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. 10.

Prob. 21. Divide the number 197 into two such parts that four times the greater may exceed five times the less by 50. Ans. 82 and 115.

Prob. 22. Divide the number a into two such parts that m times the greater may exceed n times the less by b.

When n=1, this Problem reduces to Problem 18.
When b=0, this Problem reduces to Problem 24.

Prob. 23. A prize of 2329 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?

Beginners almost invariably 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 685 and 1644 dollars.

Prob. 24. Divide the number a into two such parts that the first part may be to the second as m to n.

Prob. 25. What number is that whose third part exceeds its

Prob. 26. Find a number such that when it is divided successively by m and by n, the difference of the quotients shall be a.

Prob. 27. A gentleman has just 8 hours at his disposal; how far may he ride in a coach which travels 9 miles an hour, so as to return home in time, walking back at the rate of 3 miles an hour?. Ans. 18 miles. Prob. 28. A gentleman has just a hours at his disposal; how far may he ride in a coach which travels m miles an hour, so as to return home in time, walking back at the rate of n miles an hour?

Prob. 29. A gentleman divides a dollar among 12 children, giving to some 9 cents each, and to the rest 7 cents. How many were there of each class?

Prob. 30. Divide the number a into two such parts that if the first is multiplied by m and the second by n, the sum of the products shall be b.

Ans.

b-na ma-b

m-n m—n

Prob. 31. If the sun moves every day 1 degree, and the moon 13, and the sun is now 60 degrees in advance of the moon, when will they be in conjunction for the first time, second time, and so on?

Prob. 32. If two bodies move in the same direction upon the circumference of a circle which measures a miles, the one at the rate of n miles per day, the other pursuing at the rate of m miles per day, when will they be together for the first time, second time, etc., supposing them to be 6 miles apart at starting?

It will be seen that this Problem includes Prob. 20.

Prob. 33. Divide the number 12 into two such parts that the difference of their squares may be 48.

Prob. 34. Divide the number a into two such parts that the difference of their squares may be b.

a2-b a2+b

Ans.

; 2a

2a

Prob. 35. 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?

Prob. 36. Divide the number a into three parts, which shall be to each other as m:n: p.

Ans.

ma

na

ρα

; m+n+p' m+n+p' m+n+p'

When p=1, Prob. 36 reduces to the same form as Prob. 8. Prob. 37. 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 80 pounds, which shall be worth 60 cents?

Ans. 50 pounds at 72 cents, and 30 pounds at 40 cents. Prob. 38. A grocer has two kinds of tea, one worth a cents per pound, the other b cents. How many pounds of each must be taken to form a mixture of ʼn pounds, which shall be worth c cents?

Ans.

n (c-b)

a

-b

pounds at a cents,

Prob. 39. 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?

Prob. 40. A can perform a piece of work in a days, B in b days, and C in c days. In what time will they perform it if all work together?

Ans.

abc
ab+ac+bc

days.

Prob. 41. 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?

Therefore, A, B, and C together can perform

of the work If we put x to represent the time in which they would all finish it, then they would together perform

in one day; that is, they can finish it in 24 days.

and we should have

part of the work in one day,

27 +36 +64 = 23.

Prob. 42. A and B can perform a piece of labor in a days; A and C together in b days; and B and C together in c days. In what time could they finish it if all work together?

Ans.

2abc

ab+ac+be days. This result, it will be seen, is of the same form as that of Problem 40.

Prob. 43. A broker has two kinds of change. It takes 20 pieces of the first to make a dollar, and 4 pieces of the second to make the same. Now a person wishes to have 8 pieces for a dollar. How many of each kind must the broker give him?

Prob. 44. A has two kinds of change; there must be a pieces of the first to make a dollar, and b pieces of the second to make the same. Now B wishes to have c pieces for a dollar. How many pieces of each kind must A give him?

Ans.

a (c-b)

α

of the first kind;

b(a-c) of the second.

a-b

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 super

E

fluous. 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.

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

Prob. 46. Divide the number a into four such parts that the first increased by m, the second diminished by m, the third multiplied by m, and the fourth divided by m, shall all be equal.

Ans.

ma

ma

a

m2a

(m+1) 2 :−m; (m+1)2+m; (m+1)2; (m+1)2°

Prob. 47. A merchant maintained himself for three years at an expense of $500 a year, and each year augmented that part of his stock which was not thus expended by one third thereof. At the end of the third year his original stock was doubled. What was that stock?

Prob. 48. A merchant supported himself for three years at an expense of a dollars per year, and each year augmented that part of his stock which was not thus expended by one third thereof. At the end of the third year his original stock was doubled. What was that stock?

148a

Ans.

10

Prob. 49. 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?

Prob. 50. 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?

a-nb

Ans.