Prob. 12. A draper bought three pieces of cloth, which together measured 111 yards. The second piece was 11 yards longer than the first, and the third 17 yards longer than the second. What was the length of each?

Ans. 24, 35, and 52 yards respectively.

Prob. 13. A hogshead which held 92 gallons was filled with a mixture of brandy, wine, and water. There were 10 gallons of wine more than there were of brandy, and as much water as both wine and brandy. What quantity was there of each?

Ans. Brandy 18 gallons, wine 28 gallons, and water 46 gallons.

The following is a generalization of the preceding problem.

Prob. 14. Divide the number a into three such parts that the second shall exceed the first by m, and the third shall be equal to the sum of the first and second.

Prob. 15. A farmer employed four workmen, to the first of whom he gave 2 shillings more than to the second, to the second 3 shillings more than to the third, and to the third 4 shillings more than to the fourth. Their wages amounted to 32 shillings. What did each receive?

Ans. They received 12, 10, 7, and 3 shillings respectively.

Prob. 16. A father divided a certain sum of money among his four sons. The third had 11 shillings more than the fourth, the second 16 shillings more than the third, and the eldest 19 shillings more than the sec ond; and the whole sum was 16 shillings more than 6 times the sum which the youngest received. How much had each?

Ans. 34, 45, 61, and 80 shillings respectively. (104.) Problems which involve several unknown quantities may often be solved by the use of a single unknown letter. Most of the preceding examples are of this kind. In general, when we have given the sum or difference of two quantities, both of them may be expressed by means of the same letter. For the difference of two quantities added to the less must be equal to the greater; and if one of two quantities be subtracted from their sum, the remainder will be equal to the other.

Prob. 17. At a certain election 25,000 votes were polled, and the candidate chosen wanted but 2000 of having twice as many votes as his opponent. How many voted for each?

Let x the number of votes for the unsuccessful candidate.

Then 25,000-x= the number the successful one

had,

and

25,000-x+2000=2x.

Ans. 9000 and 16,000.

The following is a generalization of the preceding problem.

QUEST.-When we have given the sum and difference of two quan tities, how may each of them be expressed?

Prob. 18. Divide the number a into two such parts that one part increased by b shall be equal to m times the other part.

Prob. 19. A train of cars moving at the rate of 20 miles per hour, had been gone three hours when a second train followed at the rate of 25 miles per hour. In what time will the second train overtake the first?

Let x the number of hours the second train is in motion,

and x+3= the time of the first train.

Then 25x= the number of miles traveled by the second train,

and 20(x+3)= the miles traveled by the first train. But at the time of meeting they must both have traveled the same distance.

Proof. In 12 hours, at 25 miles per hour, the second train goes 300 miles, and in 15 hours, at 20 miles per hour, the first train also goes 300 miles; that is, it is overtaken by the second train.

The following is a generalization of the preceding problem.

Prob. 20. Two bodies move in the same direction from two places at a distance of a miles apart, the one at the rate of n miles per hour, the other pursuing

at the rate of m miles per hour. When will they meet?

This problem, it will be seen, is essentially the same as Prob. 8.

Prob. 21. A vintner fills a cask containing 86 gallons with a mixture of brandy, wine, and water. There are 18 gallons of water more than of brandy, and 14 more of wine than of water. How many are there of each?

Ans. 12 gallons of brandy,

30 gallons of water,

44 gallons of wine.

Prob. 22. A gentleman gave 34 shillings to two poor persons; but he gave 4 shillings more to one than to the other. What did he give to each?

Ans. 15 and 19 shillings.

Prob. 23. What two numbers are those whose sum

is 66 and difference 18?

Ans. 24 and 42.

Prob. 24. Two persons began to play with equal sums of money. The first lost 20 shillings, the other won 28 shillings; and then the second had twice as many shillings as the first. What sum had each at first? Ans. 68 shillings.

Prob. 25. A farmer has two flocks of sheep, each containing the same number. From one of them he sells 24, and from the other 129, and finds just twice as many remaining in one as in the other. How many did each flock originally contain?

Ans. 234.

Prob. 26. Divide the number 181 into two such parts that four times the greater may exceed five times the less by 49.

Ans. 75 and 106.

Prob. 27. Divide the number a into two such parts that m times the greater may exceed n times the less by b.

Prob. 28. A prize of 2125 dollars was divided between two persons, A and B, whose shares were in the ratio of 5 to 12. What was the share of each?

Ans.

(105.) Beginners uniformly put x to represent one of the quantities sought in a problem; but a solution may often be very much simplified by pursuing a different method. Thus, in the preceding problem we may put x to represent one fifth of A's share. Then 5x will be A's share, and 12x will be B's, and we shall have the equation

consequently, their shares were 625 and 1500 dollars. The following is a generalization of the preceding problem.

Prob. 29. Divide the number a into two such parts that the first part may be to the second as m to n.

Prob. 30. Divide the number 73 into two such parts

QUEST. Is it always best to represent the required quantity by