bx-9. 7. 2c* + 7d5 + c4d5.
8. 29-36 + 1 + 8 = 2.
XXXIV.
1. £1213 6s. 8d. 2. 12 po. 14 yds. 3 ft. 21£|f in.
3. 372776 oz. 4. £12 10s.
5. (i.)169; (ii.) 512; (iii.) 11. 6. -51.
7. 8a4 + 4«3 + 9a2 + 4a + l. 8.
XXXV.
1. 4500. 2. £46 18s.
3. 499 tons 18 cwt. 2 qrs. 14 Ibs. 4. £10 19s. 2£d.
5. Divide it into 7 parts, and take 5 of them. 6. If.
7. 6(a + 6)-2(z + y). 8. z*-
XXXVI.
1.12. 2. £3220 8s. 4d. 3. 2ro.4po.16isq.yds.
4. 373 tons 17 cwt. 3 qrs. 24 Ibs. 8 oz. 5. 16£.
6. 0; 2. 7. 3a-66 + 6c-6dL 8. c-d; a-i.
XXXVII.
1. 39270. 2. £5 10s. 6Jd. 3. £42.
4- 2ffff; ||; f 5. 10. 6. 56.
7. - 5Z3 + 1 6z2y-l Szy2 +3y3. 8. a3 + 6s + c3 - 3abc.
XXXVIII.
1. 306. 2. 14 Ibs. 1 oz. 3£4 drs. 3. 389 gallons.
4. Is. 6d. 5. 3^; 5£. 6. 77.
7. ^2 + 3^2 + 4. 8. y8-!.
1S2 MATHEMATICAL EXERCISES.
XXXIX.
1. 37. 2. 3 days 2 hrs. 53 min. 20 sec.
3. 13 Ibs. 8 oz. 16 dwts. 21 grs. 4. 150^y; 11.
5. 10. 6. 5b + c + 5d; -±b
7. 10a2 + 5a&-762. 8. 120 = 120.
XL.
1- T9i>; T&F- 2. 25 po. 2yds. 10 in.
3. 3-6997; 315-88568. 4. 6££; If.
5. 202£ yds. 6. I. 7. if a
8. 10z-7£; |.
XLI
1.11. 2. ^;3/T. 3.6792. 4. 3f£; 584.
5. 360-248; 24-9. 6. x4 + a2xa - 2a3x + 6a4.
7. a2-2a&-3&2. 8. 40 -z.
XLII.
1. 35280. 2. 15 tons 13 cwt. 3 qrs. 15 Ibs. 13 oz.
3. £4549 19s. 5fd. 4. 65f | ; 546.. 5. f
6. 2 + 20a; + 2z2. 7. z2 + 3ax + 3a2.
8.
XLIII.
1. 7956. 2. 190yds. 2 ft. 1 in.
3. £224 7s. 10|d. 4. 2||; 8J. 5.
6. -^ + 3H + Z2-/3. 1. y*-tfy + xf-f.
8. (a-6)(a4 + a3J + a262 + a&3 + &4); (a - 46)(a - 26) ;
ANSWERS. 183
XLIII. (a.)
1. -028 ; -27. 2. 42524 hhds. 57 gals. 3 qts. 1 pt. 2 gills.
3. 664-9244; 7-1881. 4. 8^; lf£.
5. £6 16s. 8d. 6. -31. 7.x4 -a*.
8. (i.)lf; (ii.) -14.
XLIV.
1. 26. 2. yfo; TV 3. 42256 oz.
4. 2£f; 88|. 5. 1572-5723; 11-4.
6. 3x3-3x*y + Gxy*-y3. 7. a2 + 2a& + 262.
8. *+l.
XLV.
1. 74. 2. £217 3s. 9|d. 3. 6; 2^.
4. TV 5. £457579 7s. 4^-d.
6. a3 -i- 2a'2c - ai2 + ac2. 7. a3 + 3a26 - 2ai2 + 3R
8. (m
XLVI.
1. 1111880. 2. 62857296 sq. inches.
3. £1835 2s. 2d. 4. 17|; |. 5.
6. x4-xy-i-y4-a;2 + ^2-^3.
7. z4 - ax3 - ea2^2 - 7a3z + 49a4 ; z2 - 5ax + 7a2.
8. a2 + 62 no factors, a2 - 52 = (a + J)(a - 6).
XLVII.
1. 3-625; -197530864 2. 1 ton 7 cwt. 3 Ibs. 3 oz. 9 drs.
3. 292-1468; -289918. 4.
5. 6 « ; ^ ; 2|£. 6. 0.
7. 27z3 + 8t/3 + £-lBxyz. 8. (i.) 7; (ii.; -4|.
184 MATHEMATICAL EXERCISES.
XL VIII.
1. 21. 2. (i.) 15f ; (ii'.) f
3. 162 tons 17 cwt. 3 qrs. 25 Ibs. 9 oz.
4. 1||; 7-17362 5. 73-547045; 4790.
6. 6z2 + 2y2; Qa^-lOxy + So^. 7. 3z2 -
8. 1^.
XLIX.
1. (i.) A; (»-)f^- 2. £54 15s. 6d.
3. (i.) 1A; ("•) A- 4. (i.) 7s. 3Jd.; (ii.) 12s. 4d.
5. -5; 1-5; -25; 2'25; -375; 14-263 ; 'Oil.
6. 8f+12fz + 6yz* + z*. 7. x3 - 3x2 + 3x - 6.
8. (3a6 + l)(3a&-l); (aj-6)(«-l);
(x - 2)(« + 2)(«2 - 2a; + 4)(x2 + 2x + 4).
L.
1. 4 sq. ft. 42 1» sq. in. 2. £488 Is. Ifd
3ii93.il! d 94- ^119
• 1T^IT^ *TT- *• ^4- °* li^'
6. 61tt2 + 38a& + 43J2-40a^ + 25^2. 7. 6a2-
8.
LI.
1. -72 ; 2-345. 2. 72 times £3 13s. 6d. remainder.
3. 88-1744; 184-2872108. 4. 17T8¥\; ffi-
5. 152. 6. -2. 7. a2- — +^-62.
c c2
8. (i.) 4; (ii.) 2|.
ANSWERS. 185
LIT.
1. 2003. 2. 181-759112203.
3. 125 ac. 1 ro. 25 po. 7 yds. 4 ft. 108 in.
4. 5^1; 2. 5. 34-114.
6. & + & + &; a + ^-b + ^c.
7. a4 + (a + J)x3 + (a + b + a&)z2 + (a2 + b*)x + ab.
8. a2
LIII.
1. 2223. 2. £1213 6s. 8d. 3. 4.
4 302 -20 El. 1.3. 3.16.8.1
*• 1T5T * TT- °« 2 » 4 > t * 3T > "5T J ¥ * 77'
6. 2xy-2y*. 7. x^-yyy + i. 8. 8 = 8.
LIV.
1. 9216. 2. 4 Ibs. 8 oz.
3. £158 4s. 5d.; £79 2s. 2|d. 4. 18.
g /2232479T _\ 11
W* \3 8561013 / Tff'
6. (i.) b* - ¥ + 2b5 - 4&6 + 367 - 2ft8 + #»; (ii.) &2 - 64
7. a-6. 8. x-6.
LV.
1. 17-001857142; 1-25. 2. 189 tons 19 cwt.
1 qr. 19 Ib. 3. 201-50499 ; 496-5566489.
4. 9fi; l^V 5. £2 10s. 3|d. 6. -30.
7, a3 - (a + b + c)x2 + (ab + bc + ca)x - abc.
8. (i.) 2; (ii.) 63.
LVI.
1 O _ 29 . 4
*• * TlTTr^TnT » TT-
3. 19320 grains. 4. 56f ; 180.
5. 154-26536; -0000579. 6. a3 + 53 + c3 - 3abc.
7. a2 + ab + ac + be. 8. a-1 and a + 1.
186 MATHEMATICAL EXERCISES.
LVIL
1. 9504. 2. 9300 TTt. 3. £1625 Os. Od.
4. (i.) 3»|; (ii.) 1^; (iii.) 1451282f
5. 2-^; f. 6. 7z2-2z2/2-2a%-22/3.
7. 25. 8.
LVIII.
1. 24.7.11; 26.3.11; 22.3*.7; 26.32.5.
2. 2 Ibs. 6 oz. 16 dwts. 16 grs. 3. £21 6s. 7fd.
4. llff. 5. £18 18s. Od. 6. z5 - 15z2 - 36.
7. oJ-R 8. 27/2-14/0-4<72.
LIX.
1. (i.) "02; (ii.) -Oi3i 2. 135 tons 4 cwt. 3 qrs.14 Ibs.
3. 72-678125; 2019-11556. 4. llfff; 5^T.
5. 6. 4&2-
7.
8. (i.) 1; (ii.)
66
LX.
1. £8718 15s. Od. 2. |f£l£; -657142.
3. 9 pieces. 4. ££ff. 5. 31-64625 ; -001.
6. a*-l. 7. 3a2(a-26 + 3a262); (x-y)(m-y). 8.
LXI.
1. 17-4; -006; -5; -428571. 2. 5oz. 1 dr. 2scr. 8|grs.
3. 10 feet. 4. 3£. 5. 1722 sixpences. 6.
7. 8a2Jc(2a2&2 + 3C4 - 5abc5) ; (3x - 2y)(2x - 3y).
8. Coat, £2 12s. 6d., Trousers, £1 Is. Od., Waistcoat, 14s.
ANSWERS. 187
LXII.
