(9.) Divide 40 into two parts such that if one-tenth of the less be taken from one-fifth of the greater, the remainder is 5. Ans. 10 and 30. (10.) Divide 28 into two parts, of which the one shall be three-fourths of the other. Ans. 12 and 16. (11.) Divide 36 into two parts having the ratio of 7 to 11. Ans. 14 and 22. (12.) Divide £5 among three persons, giving the first 5s. more than the second, and the second 10s. more than the third. Ans. The shares are 40s., 35s., and 25s. (13.) Divide 2 guineas among three persons, giving the first double of the second, and the third as much as them both. Ans. 7s., 14s., and 21s. (14.) Divide £1 among four persons, giving the first 1s. more than the second, the second 1s. more than the third, and the third 1s. more than the fourth.

Ans. 3s. 6d., 4s. 6d., 5s. 6d., and 6s. 6d. (15.) The sum of exactly the same number of half-crowns, shillings, and sixpences is £1; how many is there of each?

Ans. 5.

(16.) I have exactly the same number of sovereigns, halfsovereigns, shillings, and sixpences; how many is there of each, the amount being 6 guineas?

Ans. 4. (17.) In paying an account of £6 in half-sovereigns and half-crowns, how many of each must be used, so that the number of the latter shall be double that of the former?

Ans. 8 and 16. (18.) My purse and money together are worth £1, and the money is four times the value of the purse; how much is there in it? Ans. 16s.

(19.) My coat and hat together are worth £3, 10s., and the hat is one-fourth the value of the coat; find the value of each. Ans. The hat is worth 14s., and the coat, 56s. (20.) A boy's age is one-fourth of his father's, and he has a brother one-fifth of his own age; the ages of the three amount to 52: find the age of each.

Ans. 40, 10, and 2. (21.) A's age is double of B's, and B's is triple of C's; the sum of all their ages is 100: what is the age of each? Ans. A's 60, B's 30, C's 10. (22.) A son's age is one-fourth of his father's, but three years ago it was one-sixth; what is the age of each?

Ans. Son's, 7; father's, 30.

(23.) A man at the time of his marriage was twice as old as his wife, but after 20 years her age was two-thirds of his; ; what were their ages at marriage?

Ans. The woman's 20, and the man's 40. (24.) The ages of two brothers differ by 2 years, and when added together amount to the age of their father; but if the father's age be increased by half that of the elder brother and one-third that of the younger, it will amount to 71: what is the age of each? Ans. 24, 26, and 50.

(25.) A boy is one-third the age of his father, and has a brother one-sixth of his own age; what is the age of each, the ages of all three amounting to 50?

Ans. 36, 12, and 2. (26.) The age of one of two brothers is double that of the other, and the one is as much above 12 as the other is below it; what is the age of each? Ans. 8 and 16.

(27.) A's age is double that of B's, and 10 years ago he was three times as old; find A's age.

Ans. 40.

(28.) Five years ago A's age was three times that of B, but in five years more he will be only twice as old; what is A's age?

Ans. 35. (29.) The ages of A and B together amount to 100, but if B's age be doubled it will exceed A's by five years; find the age of each. Ans. A's 65, and B's 35.

(30.) A shepherd had in his flock four times as many sheep as lambs; but having sold fourscore sheep, and bought onescore of lambs, the number of the former is now only three times that of the latter; how many of each had he at first? Ans. 28 score of sheep, and 7 score of lambs. (31.) A post is one-fourth of its length in mud, one-third in water, and 10 feet above water; what is its whole length? Ans. 24 feet.

(32.) Having paid away one-fourth of my money, and then two-fifths of what remained, I had £54 left; how much had I at first? Ans. £120. (33.) Having paid away half of my money, two-thirds of the remaining half, and three-fourths of what then remained, I had £10 left; what had I at first? Ans. £240.

