that the difference between the greater and 77. may equal three times the difference between the less and 40.

Ans. 29 and 44.

Prob. 31. What number is that whose third part exceeds its fourth part by 15?

Therefore the number =12x15=180.

The following is a generalization of the preceding problem.

Prob. 32. Find a number such that when it is divided successively by m and by n, the difference of the quotients shall be a.

Prob. 33. What two numbers are as 2 to 3, to each of which, if four be added, the sums will be as 5 to 7 ? Let 2x and 3x represent the required numbers.

Then

2x+4:3x+4::5:7.

But when four quantities are proportional, the product of the extremes is equal to the product of the means. Hence

And the required numbers are 16 and 24.

Prob. 34. A sum of money is to be divided between two persons, A and B, so that as often as A receives 9 pounds, B takes 4. Now it happens that A receives

QUEST.-How may a proportion be reduced to an equation?

20 pounds more than B.

shares?

What are their respective

Ans. A receives 36 pounds and B 16 pounds. Prob. 35. A merchant bought two casks of beer, one of which held exactly three times as much as the other. From each of these he drew four gallons, and then found that there were four times as many gallons remaining in the larger as in the other. How many were there in each at first?

Ans. 36 and 12 gallons respectively

Prob. 36. A gentleman divides two dollars among 12 children, giving to some 18 cents each, and to the rest 14 cents. How many were there of each class? Ans. 8 of the first class and 4 of the second.

Prob. 37. A fish was caught whose tail weighed 9 pounds. His head weighed as much as his tail and half his body, and his body weighed as much as his head and tail. What did the fish weigh?

Ans. 72 pounds.

Prob. 38. If the sun moves every day one degree, and the moon thirteen, and the sun is now 60 degrees in advance of the moon, when will they be in conjunc tion for the first time, second time, and so on?

Ans. In 5 days, 35 days, 65 days, etc. Prob. 39. Divide the number 15 into two such parts that the difference of their squares may be 45.

Ans. 9 and 6.

The following is a generalization of the preceding problem.

Prob. 40. Divide the number a into two such parts that the difference of their squares may be b.

Prob. 41. The estate of a bankrupt, valued at 21,000 dollars, is to be divided among three creditors according to their respective claims. The debts due to A and B are as 2 to 3, while B's claims and C's are in the ratio of 4 to 5. What sum must each receive?

Ans. A receives 4800 dollars;
B receives 7200 dollars;

C receives 9000 dollars.

Prob. 42. A grocer has two kinds of tea, one worth 72 cents per pound, the other 40 cents. How many pounds of each must be taken to form a chest of 48 pounds which shall be worth 60 cents?

Ans. 30 pounds at 72 cents, and 18 pounds at 40

cents.

Prob. 43. A can perform a piece of work in 6 days, B can perform the same work in 8 days, and C can perform the same work in 24 days. In what time will they finish it if all work together?

Ans. 3 days.

Prob. 44. There are three workmen, A, B, and C. A and B together can perform a piece of work in 27 days, A and C together in 36 days, and B and C together in 54 days. In what time could they finish it if all worked together?

A and B together can perform of the work in one day;

A and C together can perform of the work in

one day;

B and C together can perform of the work in one day.

Therefore, adding these three quantities,

2A +2B+2C can perform +3+3 in one day.

27 36

= in one day.

Therefore, A, B, and C together can perform of the work in one day; that is, they can finish it in 24 days. If we put x to represent the time in which they would all finish it, then they would together perform

1

part of the work in one day, and we should have

1 1 1 2

27+36 +54 x

From which equation we find

x=24, Ans.

Prob. 45. Divide the number 45 into four such parts that the first increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2, shall all be equal.

In solving examples of this kind, several unknown quantities are usually introduced, but this practice is worse than superfluous. The four parts into which *45 is to be divided may be represented thus:

for if the first expression be increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2, the result in each case will be x. The sum of the four parts is 4x, which must equal 45. Hence,

X

10.

Therefore, the parts are 8, 12, 5, and 20.

Prob. 46. A father, aged 54 years, has a son aged

9 years. In how many years will the age of the father be four times that of the son?

Ans. 6 years.

Prob. 47. The age of a father is represented by a, the age of his son by b. In how many years will the age of the father be n times that of the son?

Prob. 48. It is required to divide the number 36 into three such parts that one half of the first, one third of the second, and one fourth of the third may be equal to each other.

Ans. 8, 12, and 16.

Prob. 49. Divide the number 50 into two such parts that the greater increased by 5, may be to the less diminished by 5, as 7 to 3.

Ans. 30 and 20.

Prob. 50. What number is that to which, if 1, 4, and 10 be severally added, the first sum shall be to, the second as the second to the third?

Ans. 2.