1. §; 100^; 1 ; f 2. 4 mi. 29 po. 1 ft. 6 in.
3. 765f miles. 4. 7ft. 5. '0003. 6.
7. (x-3)(x+l); (*+ l)(z- I)(z2 + 1);
(z+3z+l(z2-a;+l). 8. a
LXIII.
1. 9961. 2. 946 weeks 1 day 22 hours.
3. A- 4. ^3AV 5. 200 ounces.
6. a = l. 7. a + 6 + 2(c + d).
8. (i.) a9; (ii.) a"+1.
LXIV.
1. 68 cub. yds. 26 cub. ft. 1064 cub. in.
2. -013; -375; -372. 3. £518 6s. l£d.
4. 5^. 5. -092078038774.
6. z2 + 4y2 + 9z2 - ±xy +6xz-l 2yz.
a + (4-/)6-3c-8rf. ' 8. 2 + = .
LXV.
1. 26000. 2. 19fft. 3. 15 persons.
4. £10 4s. 2d. 5. 1742-914533; 1642-9145329.
6. a2
7. (x-
(x -
(2x - By + z). 8. na.
LXVI.
1. 155 mi. 7 fur. 1 po. 2 yds. 2 ft. 9 in
2. 10 tons 14 cwt. 1 qr. 4 Ibs. 3. £145 11s. O^
4. 43^. 5. 729. 6. (s* - 7)(s« - 2) ;
(x-k)(x + l). 7. a* + a
8. m5 + 3m*nz - m%3 + 2mn4 - 7i5 ; z3
188 MATHEMATICAL EXERCISES.
LXVII.
1. 34. 11 . 22. 2. 990 tons 7 cwt. 2 qrs. 0 Ibs.
3. £6 18s. 9|d. 4. 11%$%; 4£.
5. 398 Ibs. ( = 3 cwt. 2 qrs. 6 Ibs.) 6. 114£.
7. -a&2. 8. 144a2-169&2.
LXVIII.
1. 1 ac. 3 ro. 33 po. 4 sq. yds. 2. £2340 12s. 4;
3. 8i£; ||. 4. 1. 5. £45 10s. Od.
7. (m - n)(m + n)(m2 + mn + n2)(m2 - mn + n2) ;
(m - n)(m + n)(m2 + w2) ; (m - w)(m2 + mn + n2).
8. z=H.
LXIX.
1. 432432. 2. 85296 oz. 3. T7¥; 2|
4. £1 6s. 9d. 5. f£. 6. 5a - 9x.
7. 4z2 - 3az + a- ; 8^-4 = 2. 8.
LXX.
1. 278. 2. £381 16s. 5d. 3. Iff;
4. 3534 grains. 5. £245 18s. ll£d.
6n& _ Q/r2/j. 7 &2 -t- 4
. «(/ «7lt ^/. | . A< i T.
8. (a - 3x)(a + x) ; (a + 2x)(a + x); (a + b + c)(a + b-c).
LXXL
1. 49. 2. £58377 2s. Od. 3. lf|.
4. 4; |§. 5. 7899-518475.
6. 4a3 + 4a2 - a + Gab - 13ab2 - 6i3 + 762 - 1 ; 1 x 4 x 8 - 32.
7. a + 6-c + d. 8. (A;-0(« + y); («2- 3)(a-3)(« + 3).
ANSWERS. 189
LXXII
1. .£20 12s. 2fd. 2. £21579 15s. Od.
3. 10^; 4TV 4. & 5. £1 2s. 9d.
6. 0. 7. c-d. 8. 9z2- 647/2.
LXXIIL
1. 202125. 2. 10340 yards. 3. £19 2s. 4|d.
4. 4^. 5. Iff ; f ' 6- *-aW-a*x*-a*x.
7. 2&2z2 + 24CX5 - 4Z6. 8. y%-3.
LXXIV.
1. 31. 2. £4414 19s. Ofd. 3. V/T J 45-
4. if. 5. £11 14s.
6. 3z2 - 4a;y + 8a» + 8yz - ±f - 302.
7. a? + x-y + vy* + y* ; a.
8. (win + 7)(wn - 4) ; (x-a)(x + a)(x* + a2).
LXXV.
1. 16 mi. 5 fur. 7 po. 2. £2045 4s. 7|d.
3. 10||;2TV 4. T9A- 5. 3^1bs.
6. 2(c - 6). 7. (2 + 3s)(l + 5z) ; (x - 3y)(x - y) ;
8. «=6.
LXXVI.
1. 3x5x7; 24x7; 53; 22 x 5 x 7.
2. 4 cub. yds. 5 cub. ft. 192 cub. in. 3. £21 3s. 2*d.
4. *; U; 7|; J. 5. £1450.
6. (i.) &2z2 + 6frfx2 - 6^2x2 -; (ii.) Put b = c = d = x = 1.
7. 3a;+8?/-5. 8. 12.
190 MATHEMATICAL EXERCISES.
LXXVII.
1. 888. 2. 6 ac. 1 sq. ch. 247 sq. Iks.
3. £159 18s. l|d.
4. (i.) £&; (»•) fs', (i«-) 14512821 5. £.
6. a2 + 2a& + Z>2. 7. 1 - x*y*. S. xxy, x.y, xy.
LXXVIII.
1. 34. 23.112.29. 2.525016. 3. £4 4s. 8|d.
4 14^; |§. 5. £60. 6. 8; 18.
8. 9Jfea - 16/2. They are to be multiplied together.
LXXIX.
1. 20493. 2. £12 18s. l|d. 3. l-ffc-, 15.
4. f 5. £5 3s. 0^. 6. a4 - b*.
7. l-3x+2a;2-^; - 100309-=- 121 = -829.
8. (x + Sa)(x - a) ; (a; + a)(a; + &);
(a-2/)(o; + y)(a;2 +
LXXX.
1. 211. 3; 38; G.C.M. 3. 2. 252 mi. 5 fur. 20 po.
3. £18 8s. lO^d. 4. |- greatest, £ least.
5. 9| hours. 6. a - 2b + 3c. 7.
8. 8a;y.
LXXXI.
1. 183^1 packets. 2. £184 9s. 9|d.
^ 21489.
¥T»
5. £286 16s. 4d. 6. x* - x*z - y* + yzz.
7.
8. a + J-c.
ANSWERS. 191
LXXXII.
1. 2184. 2. -£-s\ 37-598206349
3. 142686 sq. ft. 4. 4||; 20£.
5. -0583; -0083; "00083. 6.
7. z2 + te + a. 8. z3 -
LXXXIII.
1. £16 11s. ll|d. 2. £2280 19s. 4d.
3. H^; |4. 4 f. 5. 19s.
6. 3a- + ^C-. 7.
(a + b + c - d)(a + b-c + d); (x- y)(x2
8. x=7.
LXXXIV.
1. 137-9451. 2. £42081. 3. 4££
1 1 fur. 4 po. 3 yds. 5. £21 13s. 4d. 6. a + b-c.
7. (x -4)(aj+l); (x-l)(x+l)(x2+l);
z+l). . 8. x=2.
LXXXV.
1. 5 . IP ; 2W. 7. 2. 4 gall. 1 pt. 4 fl. 5. 1 fl. 5.
3. £16 11s. 8^d. 4. 30f. 5. 13 da. 6 hrs.
6. ic6 - ^2-a;y + f • 3 x 1 x 5 x 3 = 45. 7. 62 - a2.
8. &5. Any even number.
LXXXVI.
1. 17780. 2. j. 3. £8 7s. 6d.
4. 6^; 91i§. 5. |ft
6. a4 + a%2 + M-a2-aft-62. 7. a2-a6 + 62. 8.
192 MATHEMATICAL EXERCISES.
LXXXVIL
1. 24 . 32 . 5 . 7. 2. 12356 Ibs; 3 oz. 6 dwts. 4 grs.
3. £94 Os. l|d. 4. 4f ; 3. 5. £10560.
6. 5. 7. f z2 + &xy + \f. 8. (m - nf -(p- q)*.
LXXXVIII.
1. 563 days. 2. £15 4s. Od. 3. 19775^-.
4. 1. 5. £231 8s. Od. 6. 4s3 - 5x2 + 6x - 5.
7. 4<Z/ - 2d2 - 2/2 - 2de + 2ef. 8. 5.x2 + 2xy ; 20 - z.
LXXXIX.
1. 2, 3, 6. 2. 55 qrs. 7 bush. 3 pks. 3 qts.
3. £884 9s. 4R 4. 7^; TVr- 5. 3150.
6. -V-- 7.
8. W -
XC.
1. 5. 2. £3337 13s. 3fd. 3. 4§f ; l-\.
4. /^. 5. £45 2s. ll^d.
6. a6 + 2a5b - a4J2 - 4a363 - a26* + 2a&5 + 66. 7. 3o; + 1 5y.
8. (i.) (2z + 3y)(2a; - 3y) ; (ii) (a - 3J)(a + 2J) ;
(iii.) 3y(a;
XCI.
1. 7 Ibs. 10 oz. 18 dwts. 21 grs. 2. £471 11s. 3d.
a 823 .21 A 41
°T~2 > TT- *• IT-
5. 3| fields of the same size. 6. llx + y.
7. (i.) (x-5)(*-l);
(ii. ) (ax - by)(ax + Z>y)(a2x2 + abxy + b*y-)(a?x2 - abxy + Z>2?/2);
(iii.) (mntl)(ni»-'l); (iv.) (&+12)(&-8).
8. (i.)x = 8; (ii.)^ = 6.
ANSWERS.
193
1.
3.
5.
7.