(34.) A class consists of three times as many boys as girls; but when three of each have left, there remain four times as many boys as girls: how many of each are in the class? Ans. 27 boys and 9 girls. (35.) The product of two numbers is 80; but if the greater

be increased by 1, the product is increased by 8: what are the numbers? Ans. 10 and 8. (36.) Find two consecutive numbers such that threefourths of the less shall be equal to two-thirds of the greater. Ans. 8 and 9. (37.) Find two consecutive numbers such that the sum of the half and fifth parts of the first shall be equal to the sum of the third and fourth parts of the second.

Ans. 5 and 6. (38.) Find a fraction such that if 3 be added to its numerator it becomes, but if 1 be taken from its denominator it becomes. Ans.

(39.) Find a fraction such that if 1 be added to the denominator, or if 1 be taken from both numerator and denominator, the resulting fractions are each = 1. Ans. 13.

Ans..

(40.) Required a fraction such that if its numerator be added to both numerator and denominator, the resulting fraction = 1/ (41.) Required a fraction such that if its numerator be increased by half its denominator, and its denominator by half its numerator, the resulting fraction.

Ans. .

(42.) Required a fraction such that if m be added to its numerator it becomes, but if n be added to its denominator

it becomes

с

ď

Ans.

bcm + acn

bdm + bcn

(43.) Required a fraction such that if its numerator be increased by m times its denominator, and its denominator by n times its numerator, the resulting fraction =

Ans.

a

(44.) Find two numbers whose sum = 40, and difference Ans. 14 and 26.

= d.

= 12. (45.) Find two numbers whose sums, and difference Ans. (s+d), (s — d). (46.) Divide a into three parts such that the first shall be m times as great as the second, and the second m times as great as the third.

the men

(47.) A farmer has 50 reapers, men and women; are to receive 3s. each per day, and the women 2s. 6d.; their wages altogether amount to £7 per day: how many are there of each? Ans. 30 men and 20 women.

(48.) Give a general solution of the last question, supposing the number of reapers to be a, the wages of the men m shillings, and of the women n shillings per day, and the total amount p shillings per day.

α

P

women.

(49.) Divide the fraction into two parts, the sum of whose numerators shall be equal to the sum of their deno

(50.) A certain number of pounds, half-crowns, and shillings together amount to £14, 10s., and the number of halfcrowns is double that of the pounds, and half that of the shillings; find the number of each.

Ans. 10 pounds, 20 half-cr., and 40 shillings. (51.) If a be taken from n times a certain number, the remainder is = b; find the number.

a+b

Ans.

n

(52.) Find a number whose mth and nth parts together

= a.

Ans.

amn m + n

(53.) Find a number the difference of whose mth and th parts = a, m being greater than n.

Ans.

amn

m -n

(54.) Find a number such that if a be subtracted, and the remainder multiplied by a, the product is the same as when b is subtracted from the number, and the remainder multiplied by b. Ans. a + b.

(55.) Divide a into two parts, of which the one shall be the part of the other.

mth

a

Ans. + 1, and

m

am

m + 1'

(56.) Divide p pounds among three persons, giving the second m times, and the third n times as much as the first.

(57.) A labourer is engaged for n days, on condition that he receives p pence for every day he works, and pays q pence for every day he is idle. At the end of the time he receives a pence; how many days did he work, and how many was he idle?

Ans. He worked "q+a

and was idle

пр a
p+q

p + q'

days.

(58.) At a certain election a persons voted, and the successful candidate had a majority of b votes; how many voted for each? Ans. (a+b), and 1⁄2 (a—b).

(59.) If A can perform a piece of work in 8 days, and B in 12 days, how long will they take to finish it if they work both together? Ans. 4 days. (60.) If A can perform a piece of work in a days, B in b days, and C in c days, how long will they take working all together?

Ans.

abc ab + ac + be

CHAPTER V.

INVOLUTION.

43. Involution is the raising of a quantity to any required power. This is done by multiplying the quantity continually by itself, till it has been used as factor as often as there are units in the index of the required power.

EXERCISES.

Raise each of the following quantities to the fourth power: