12. A man meeting some beggars, gave each of them 4 pence, and had 16 pence left; if he had given them 6 pence apiece, he would have wanted 12 pence more for that purpose. How many beggars were there, and how much money had he?

Let x represent the number of beggars.

Ans. There were 14 beggars, and he had 6s. 13. A man had six sons, each of whom is 4 years older than his next younger brother; and the eldest is three times as old as the youngest. Required their ages. Ans. 10, 14, 18, 22, 26, 30.

14. Three persons, A, B, and C, make a joint contribution, which in the whole amounts to £76, of which A contributes a certain sum, B contributes as much as A and £10 more, and C as much as A and B both. Required their several contributions.

Ans. £14, £24, £38, respectively. 15. A boy, being sent to market to buy a certain quantity of meat, found that if he bought beef, which was 4 pence per pound, he would lay out all the money he was entrusted with; but if he bought mutton, which was 31 pence per pound, he would have 2 shillings left. How much meat was he sent for.

Ans. 48 pounds.

16. A man lying at the point of death left all his estate to his three sons, to be divided as follows: to A he gave one half of the whole wanting $500; to B one third; and to C the rest, which was $100 less than the share of B. What was the whole estate, and what was each son's share?

Let a represent the whole estate.

These together will be equal to the whole estate, which was represented by x.

I 2-500+ 十

3

.100 = x

3

Uniting 's and numbers in the first member,

The whole estate is $3600; the shares are $1300, $1200, and $1100, respectively.

17. A father intends by his will, that his three sons shall share his property in the following manner; the eldest is to receive 1000 crowns less than half the whole fortune; the second is to receive 800 crowns less than

of the whole; and the third is to receive 600 crowns less than of the whole. Required the amount of the whole fortune, and the share of each.

Ans. The whole fortune 28800 crowns, the shares 13400, 8800, and 6600, respectively.

18. A father leaves four sons, who share his property

in the following manner; the first takes 3000 livres less than one half the fortune; the second, 1000 livres less than one third of the whole; the third, exactly one fourth; and the fourth takes 600 livres more than one fifth of the whole. What was the whole fortune, and what did each receive?

Ans. The whole fortune was 12000 livres; the shares 3000 livres each.

19. In a mixture of copper, tin, and lead; 16 lbs. less than one half of the whole was copper; 12 lbs. less than one third of the whole was tin, and 4 lbs. more than one fourth of the whole was lead. of each was there in the mixture?

What quantity

Ans. 128 lbs. copper, 84 lbs. tin, and 76 lbs. lead. 20. A general having lost a battle, found that he had only 3600 men more than one half of his army left, fit for action; 600 more than one eighth of them being wounded, and the rest, which amounted to one fifth of the whole army, either slain or taken prisoners. Of how many men did his army consist before the battle ? Ans. 24000 men.

21. Seven eighths of a certain number exceeds four fifths of it by 6. What is that number? Ans. 80.

22. A and B talking of their ages, A says to B, one third of my age exceeds its fourth by 5 years. What was his age? Ans. 60 years.

23. A sum of money is to be divided between two persons, A and B, so that as often as A takes £9, B takes £4. Now it happens that A receives £15 more than B. What is the share of each?

Ans. A £27, and B £12.

24. In a mixture of wine and cider, 25 gallons more than half the whole was wine, and 5 gallons less than one third of the whole was cider. How many gallons were there of each ? Ans. 85 of wine, 35 of cider.

IV. 1. A man having some calves and some sheep, and being asked how many he had of each sort, answered, that he had 20 more sheep than calves, and that three times the number of sheep was equal to seven times the number of calves. How many were there of .each?

Let x denote the number of calves.

Then x + 20 will denote the number of sheep.

7 times the number of calves is 7 x; 3 times the number of sheep is 3x+60; for it is evident that to take 3 times x + 20, it is necessary to multiply both terms by 3.

By the conditions these must be equal,

7x=3x+60.

Subtracting 3x from both members,

4 x = 60

x= 15 number of calves. x+2035= number of sheep. Ans. 15 calves, and 35 sheep.

2. Two men talking of their ages, the first says,

your age is 18 years more than age is equal to three times mine. each.

mine, and twice your

Required the age of Ans. 36 and 54 years.

3. Three men, A, B, and C, make a joint contribution, which in the whole amounts to £276.

A con

tributes a certain sum, B twice as much as A and £12 more, and C three times as much as B and £12 more. Required their several contributions.

Ans. £24, £60, and £192, respectively:

4. A man bought 7 oxen and 11 cows for $591. For the oxen he gave $15 apiece more than for the cows. How much did he given apiece for each?

Let a denote the price of a cow.

Then the price of an ox will be x + 15.

11 cows at x dollars apiece will come to 11 x dollars. If one ox cost x + 15 dollars, 7 oxen will cost 7 times x+15, which is 7x+105.

The price of the oxen and of the cows added together will make $591, the whole price.

Uniting x's,

11x+7x+105 = 591

18x+105 591

Subtracting 105 from both members,

Dividing by 18,

18 x 486

x = 27= price of cows. x+15= 42 = price of oxen.

5. A man bought 20 pears and 7 oranges for 95 cents. For the oranges he gave 2 cents apiece more than for the pears. What did he give apiece for each? Ans. 3 cents apiece for the pears, and 5 for the oranges.

6. A man bought 20 oranges and 25 lemons for $1.95. For the oranges he gave 3 cents apiece more than for the lemons. What did he give apiece for each?

Ans. 6 cents for oranges, and 3 cents for lemons. 7. Two persons engage at play, A has 76 guineas,