1.
3.
6.
8.
1.
3.
6.
8.
1.
4.
7.
54*1 cubic feet.
£4414 19s. 0|d.
(i.) -07504; (ii.) 490000.
XCII.
2. -1619047; 5-285714"; 1-6.
6. 9w2-16n2.
8. (i.) 5; (H.) 3£.
306495278.
£30 13s. 6d.
(8x0 = ) 0.
XCIIL
2. 363 Ibs. 11 oz. 8 dwt. 6 grs.
4.
7.
XCIV.
14 days 7 hrs. 1 1 min. 17 sees.
115 . 39
TIT-
x =
_ e
T
300.
981
7.
xcv.
2. ^; -5439566.
5. 12500.
8. 2 = 2.
5. £105.
6 + 7-1=17.
2. £839 14s. 4£
K 3fi03fiO
°- -JOUODU.
; x(x - 2a)(x - a).
3. £518 6s. l|d.
6. 4a(8c-66).
XCVI.
1. £10 12s. lid. 2. £
3. 227178 half-crowns; 358 pints. 4. 3s. 7£d.
5. 44-52494221.
6. 14x4-3o^ + x2 + a;-45; -2x16= -32.
7. a + b and a-b. 8.
XCVII.
1. £9 12s. 7£d. 2. T^^^D 3. £12 Os. 3* *d.
4. -375. 5. 7360. 6. x -17.
7. o^-zy + y2; (z* - afy2 + y*}(xi - xy + y2)(a;2 + zy + y2).
8.
194 MATHEMATICAL EXERCISES.
XCVIII.
1. £8 8s. Ifd. 2. f^V; T^O ; 5 Ibs. 4 oz.
3. £9 7s. 5d. 4. f- ; -285714. 5. '177.
6. x = 3. 7. 2x-ty; 576-r(-96) = -6.
8. oW.
XCIX.
1. -02875. 2. 933 tons 9 cwt. 2 qrs. 1 Ib.
3. 4739^7T revolutions. 4. 6^|. 5. £|.
6.
7. (z-10?/)(a;~?/); (a + &)(a2 - ab + &2) ; (x - y + z)(x - y - z).
8. 60.
C.
1. £12 15s. Od. 2. 2-0272; 1^.
3. £4147 19s. Ofd. 4. 14^;-2|f.
5. -0619628; 5. 6. 64(a + bf- 8I(c + df.
7. 7z2-2? + i; a;24.6?_l 8.^ = 2.
y y2 y f
ci.
1. 164 yards 2}| feet. 2. -14285? ; -523809; -00971
3. 3 yds. 3 qrs.; 29 Ibs. 13 oz. 13ff drs.
4. -re ; IT ; I ; A and HI e<iual- 1 A-
5. -0147; 7-137. 6. y^ + H2 + Z4.
7. _ia + -8c+-6e. 8. a: = 5.
franpQ nparlv 817 . 197 . 107
iiancs, nearly. -- ~~ "
GIL
.
^(5-575-^ ~s^~s > TTO"-
3. 4 tons 3 cwt. 2 qrs. 11 Ibs. 4. £3 4s. 2Jd.
5. 692-8166. 6. 2a4 - 7a?b - 1 1 ab3 - 2064 ; a°; a20.
7. »2 + 8a^-7y2.
8. 74; (a + 2& + c)(ft2 + 462 + c2 - 2ab - ac - 2Jc).
ANSWEES. 195
GUI.
1. £20 19s. l|d. 2. i£r3-. 3. £86 19s. 6d.
4. 894-6448 Ibs. ; 10 mi. 4 fur. 4 po. 1 yd. 2 ft. 3^ in.
5- ™fr or '93945109.... 6. x = 7.
7. tf + y3; a*. 8. 30a4Z>4.
CIV.
1. Is. lOd. 2. 84 6306,
3. 3 cub. yds. 15 cub. ft. 864 cub. in. 4.
5. 646-738 ; 7'6. 6. 42x2+ 21Gxy + 30«/
7. (i.) (3as-y)(as-3y); (ii.) (3x - y)(x - y) ;
(iii.) (x-3y)(x-y); (iv.) 3«-y.
8. a2-62.
CV.
1PQ RQ QJ <> 657 . 1633
. AO os. ya.
1000000 > T-
3. £524 13s. 9d. 4. -821875.
5. 286-5; £81. 6. £=13.
7. x6 - 3s5 + 9x* - 27s3 + 81a2 - 243« + 729; «n.
8. 36.
CVI.
1. HI; 5^; 51f£f. 2. 1745 qrs. 2 pks.
3. £6 14s. 6d. 4. 1^. 5. 43-039783 feet.
6.
7. (a + &)(c-«Z); 3a2k(a3 - 2i3 + 3c3) ; (x-5)(x-2).
8.
CVII.
1. 580 tons 10 cwt. 2 qrs. 2. 2037 cub. ft. 189 cub. in.
3. 2 qrs. 17 Ibs. 1^ oz. 4. 5 dwts. 3T\VV grs-
5. -004399455 ; -0016. 6. a3 + a2 + a + I ; a3 - 1.
a3
7. -2«3-TV22y + |xy2-|y3.
1«..T, ««
196 MATHEMATICAL EXERCISES.
CVIII.
1. £921506 5s. 6d. 2. 9ff ; f££.
3. -2671875. 4. 74-8 yards.
5. wwx5 + mV + wV - w V - 2W7UC3 - nW - 2mnx2
+ m?x + mn.
6. (i.) (a;-l)(a;+l)(a;2+l);
(ii.) (x + 2)(x - l)(z2 - 2z + 4)(z2 + z + 1). 7.
8. 72x3yV(l + 2a»).
CIX.
1. 350 ac. 1 ro. 11 po. 29 yds. 7 ft. 64 in.
2. 4502 links. 3. 1. 4. 4 horses.
5. 3-2304328; -000001001. 6. 1.
7. i)*=10; <ii.)*=--
CX.
1. 453492; JSfa. 2. H;34f. 3. "415.
4. 4-16; -1984126.
5. 6. x2-3a;+ll.
7. z3-3a;2 + 3a;-l=(a;-l)3. 8. x.
CXI.
1. 149328 oz. 2. £65 Os. O^d. 3. f
4. 2s. 6d. 5. 3332-38699. 6. a- 6; 2-fl = 2.
7. (x2 + y2)2 - 2* = x4 + 2zY + y4 - a*.
8. a = 5|; « = 3, y = 4.
CXII.
1. £7 12s. 4|d. 2. 8^; 26. 3. $; "78516.
4. 1 hr. 23 mins. 15 sees. 5. 0.
6. (p-r)(q + r); 3xz(x-3z)
7. 24. 8- a2 -a;2.
ANSWERS.
197
CXIII.
1. 15 mi. 6 fur. 10 po. 2 yds. 2 ft. 4 in. 2. 64.
3. 6f£. 4. £7 8s. 6|d. 5. 9-8202; 148-279.
CXIV.
2. 25f£i; 84|. 3. '053571428.
5. (i.) «2(a;-a)2; (ii.) (a - 4&)(a + 35) ;
(iii.) (m - n + & - l}(m -n-k + l). Q. x-3.
8. ay.
cxv.
3636 ac. 1 ro. 9 po. 26 sq. yds. 8 sq. ft. 2^ sq. in.
£3 5s. 9£d. 3. ^. 4. £25600.
-010101; 32-114. 6. z2 - (c + d)x + cd.
- 2x + 2y. 8. (i.) x= - 2 ; (ii.) z = 1.
CXVI.
2. 6j|; 33V 3. '354.
5. x3 - 5x2y + Sxf - if ; ab20.
a3?/3 -t- a;2?/4 + xy5 + y6) ;
7. 42 or 84.
CXVII.
489 tons 14 cwt. 1 qr. 2. 5d. 3. 10f.
£40 18s. 7£d. 5. 36500 ; 9 quarts.
£1621 9s. lOd.
£2 Is. 8d.
(i.)
(ii. ) (3; -
(iii.) (2x-y)(x-2y).
6. 1; «&3(l-a2).
8. (i.) a;=3; (ii.) x= -b.
198 MATHEMATICAL EXERCISES.
CXVIIL
1. J*s. 2. f£ yard. 3. 1281 minutes.
4. 1-0490734.... 5. 6. a(a + b).
7. fl2 + a& + &2. 8. (a2 + 1)(62 + 1) - (aft - I)2 = (a + &)2.
a
CXIX.
1. 3 fur. 27 po. 3 yds. 2. £6 17s. 9|d.
3. 22^ Ibs. 4. £3 15s.
5. 343-780696; -064. 6. T^ + irr64-
7. 26s3 - (a2 - 62)x2 + (a'26 + 2a62 - 63)x.
8. (i.)x = 5; (ii.)*-S-
Cv
cxx.
1. £14 3s. l|d. 2. 8l||; 1J.
3. ||; -527. 4. 15 oz. 9-34 drs.
5. a4-5rcy + 4i/4; ku.
6. 3x2 - 8xy - 3y2 ; ifc* - Prnz ; a62c. 7. 30.
8. oV*3.
CXXI.
1. 3 sq. po. 30 sq. yds. 1 sq. ft. 68 sq. in. 2. f yard.
3. li. 4. 17£ mins. past 12.
5. 2-4142003; 440. 6- (i.) z = 6; (ii.) z = ^|-
Q 26-23a
7. 0; x-y. 8. —ft —
CXXII.
1. 2. (i.) 6||f ; (ii.) 1A5A-
3. 137-34d. 4. 91-8638257.
5. (i.) Those with even indices ; (ii.) All.
Ct3j
6. z2-a;+l. 7. (x + l)2-a;2 = a; + a;+ 1. 8.
ANSWERS. 199
CXXIII.
1. 11 tons 17 cwt. 4 Ibs. 2. £3 5s. 4d.
a TV- 4. 13 cwt. 3 qrs. ( = |J ton.)
5. -4487775; 448-7775; 448777'5
6. afi-2a*a? + a*. 7- x* - x"*y2 + y* - x* + xy - y2.
8. (i.) <c= -21; (ii.) x = a-b.
CXXIV.
1. £9 5s. 3d. 2. (i.) 17J&; (ii.) 2£.
3- ^fr; A- 4- HOd. 5. 0.
6. (*+2jr)(*-2y).
7. Men, Is. 6d. each; Women, Is. each. 8. x*-a*.
cxxv.
1. 7355f sq. feet; 50000 sq. links. 2. £7 4s. 5|d.
3. 3s. 3^d. 4. 11 Ibs. 7 oz. 16 dwts.
5. 1 -514823& j -00276.
6. a3 + J3 + c3-f d3-3(abc + bcd + acd + abd).
7. -a3 + a26 + ^2-263. 8. (i.) as = 5; (ii.) « = 5.
CXXVI.
. £113 5s. 9d. 2. x
3. ^ 4. 18-522d. 5. (i.) - ; (ii.) b-c.
6. (i.) «6(2a + &)(a + 26); (ii.) 2(«-2)
(iii.) (Z-2XSB-1).
7. 11,13; 2ra-l and 2w + l. 8.
CXXVII.
1. 8 oz. 14 dwts. 12 grs.
2. 174 cub. ft. 617^1 cub- in- 3- 1
4. 1250 men. ' 5. ^6T. 6. 1\.
7. (i.) x=l3- (ii.) x = 5, y=l. 8. -
200 MATHEMATICAL EXERCISES.
CXXVIII.
& S. D.
1. Kibbon, - 0 18 6|
Calico, - 1 13 3|
Gloves, - 5 10 0
Tablecloths, 7 17 6
Cotton, - 0 1 6f
£16 0 11
2. 3||§; 1. 3. 12s. 9-63d. 4. M.
5. Identically equal. 6. 4x2-6«+9. 7. x+1.
a? + ax + x2 ," \ bc + ca + ab
0 r \
o. (i.)
- - - =
a + x abc
CXXIX.
1. 123 acres 6 sq. chains 3636T4T sq. links.
2- 52| square feet. 3. y^-.
4. 1882 tons 13 cwt. 2 qrs. 7 Ibs. 5. '00002 ; 628000.
6. a2 + 462 + <? - 2ab - 2bc - ac. 7. a2 - ft2 - 2ac + c2.
8. (i.) s = i; (ii.) <K=1.
cxxx.
1. £1724 17s. 2|d. 2. -fa. 3. '4; -75; 1-6.
4. £4 8s. 7-56d. 5.
61* \ / O . O\ / \ /*"\/O
. (i.) (x2 + f)(x + y); (11.) (c2
(iii.) (a-b + c)(a-b-c). 1. 6.
CXXXI.
1. 58 mi. 3 fur. 39 po. 4 yds. 1 ft. 11 in.
2. £2 5s. 3. ff£g. 4. 140, 170, 190.
5. 4188-63. 6. 116f.
7. (i.) z = 6; (ii.) * = *,= !. 8. p^|.
ANSWERS. 201
CXXXII.
1. 103 soldiers. 2. 13f. 3. 2541 sq. yards.
4. -0333. 5. 6. 3-2.
7. a-b-5. a 1.
CXXXIII.
1. 694 tons 5 cwt. 2 qrs. 24 Ibs. 2. 4s. 10|d.
3. U£. 4. -Mf. 5. -9638.
y O 0 5 1 O
7. (a:2 + a^/ + y2)^ - a%2 + y4) or, ic6 + tfy - x3^ + xy5 -f tf.
8. (i.) z = 5; (ii) 3 = 0.
CXXXIV.
1. £777 5s. 8Jd. 2. 6T5^; TV
3. -008075; (ii.) ^f^ 4 -048828125.
6. (i.)
(ii.)
(HI) (x + y)(a-b). 7. 14. 8.
cxxxv.
1. 66. 2. £3 2s. 2|d. 3. 1^; 1-0714286.
4. 30 days. 5. 20166-0228203; 4320-99900099.
6. -1. 7. z = 2. 8. 1; a1.
CXXXVI.
1. 3 Ibs. 2. 96 J; <^. 3. 13968 grains.
4. -7296. 5. 6. xi -3y.
7. 12, 6. 8.
202 MATHEMATICAL EXERCISES.
CXXXVII.
1. 186636 cub. in. 2. 1TVAV rods.
4. £12. 5. -056068246; 575.
6. 1 - 2x - 31z2 + 72z3 - SOz4 ; (-5)x(-2) = 10.
7. 5(a2-J)-6(o8-52); 3.*
8. (i.) x=|; (ii.) a; =2, y=13.
CXXXVIII.
1. £2341 11s. lljd. 2. 3^.
4. 3-8. 5. (i.) 0; (ii.) 1.
6. (i.) (k-z)(l + y)-, (ii.) (x-9y)(a; + 9y); (iii.)
7. 4 + 12a:-7x2-24x3 + 16a,4. 8. (x2-
CXXXIX.
1. 8640 grains. 2. 6 cub. ft. 192 cub. in. 3.
4. A, £501 16s. 6d., £, £250 18s. 3d., C, £83 12s 9d.
D, £20 18s. 2£d. 5. -00004093; 370-6.
6. 4£. 7. (i.) o; = 6; (ii.) x=2a + 3b.
8. (i.) 0; (iL)o^; (iii.) 1.
\
CXL.
& S. D.
1. Raisins, - 089
Nuts, 0 10 6
Figs, 2 13 1-|
Figs, 074
Currants. - 0 3 10
Oranges, - 0 12 6
£4 16 0|
ANSWERS. 203
9 647 • J}5
3. 239-97 dwts. = 11 oz. 19 dwts. 23-28 grs.
4. Terminating, if the denominator contains no factors but
2 and 5. 5. (i.) Bx2(x2 - 2xy + 3y2) ;
(ii.) (z-l)(z2 + £+l); (iii.) (m +p + q)(m -p - q).
6. 4o62 7. 21, and 3. 8. — .
a-x
CXLI.
1. 71 Ibs. 0 oz. 2 drs. 1 scr. 14 grs. 2 729 yards.
3. -j4-g. 4. £1 Is. 4d.
5. 1669-94885; 31-0539544. 6. z*-z
7. (a - 3c)2 - 462 = a2 - 4&2 - 6ac + 9c2.
8. (i.) a = |; (ii.) x = a + b.
CXLII.
1. £58 10s. 9d. 2. 7^; 4|.
3. 5-1825; 3-136. 4. 50-4d.
5. 2&2-<Z2 + 2a5+2&c; -^-. 6. £a.
7- (i.) (k - 1 + m - ri)(k - 1 - m + ri) ;
(ii.
8. x3
CXLIII.
1. 3998936616. 2. 5.12 P.M. on Thursday.
3. 1. 4. 21 days. 5. 35164-8 grains.
6. If. 7. ™~f'>
8. (i.) x = l; (ii.) x = 4.
204 MATHEMATICAL EXERCISES.
CXLIV.
1. 11 hours. 2. 4f; 9. 3. £10 6s.
4. -16. 5. 6.
7. a;2
CXLV.
1. 36229680 sq. in. 2. 40 yards. a
4. £2835. 5. -136; 1240.
6. l+a-a*-a*; ax+!l. 7. f a3 - f a26 - 1«62 +
8. (i.) x = 7 ; (ii.) z = c.
CXLVI.
1. £61 6s. lOff 2. 3HI; If.
a M031313131 ...; Itffa 4. 547'01425d.
5. (i.) Sz4-!; (ii.) - 2abc.
6. (L) (a + b + c)(a2 + tf + c*-ab-bc-ca);
(ii.) (m + n)(m? -mn + n2) ; (iii. ) ax(x - Sax2 + 3a2).
7. 16 = 16. 8.
CXLVIL
1. 7 sq. po. 20 sq. yds. 7 sq. ft. 69 sq. in.
2. 4 inches. 3. 63^|f. 4. 13s. 4d.
5. 6-9193. 6. -1£.
7. (i.) z=2; (ii) 3 =
ANSWERS.
205
CXLVIII.
1. 29393 ; jfif^ 2. 3 miles an hour.
3. 3-4992 farthings.
4. 4411., or 44-578947368421052631.
5. (i.) (5x + y)(5y - x) ; (ii.) (a2 + 1366)(a2 - 8i6) ;
(iii.) (5y - x)(y + 5z). 6.
1 a
7. £16s. 8d, 8. -- ^ §
1. £25 2s. 3|d.
a 2-23608 Ibs.
5. £409 10s.
CXLIX.
2. 3-539142857.
4. 10TV = (3£)2.
6. ac - ad - be + bd.
CL.
2.
-xy + y
•C ^
1. 4 mins. 42 sees.
4. 2£ years.
5. (i.) (x + y)(x-
(ii.) (x - y)(x
(iii.) (x -
6. a2-62 + c
7. (9x2 - 4)(9cc2 + Qx + 4)
8. 33 and 18.
CLI.
1. 90-909. 2 -009375.
4. 564 passengers. 5.
7. (i.) z = 4; (ii.) x = a + b.
. 93_03
3. £6 19s. Od.
)(x* + xy + y*
+ f) ;
xy3 + y4).
;- 16).
3. 90 yards.
6.
4(a;2-4)
206 MATHEMATICAL EXERCISES.
CLIL
1. 1949-6 ft. or 2 fur. 37 po. 1 yd. M ft.
2. -015972. 3. 7f|£°; ^ 4. £4425.
5. 6. 2a. 7-
8. (a + 2&)(2a2 - ab - 3i2), = 2ci3 + 3a2Z>-
CLIII.
1. £1 13s. 4d. 2. 25200-288 grains.
3. 5 acres 2 roods 10 sq. yds. 2 sq. ft. 108 sq. in,
4. -062068 ; -0000589. ' 5. 52| days.
6. 1 2a2 + 1 2ab + 2862. 7. z2 - 5.
8. x = a+2b + 2.
CLIV.
1. 65120 bricks. 2. 7£|. a 2s. 3d.
4. 26722-36. 5. - 6z2.
6. (i.) (m + n + a + b)(m + n- a-b); (ii.) (3x - y)(x + 2y)
(iii.) (a + 8)(a2 - 8a + 64). 7. 72(a^ + f)(x - y
8. A has 3s., J5 5s., C 7s.
CLV.
1. £653 4s. 4|f. 2. 26 Mch., 1790.
3. ^r = 22^|. 4. 26-1509125.
5. 8if hours. 6. 118.
A /* \ /* "
i _ i
--- ~-
ANSWERS. 207
CLVI.
1. sl-a of an acre. 2. 32T8T past 12.
3. 5 per cent. 4. 57-85065 pence. 5.
a r, «2 + 2
7-^T3)'
8. A, 40 years ; B, 20 years.
CLVII.
1. ISffl- mins to 3.
2. 725 posts. (Since there are 726 spaces.)
3. £502 6s. 10|d. 4. -248.
5. (i.) || ; (ii.) £. 6. or* + 2aa? + 3«2z2 + 2a3a; + 8a4.
7. f
8. (i.) x= ..
aft -f ic + ca
CLVIII.
1. 795-965 ... yards. 2. 180. 3. '007954.
4. 1 ton 7 cwt. 3 qrs. 27 Ibs. 8 oz.
5. (i.) y*x-x2-x + y; (ii.) a9.
6. (i.) (x + y + z)(x -y-z); (ii.) (x + 2a)(x - 2a)(x - a) ;
(in.) (f -60)2. 7. 23Jdays.
8. 24(a4-64)(a_J).
CLIX.
1. £7738 16s. 5d. 2. 960 men. 3. £279 17s. 3d.
4.^ has £400; £,£160; C, £140.
5. -0155754495. 6. 18; x.
7. (i.) x = 5; (ii.) z-2, y=3.
a (i.) 1±»; rf.
208
MATHEMATICAL EXERCISES.
1. 121695.
3. £5455 11s. lOd.
5. (i.) x(3x - 4y - 6) ;
(iii.) (x+10)(a;-2).
7. 64 square yards.
CLX.
2. 88345-6.
4. 28 J hours.
6. By a -5.
3a
8.
W
1. £2 5s. lOd.
3. 7 cwt. 3 qrs. 11 Ibs.
5. £105 Os. 4|d.
7. z*
CLXI.
2. 65-062 pints.
4. 25548-9922; -582.
6. (i.) z2 + 3y2; (ii.) x-5.
(• \ j-i /•• \ CtC\ i. — tjCvO)
i \ /¥• M • in i o* \ »
1. I *v — ty . V 11. / «v — j •
CLXII.
1. 53^ yards. 2. (i.) 27225; (ii.) 165.
3. 5 oz. 10 dwt. 3-552 grs. 4. £43 17s. 8™d.
5. a - b. 6. (i.) (a2 + a& + 62)(a2 - ab + i2) ;
(ii.) (4x- 3) (2x - 5) ; (iii.) x2(x + y) (x- y).
7. In 2| hours. 8. xy (x2 - y2).
1. 54^ mins. past 4.
3. 7372-4489.
CLXIIL
2.
6. -27f.
1
4. 70 more.
7. (Da? = 2
5. 63.
8. (i.)
xyz
(ii.)
CLXIV.
1. £3 5s. 4d. 2. 1256-64 yards. 3. £8099 Os. 4-992.
4. £1221 19s. llfd. ; £720 5s. OJd.
5. 6. x2 + x + 1. 7. ^ = 640, B = 720, (7= 840.
8. (i.) ?; (ii.) _^_. (i.) 4 = 4; (ii.) 2 = 2.
y •-!
ANSWERS. 209
CLXV.
1. £3 16s. 8±d. 2. T\V 3. 126 yds.
4. -3183. 5. 29^ yards.
6. 4z* + 4oz* + cfo? - a4. 7- az2 + 2a% - 63.
8. (i.) * = 0; (ii.) x=l.
CLXVI.
1. 11 cub. yds. 6 cub. ft. 1113 cub. in. 2. -00379.
3. -09583. 4. £4100. 5. 9 (a?-f).
6. (i.) (x* + y^(x + y)*(x - yf ; (ii.) y(x - y)(x* + xy + f) ;
(Hi.) (x + y + z)(x2 + y2 + z2-xy-yz- zx).
7. (x+l)*-x* = x + x+l. 8. 21 Oar*//2.?-1'.
CLXVIL
1. 108^ tons. 2.
a -2237 ...... ; 563-1860 ...... 4.253; 44.
5. It will be filled in 6 hours. 6. 0 = 0
7. (i.) «- -5; (ii.) a = 4. 8.
CLXVIII.
1. 1'240 grains. 2. 314-16 square yards.
3 147-186 yards; -0955753246. 4. 7 £ years.
5. (i) xy(a?x-b2y); (ii.)
(iii.) (m + n+p + q)(m-n-p + q).
CLXIX.
1. 24 tons 18 cwt. 3 qrs. 1 Ib. 8 oz.
2. 100 acres 3 roods 17 perches. 3. £ hour a day.
4. 7-1749709428571. 5- 5^-mins., or 38T2T mins.past 4.
6. x =2. 7. a6 - «462 + 2ft3&3 - a264.
8,. , _ /.. v 4a — &
•(l')a; = 5; (u-)Zss
210
MATHEMATICAL EXERCISES.
CLXX.
1. 14838-18 square feet. 2. 1088 inches.
3. A, £63 ; B, £218 8s. ; C, £138 12s.
4. £2601 9s. 5. 3xy + 3ay- 3bx- 2ab-ac + ad.
6. (i.) (3ad - b)(3a - be) ; (ii.) (x + 7)(x- 1) ;
7( ff£t - - <l/2\ * ( rp _i_ ni\ — ff ^ nt W ftt*1t ISY* - Qi^\
• Ii*/ If I ~ lw T tf I — w V» O« \.wjU \Jj */ /•
CLXXI.
2. 18-2083; -02957.
4. 2i£f£. 5. 0.
c2 — a6
1. 17660 times.
3. 7 per cent.
.... /..
6. W.-TI (".
7. (i.) 4a2 + fl&-&2 ; (ii.)
8. 24, and 36.
CLXXII.
1. £3 4s. lid. 2. 1 cwt. 2 qrs. 10 Ibs. 13-5552 oz.
3. £4285 19s. 7d. 4. 19£ days.
5. ^(2 + r) =^(^> + g + r) - p2, etc. 6. 3x 4- 5.
9icrf 213 °3
7. ax - ay is divisible by a.
8.
5-
CLXXIII.
1. £4 9s. Ofd.
3. £12 12s.
5.
2. 1152 pence.
4. 14-05.
7. (i.) x = a-, (ii.) a? = 5, y=l.
8- Euc. I. 3.
ANSWERS. 211
CLXXIV.
1. 58f yards. 2. ff. 3. "3125. 4. £2227 10s.
5. (i. ) (a - b)(a + b)(a* - ab + 62)(ft2 + ab + Z>2) ;
(ii.) (a-b + c + d)(a-b-c-d).
6. (a-b)(b-c)(c-a).
13
7. x* + x*f + y*; x*~. 8. 3.
CLXXV.
1. 17^ days. 2. 230-039346; 4-1971062.
3. 30507. 4. 7s. 5. 5.
6. a ; One factor is (a - b + c + b - a - c) = 0.
7. (i.) z = =^z!4; (ii.) a; = 6f. 8. Euc. I 4.
4fl — D
CLXXVI.
1. £5 17s. ll^gd. 2. 7s. 6-888d.
3. 10 days. 4. At 5 per cent.
5. (i.) aaabbcccc ; (ii.) xf(3x + 4y) • (iii.) (2a; + y - z)(y + z).
6. 7a>-2x). 7. (i.) fty~M).: (ii.) a'.
mA(m - n)
8. 28 and 36.
CLXXVII.
1. £3696 12s. 9d. 2. 25800 pence.
3. 49^ minutes past 9. 4.4027-428609; -2094.
5. 4^
8. Euc. I. 4.
212 -MATHEMATICAL EXERCISES.
CLXXVIII.
1. 149 yards 8 inches; £33 11s. 6d. 2. f£.
3. £29 8s. 3|f 4. £612 8s. 6d.
5. (i.) (x-y)(x* + xy + y*)(x* + a?f + f);
(ii.) (x + 5)(« + 2) ; (iii.) a?(x + y)(x - y).
6. bxy(b2 - x*y2). 7- (i.) -4a6(a2 + 62); (ii.) a2r-a2'.
8. -4, 36 years; 1?, 18 years.
CLXXIX.
1. 12 minutes. 2. 24707221. 3. 1305 yards.
4. 7699-524 metres; 7 kilom. 6 hectom. 9 decani. 9 metres.
5 decim. 2 centim. 4 millim.
K R 7 4-5r2 _ 9 7 r
5' 9J6-06' 2^'
8. Euc. I. 32.
CLXXX.
1. £5 18s. ll|d. 2. -1954
3. 1256-5. 4. 7 years.
5. (i.) 3.5.7. aabccc ; (ii.) No real factors ;
(iii.) (k-l)(k + l)(k*
6. 2a(a2-a& + &2). 7. 8. ? + ---.
b ax ax
CLXXXI.
1. £28 14s. 8Jd. 2. Is. 6|^d.
3. 28| minutes to 6. 4. '001 ; 0.
. _. /
' 27 "64~- '
7. (i.) z=13; (ii.) x = 12,y=3. 8. Euc. I. 3, 23.
ANSWERS. 213
CLXXXII.
1. 4s. 8d. 2. 5£|. 3. -43625. 4. £585 4s. 10|d.
5. (i.) (7x-3)(3x+l); (ii.) xt/(x + 2y)(2x + y) ;
(Hi.) 9(a + i)(a2 + a&H-&2).
6. 30z2(z2-l)2. 7- a2- 4&2+ 126c-9c2.
8. 16 yards.
CLXXXIII.
1. £717388 Os. 6d. 2. 4-03. 3. 1458 yards.
4. -9941 metre. 5.
6. (i)a?*+*+*-; (»-)^|;
7. (i.) £ = 5; (ii.) x = 2b-a, y = 2a-b.
8. A right-angled triangle.
CLXXXIV.
1. 32 men. 2. 2938-402 Ibs.
3. £17 14s. 4. £445.
5. (i.) (x* - tf)(x» + f) ; (ii.) («* + f)(tf + y*)(x + y)(x - y).
6. x* - 3aV - 4o*. 7. 8. 126 gallons.
CLXXXV.
1. £25 17s. 3M. 2. £7 16s. Offd.
3. 32^- mins. past 12, and 38T\ mins. past 1.
4. 18-203. 5. ^~. 6. (i.) * + 4; (iL) a2.
CL
7. (i.) z=14i£f ; (ii.) x = y = 4. 8. Not always.
CLXXXVI.
1. 3 feet 8 inches; 18333-7590857142 Ibs. avoirdupois.
2. |4- 3. '025. 4. £248 19s. 2|d.
214 MATHEMATICAL EXERCISES.
5. (i.) x»(f - z") ; (ii.) (a + b + c)(a + b-c);
(iii.) (a + b-c)(a-b + c). 6. 2(3x-5).
rr /• \ ft2 + 6- /•• \ Or~b~ >••• \ i /• \ 2
7. (>-)-r; ("-)-^-; (m-)i; (iv.)
8. A spends £370, B spends £330.
CLXXXVII.
1. £8 14s. 2d. 2. 8-525.
3. (i.) 130 yards 5 inches ; (ii.) 184 yards 2 inches.
4. 9f!~| hours. 5. a2 - 5ab + 462 ; x
6 (' } • (" ) 6k + Sac + 2a5 7 •"> 4 6
^ ' 66c-2oc+3o6'
&/ * \ ' /"*\ 1 * PC
(i.) x= — — ; ^n.; x= i, y = o.
CLXXXVIII.
1. |. 2. -0028125.
3. £50 12s. 6d. 4. 2£ per cent.
5. (i.) (3x-2)(x+2); (ii.) 4a6 ;
(iii.) a3(l + 5a)(l - 5a+ 25aL').
a Euc. I. 28.
CLXXXIX.
1. 25313-48. 2. -0027. 3. £624 6s. 4-8d.
4 A c ~~ x " i o
. -6. 5. — r =13.
6. (i.) 4-x-2y; (ii.) o'+ir+t.
7. (i.) « = 9 ; (ii.) a; = 6, ?/ = 4.
8. Two-thirds of a right angle.
ANSWERS. -Jlo
cxc.
1. B won. B's time 5 min. 1 G sees., A's time 5 min. 20 sees.
2. 63. 3. 19s. l|d. 4. £594 4s. 10725.
5. (i.) x-3 and x - 4 ; (ii.) 4xy and x - 3y ;
(ii\.) x+1 and x2 - 2. 6. 3a + b.
7. (i.) i; (ii.) «2 + 6«& + 9&2.
8. 75 quartos, 375 octavos, 750 duodecimos.
CXCI.
1. £2734 15s. S&d. 2. T^.
a £7699 13s. 10-5792d. 4. 230-039346 ; 7436-1.
5. x*- (p- I)x3 + (q-p + l)x2- (p- l)x + 1.
7. (i.) z = f ; (ii.) a;= -1, y = 8. 8 Euc. I. 4.
CXCII.
1. 96 planks. 2. l^VV
3. 3s. 8£d. 4. £543 3s. l|d.
5. (i.) (a + & + c)(a
(ii.) (a + y)(a -
(iii.) (x - ay)(x + by).
6. (i.) (2« - 3)(a;2 + 2x + 4)(a;2 - 2x + 4) ;
(ii.) x2 - 2a; + 4 and x2 + 2x + 4.
7. y8 + m*y* + m8. 8. -4, £2400 ; .B, £2500.
CXCIII
1. £200 16s. 4£d. 2. 1-71428S. 3. £2350.
4. 358-4 pages. 5. (i.) 1 ; (ii.) - 4.
6- (i.) \ } ; (ii.)
7. (i.) *-!•; (ii.) *=±4.
8- Euc. I. 26 ; or I. 8 may be regarded as a converse.
216 MATHEMATICAL EXERCISES.
CXCIV.
1. £U 8s. 2. -3140625.
3. 18 youths. 4. 3£ per cent.
5. (i.) (z2 + xy + f)(xz - xy + f) • (ii.) (2
(iii.)
6. a. 7. ??_+£..
y y xy
8. (i.) ^^ persons had 9d. each, and ^~'6n had 1 5d. each ;
(ii.) 6 had 9d., and 14 had 15d.
CXCV.
1. £27 19s. 7d. 2. £i 3s. l|d.
3- £103 12s. l-785d. 4. M126244825.
7. (i.) « = f ; (ii.) a = ±l*. 8. Euc. I. 6.
rt
CXCVI.
1. 3-13. 2. 3|.
3. -0431365740. 4. £6 8s.
5. a,4(d - 5) + z3(a - d) + a;2(c -f) + ex.
6. 1 2a;y (a3 - x*y - xf + ys).
7. 2a2&2 + 2&2c2 + 2c2a2-a4-64-c4. 8- 3000 men.
CXCVII.
1. At 8 p.m. on April 22, they will be at 7 hrs. 48| mins.,
and 8 hrs. 18| mins. respectively.
2. 4374 yards, = 2 mi. 3 fur. 35 po. 1£ yds.
3. -7344. 4. £893 19s. 5. 0.
6. (i.) aW - ahy? + abx + Wx - W ; (ii.
x + a
7. (i.) x= - Y ; (ii.) a= 4, y = 2. 8- Euc. I. 13.
ANSWERS. 217
CXCVIII.
1. 40 yards long, and 30 yards wide ; £24 10s.
2. 30-8 pence. 3. 24 hours. 4. At 8£ per cent.
5. (i.) (x - y)(x* + y?y*+ x*y'2 + xys + y4) ;
(ii. ) (x - y)(x + y) (x2 -xy + y2)(x2 + xy + yz) ;
(iii.) (x + y)(xP — x5y + x*y2 — x3y3 + x*y* — xyb + y6).
6. x-l. 7. (i.) — ; (ii.) 2 + 3x.
B. 8 half-crowns.
CXCIX.
1. £10 3s. 4d. 2. 658jVT oz.
3. £2727 15s. 10-848d. 4.381-096704; 18-903296.
5. £a3 - 4#2« + iz3. 6.
7. e^ + ^_f+_L. 8. Euc. I. 32.
y y y xy
CO.
1. 3-239. 2. £675.
3. 846 lengths, -0046875 ft. left. 4. £720 9s. 9|d.
5. (i.) (x-l)(x-2)(x-3); (ii.) (3« - 2?/)(2z2 - y2).
6. 6(x2 - l)(a;2 - 4). 7- a = 13, &= 1.
8. 28 days.
CGI.
1. ST\ hours. 2.6-5793741. 3. £1023 6s. 8d.
4. £7173 6s. 8d. ; £8070; £8608; £8966 13s. 4d.
52+1U X _ a /• \ 9 . 19 /" \ &
. ___-_ = 5. 6. (i.) a2 + 62; (n.) T.
7. „,, 8. Euc. I. 19.
21S MATHEMATICAL EXERCISES.
CCII.
1. 11 feet 6 inches. 2. 6 shillings.
3. £446 10s. 4. £3570.
5. (i.) (2x - 3y}(2x + 3y) ; (ii.) (2z - 3y)(4z2 + I2xy
(iii.) (a-3b)(a
7. a:2 + a2. 8. 42.
coin.
1. £38 16s. 9d. 2. 21^4. 3. £2857 3s. Id.
4. -9999998. 5. 42.x2 + 216o;y + 30y2.
Mi.)^^-0; di.)^.
8. Euc. I. 28.
CCIV.
1. £2 4s. 6d. 2. 946 feet.
3. 66| hours. 4. £18 17s. 7^d.
5- (i.) (db + be - ac)(ab - be + ac) : (ii.) (a + J + c)2.
6. b + c-a. 7. sr-v 8. £45.
3(x+ 1)
ccv.
1 44Jlbs. 2. 7-875.
3. £93 11s. 3|. 4. 2 mins. 37£ sees, past 3.
5. 43. 6. 4 + x\ 4 + 4x-8a.
7. a?& - b*y3 + c3^3 - 3a2ta2?/ + 3a"cx2z +
-*- 3&2cy2^ - 36cV2 - Gdbcxyz ; x.
8 Euc. I. 37.
ANSWERS. 219
CCVI.
1. £14 10s. 3Jd. 2. 7i-*§d. 3. 18 hours.
4. (i.) 4£ years ; (ii.) If the Present Worth of £1493 7s. 6d.
is £1300, and the interest be reckoned at 3J per
cent., when is the sum due ?
5. If n be an even integer. 6. 2axy.
7. 5x + 5y. 8. A, 66 years; B, 33 years.
CCVII.
1. £12 7s. 2£d. 2. T5T. 3. £1 3s. 5fd.
4. -273688. 5. 3a2 - 2ab + 5&2.
2 ... . .
7- (i.) 2=2; (ii.) z = 3, y = 4, ^=6. 8. Euc. I. 20.
CCVIII
1. 8 feet ; 64 square feet. 2. f .
3. '028. 4. In 7| years.
5. (i.) (a - x)*(a + x) ; (ii.)
(iii.) 2a(5x - 3a)(2x + 3a).
6.
7. Put % = - = k, and substitute for a and c.
o a
8. (i.) x - y ; (ii.) (x + 5)(x + 6)(9a;2 - x - 3).
CCIX.
1. 01849; -629. 2. '018; 180; -00000018.
3. £600. 4. A and B £333 6s. 8d. each, C £150.
5. J-fr 6. (i.)*-*; («•) 0.
8. 20 years.
•220 MATHEMATICAL EXERCISES.
OCX.
1. £3 4s. lid. 2. -16953125.
3. 5 men. 4. 50f years.
5. (i.) (a* + f)(x* + y4)(3» + f)(x + y}(x - y) ;
(ii. ) (5x - 1y}(x + 3y) ; (iii. ) (x + 1 )3
6. a-2x + 3. 7. (i.) 0; (ii.) 1. 8- Euc. I 26.
CCXL
1. £2418 17s. 3|d. 2. £1 5s. 8£d.
3. £10 5s. Oid. 4. -5204397.
5. (i.) 2x - ty, remainder - 4y2(x2 + xy + Gy2) ; (ii.) ax*.
9,,;,
6. 0. 7. (i-)^ = ^J (ii,)a> = 2,y=l.
8. Euc. I. 41.
CCXII.
1. 'f. • 20. 2. £640. a £1 5s.
4. No difference. Each is £3.
5- (i.) i ; (ii.) Ir.
6.
7. taxb-cz. 8.
CCXIII.
1. 276 tons. 2. 30000. 3. £2394 14s.
4 38j2T minutes past 7. 5. - 3|.
6. (i.) ^(x + y + z); (ii.)
7. (i.) z = 7; (ii.) a; = 96, y -- 48. 8. x3 - 2a;2 + a - 2.
ANSWERS. £.'1
CCXIV.
1. 5 ac. 1 ro. 33-41424 po. ; 162 yds. 1 ft. 7 in.
2. 1-87104 pints. 3. 480 roubles. 4. 5 per cent
5. (i.)*-l; (ii.) ax(tf-ae). 6- ^.
7. (i.) 0 ; (ii.) 2£G ; (iii.) 0, or 2bm, as m is odd or even.
8.
CCXV.
1. J|f 2. 1-407. 3. 1-414213.
4. £22668 8s. lOd. 5. a'"a' + ?.
a8 + a4 + 1
6. a2 + b2. 7. After B has gone 6 £ hours.
8. Euc. I. 6.
CCXVI.
1. £386 11s. ll£d. 2. 101-310.
3. £23 9s. 4d. 4. 1. 5. 5.
6. G.C.M. isic3-!, LC.M. is^-^ + ^-^ + z-l.
7. Each would take 50 days.
8. The digits are 714285. After the 7, each pair are ob-
tained by doubling the preceding pair, the figures in
the hundreds place, when they occur, being carried on.
CCXVII.
1. 1309 miles 4 fur. 32 po. 4yds. 2ft. Sin. ; 446418 inches.
2. -64125. 3. 1-4422.
£ s. d.
4. 5 pairs blankets, 600
12| yards merino, 1 16 5£
15 1 yards cloth, - 7 9 1\
5^ yards flannel, 0 9 7|
2 counterpanes, 276
25 £ yards calico, - 0 18 11^
£19 2 "If
222 MATHEMATICAL EXERCISES.
- 2a+2b-ab-b2 0 n
5. x = - 5 — -, y = - Z- — - - . 6. 9.
a-b + 2 a-b + 2
7. A, 36 days ; B, 60 days ; C, 15 days.
8. Euc. I 32, 9.
CCXVIII.
1. 4497 times. 2. 1200 gallons. 3. £494 4s. 5d.
4" 5. W
6. (i.) z = 5, y = 8; (ii.) a; = 7, or - 2|.
7. x2 - a2 + 2a6 - b2. 8. Euc. I. 38.
CCXIX.
1. 10s. 2. 1 -84375. 3. 4 tons 6 cwt. 2 qrs. 12 Ibs.
4. 2*2 - 2xy + 2y2. 5. a2 + i2 - c2 - d2.
6. (i.) «; = £; (ii.) « = 0. 7. o-l.
8. Bisect the angle at the centre of the circle, and use I. 4.
ccxx.
1. Tin 37| Ibs. Copper 112J Ibs.
2. £5 8s. 3T\VL 3." ^, £10 ; £, £18 ; (7, £15.
4. (i.) x= -13; (ii.) z = 5, y = 2.
5. (i.) 1 ; (ii.) w. 6. 14 sheep.
7. 144a2&2(a2-z2). 8. Euc. I. 38.
CCXXI.
1. £4 3s. 9d. and £3 2s. 9fd. 2. £3545 Is. 2d.
3. '9386 kilometre, or §54 kilometre.
4. (i.) « = f, y = §; (ii.) z = 3or -4T8T. 5. 1/V-
6. A, 30 days ; £, 60 days ; C, 20 days.
7. as"(z - 2)2(a; - 1). 8. Euc. 1. 1 1, 19.
ANSWERS. 223
CCXXII.
1. 2f. 2. 4217.
3. £12 15s., and £19 2s. 6d. 4. ax -by.
5. z = ^i 6. 50 and 15.
2i
7. l-2a + 3a2. 8. Euc. I. 32.
CCXXIII.
1. 14 days. 2. £19 18s.
3. 14s. per cwt. ; 20 per cent.
4. (i.) x = 5 • (ii.) x = 4. 5. a16 - y16.
6. x* - y?y + «V - xys + y* • «3 + a + - + .
a (i
7. £100, £100, and £70. 8 Euc. I. 4, 27.
CCXXIV.
1. MTTT- 2. -8. 3. £400.
4. 33Tyj3T tons = 33 tons 11 cwt. 2 qrs. 5£ oz.
5. (i.) x = 13; (ii.) z = -|, 2/ = f
8. 85 for, 70 against.
ccxxv.
V2r; TS- 2. £1 19s. 3. 36, and 45.
4. (~z2-y2)2 + (z2-2/2)22 + z4- 5.
6. ± 7. (i.)«-0,or^j (ii.)a-o,y=
8. 1. 9. 7 and 10.
224 MATHEMATICAL EXERCISES.
CCXXVI.
1- *TT( = *XTXT)- 2. 2 quarts.
3. 4 months. 4. a= -c, b = c2.
6. (i.) x = a; (ii.) « = f, y = |. 7. T4T.
8. Euc. I. 19.
CCXXVII.
1. 2| miles. 2. 1250000000. 3. £451 9s. 3d.
4. 21. 5. (i.) z = 8; (ii.) 3= », or -£.
6. W + ml-Jcm; f«3 - 5a2 + ^a + 9.
8.
CCXXVIII.
1. 7702-2 inches. 2. 3738 sovereigns.
3. True disc. £99 15s. 2££fd. Merc. disc. £105.
4. £10. 5. --i-|-.
y 2z
6. 9^ miles from A. 7. x = ^,y = \.
8. Euc. I. 15.
CCXXIX.
1. £1 5s. 2. Debts £30,000, assets £20,000.
Kr i Q
3. £13 15s. ; 45| per cent. 4. J - -J.
5. (i.) x = |; (ii) a=ll, y = 6; (iii.) a=lll, or 12.
6. 6 half-crowns, 15 shillings, 30 sixpences.
7. 0. 8. Euc. I. 28.
ANSWERS. 225
ccxxx.
1. 89^-. 2. 40 years.
3. 3£. 4. x + y + z.
5. (i.) 3=111; (ii.) x = 4, or 1.
6. 5J miles. (2 miles up hill, 2f miles down hill).
7. 8. Euc. III. 12.
CCXXXI.
1. f. 2. 1609-306 metres. 3. By 30.
4. (a + b + c}(a + b - c)(a - b + c)( - a + b + c).
5. G.C.M. a3 - 1 ; L.C.M. rt5 - a4 + a3 - a2 + a - 1.
6. (i.) x -30; (ii.) x = b.
7. Man £4 4s., woman £3, child £1 16s.
8. Euc. I. 26.
CCXXXII.
1. 1-126. 2. -1. 3. 16 men.
4. 1J. 5. (x*
6. , and 7.
b-a a-b
8. a3 + «&c + a2b - ac2 = a(a + c)(a + b-c) = 0.
CCXXXIII.
1. 5s. Id. 2. £6 17s. 4
3. £28 Is. 4. TO.
5. (i.)« = 7; (ii.) w = 4, x=12, y = 5, «
6. The square of x3 + ys + & - 3xyz.
7. x* - x - 1 ; 1 80a3z3(«2 - x2)3.
8. Equal to twice one side of the triangle.
226 MATHEMATICAL EXERCISES.
CCXXXIV.
1. 1-007; 13£. 2. £1 10s.
3. 900. 4. (z2-l)(z2-9)(z + 7).
5. (i.) x=-3; (ii.) z=30, y=20. 6. «4-l.
7. (a4 + ^)(a2 + a;2)(a + a;)(a-a;); (2ft - x)\a - xf ;
(a2 + a + 1 )(a2 - a + 1 ). 8. Euc. I. 47.
ccxxxv.
1. 31-87; 1T9T. 2. £240. 3. £4 Is. 8d.
z2(a2 + z2) 1
4. ~/ o T> o\' 0' *^ T * H 5.
«(a2 + ^x2) x2
6- (i.) x = 0; (ii.) z = 4, y = 6.
7. lOif mins. past 2. 8. Euc. III. 22.
CCXXXVI.
1. £126 ; 3§5- per cent. 2. -26457.
3. £12,000. 4. 0.
6. (i.) .« = 24; (ii.) ^=2,^ = 4,^6.
7. £52 and £2 12s. 8. Euc. I. 32.
CCXXXVII.
1. 2|f| per cent. 2. £1283 16s. 6|d.
3. £720. 4. a2-2 + 4ar2.
5. 90o6(a6 - i6). . 6. (i.) x = | ; (ii.) a; = 2, y = 3.
7. 5 and 6. 8.
CCXXXVIII.
1. 250 cub. chs. 47000 cub. Iks.
2. 22-289473684210526315?.
3. 11.5.22.3.132; 25.33.73; 2".35.13; 2,3,4,6,12.
ANSWERS. 227
4. «2-rt^ + F. 5. 6. x + ?-b.
o 2i
7. (i.) ;>' = 0, ora+b; (ii.) a; = 4, y = 5.
8. Euc. II. 12, 13.
CCXXXIX.
1. TV sq. in. 2. '6324 j -2.
3. 80 per cent. loss. 4. (2x + 3y)(x + y). 5. a; = 9, or -3.
0-5 11 r3 77r 9<* 4 31 2 is
6. * - - + - ;
7. 12. 8. Euc. III. 22.
CCXL.
1. 8TfT. 2. 8£ per cent.
a 4 fur. 7 po. 5-02 yds. 4- 2a- 1.
5. (i.) a; =7, y = 8- (ii.) a; = 8.
6. £150. 7. 8. Euc. I. 34.
CCXLI.
1. ISOlbs. 2. 32s. 0-75602vd.
3. 8448000 galls. 4. 4 + a-a3.
5. 6. 7. 84o%%3. 8. Euc. III. 17.
CCXLII.
1. 913. 2. 33 horses.
3. (i.) ^; (ii.) TVWV 4. 3rf-8x-3 = 0.
5. 1 and 3. 6. f 7. (i) a; = 13 ; (ii.) a; = 5, y = - 2.
8. Euc. I. 23.
CCXLIII.
1. 1-4410. 2. £3 17s. 9d. 3. £1400.
5. .r + 2 ; 72«2i3(^2 _ &2)(as _ js). g j 5 and 20
7. (i.) ^ ; (ii.) 0. 8. Euc. I. 20.
228 MATHEMATICAL EXERCISES.
CCXLIV.
1. £520. 2. 503; -1123.
3- £199 11s. 5id. 4. 1 + — + ^ + ^.
a a* a6
6. (i.)4ftz; (ii.) (x - 5)(z + 4) ; (iii.) (1 + 3a)(l - 3a+ 9ft2).
7- (i-) ^fA ; ("•) o^r?-,- 8. Euc. I. 32.
CCXLV.
1. -4$. 2. £1 3s. lOid. more. 3. 4 days.
4. 57. 5. (x-62)(.i--26); (9z2-5)(4z+ 3).
6. 2z2 + 4ca-3c2. 7. a; = 5i. 8- Euc. I. 20, 21.
CCXLVI.
1. 63^- per cent. 2. £58 Is. 4d.
3. 3 yds. 2 ft. 6 in. 4. z2 + x + 1.
5- <l> ; <a>»i- 6-
7. (2x + 3y + z)(2x -3y + z)(2x + 3y - g)(2x -By- z).
8. Similar figure to Euc. II. 5.
CCXLYII
1. 2319. 2. 4 per cent.
3. 56 florins, 40 shillings, 24 sixpences.
6. (.-c-2/)2. 7. 180 yards, and 205 yards.
8. Euc. I. 1.
ANSWERS. 229
CCXLVIII.
1. £4 7s. 6d. 2. £105. 3. 5 per cent.
4. The remainder ak2 + bk + c must = 0. 5. 0.
6. (i.) x= -5 ; (ii.) x = 2, or 4f ;
7. (« + &)(« - c). 8. Euc. I. 32.
CCXLIX.
1. £900, £1120, £750. 2. £14 17s. 3. 108|.
4. (i.) 2
5. (i.) 1; (ii.) -j-
6 (i) x-2a~b or 2b~a- (H\ X=R ori
\ / Q * i) ^ V * ' * 2
7- 8. Euc. I. 32.
CCL.
1. 6. 2. 36 days. 3. 5147.
4. (i.)s = 6, or-S; (ii ) »- 7±5 v/2. 5.
6. 48 of each kind. 7- 15«4J2a%3(2#?/2-aa;2).
8. Euc. II. 12, 13.
CCLL
1. 1TV hrs. 2. £10 10s. 3. 5-5747.
4y t\ -v3 1 <?o. 9
. 4; . 5. ^»-a
6. (i.)a; = 2; (ii.) o;^ '^ ; ("i.).^25^3.
7. 8. Euc. III. 26.
I'M MATHEMATICAL EXERCISES.
CCLII.
1. In 12| mins. 2. 15 yards. 3. As 32 : 35.
4. (i.) z=ll; (ii.) x=l, y=l. 5. 2x3-.r2 -3.
6. 7. A £800, B £320. 8. Euc. I. 24, 47.
CCLIII.
1. 1 ft. 3^ in. 2. He loses £13.
3. 32584734. 4. l-~
1 -a;10
5/1 \ />« — , /» f\V f* _ ' i 11 1 'T* — ^ 4J — fi
. 1 1* / .6 — (, . U L C ~* — . 1 11. i Ju — v. f/ — U.
\ / 2
6. 15, 20, 24.
7. (i.) (.T + 3«/)(.T2-3a-z/ + 9?/2) ; (ii.) (x - 5)(x - 8) ;
(iii.) (.c + l)(x - m).
8- Join the point to the intersection of diagonals.
CCLIV.
1. 953-05. 2. £6 9s. 9f£d.
3. £72. 4. 6|days.
5. (i.) x .= 3i ; (ii.) x = 2, y = 4, z = f . 6. 2r - 9.
\/ A ' \ / **/o' O
7. a = L_ 8. Euc. I. 26.
1 — e
CCLV.
1. 6 per cent. (Ratio is as 1176 : 1135).
2. 3. 3. £5 3s. O^d.
5 «2 + 3a + 2. 6. 25 per cent.
7 8. Euc. I. 1.
ANSWERS. 231
CCLVI.
1.3.29.31; 3.37.31; 31.41; G.C.M. 31.
2. 7 o'clock. 3. £12 Os. 3f*d.
4. .' - - 1 ; (x10 - 1 )(a* + 1 )(x* + x* + 1 ).
5. (i.) x = 5,y = 4; (ii.) # = -£, or J.
6. — ?-„ 7. 1800. 8. Euc. I. 12.
a - 4&
CCLVIL
1. T%. 2. "1883; 30-30.
3. 6.30 p.m. 4. 999-41 oz.
7. £420 10s. 8. I = a2 + C\
2ft
CCLVIII.
1. 6-683. 2. (i.) 6 hours ; (ii.) 6 hours.
a 269-443872... yards, or 110 JG yards.
4^«// "*• O*C *~" *j</ ?/ -~ OX'w
-y T
5. x* + x- \. 6. 32T8T mins. past 9.
7. <i.)* = f; (ii.)x = 4, or 2.
8. (i.) Euc. I. 4 ; (ii.) It is a parallelogram.
CCLIX.
1. £275.
2. ./ £234, B £266 8s., (7 £306, D £345 12s.
3. 3-4199. 4. (i.) 1 ; (ii.) xy. •
5. 6. 140 Ibs., and 60 Ibs.
7. G.C.M. 3a?-2x+l;
L.C.M. (3x2 - 2z + l)(2x2 + .r - l)(3a;2 + x - I).
8. Euc. I. 39.
232 MATHEMATICAL EXERCISES.
CCLX.
1. 603729. 2. 699-84.
3. 4|d. 4. (i.) x = 15; (ii.) .« = 2, orf.
°x2
5. 20 inches, and 16 inches. 6. ^ - 6x+ 1.
o
7. 8. Euc. III. 22.
THE END.
I'RISTK.D BY ROBERT MACI.E11OSE, USIVERSITT PRESS, GLASGOW.
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