30. How much loaf-sugar, at 25 cents a pound, .nust be given in barter for 625 yards of cotton, at 13 cents a yard?

31. A man sells 17 acres of land at 30 dollars per acre, and is to receive his pay in flour at 5 dollars a barrel. How many barrels must he receive?

32. How many gallons of wine, at $1,75, must I give in exchange for 1295 pounds of shot, at 5 cents a pound?

33. Two merchants, A and B, engage in a speculation, and gain $750, of which B is to have 4 times What is the share of each?

as much as A.

In this question, a given sum is to be divided between two persons in unequal proportions. It may be generalized as follows:

Represent the sum to be divided by a.

Let x A's share,

Now, whatever sum may be represented by a, it is evident that A is to receive part of it, and B & of it If a, the sum to be divided, be $1000,200,

4

If a represent $1500, = 300, and 1a

= 1200. This question may be still farther generalized, if it be announced as follows:

Two merchants, A and B, engage in a speculation, and gain a dollars, of which B is to have b times as much as A. What is the share of each?

Let x A's part of the gain.
Then bx = B's part.

And xbxa, by the question,
or (1 + b) x = α,

In the equation x + bxa, the first member, x + bx, is equal to (1 + b) x; therefore, (1 + b) may be regarded as the coefficient of x.

34. By this formula, divide $1200 between two men in such a manner, that one shall have 5 times as much as the other.

Here, a 1200, and b = 5.

35. A father says to his son, "I am four times as old as you, and the sum of our ages is 85 years." Required the age of each.

36. Divide 2592 dollars between two men in such a manner, that one shall have 7 times as much as the other.

37. Says A to B, "Our estates together are worth 36465 dollars; but my estate is worth only one fourth part as much as yours." What is the value of each estate?

38. A merchant sold a quantity of blue, black and mixed broadcloths, for $5000; the black at $10, the blue at $11, and the mixed at $7 per yard. There were twice as many yards of the blue as of the black, and as many yards of the mixed as of both the others. How many yards of each color did he sell?

Let x = the yards of black,

Then 10 x 22 x + 21 x = 5000, by the question,

And

x =

2 x

5g 943, yards of black.
1883, yards of blue.

3 x 283, yards of mixed.

All the answers to the above question contain fractions. I now wish to ascertain what sum I must substitute for $5000, in order that the answers may consist of whole numbers. For this purpose, I will solve the question by using a instead of the given sum. If I denote the yards of black by x, the yards of

blue by 2 x, and the yards of mixed by 3 x, as above, I shall have the following equation:

Any number, divisible by 53 without a remainder,— that is, the product of 53 multiplied by any number whatever, may be put in the place of a, and the answer will be free from fractions.

Suppose, for instance, the value of the whole quantity sold to be 2491 dollars, which is the product of 53 multiplied by 47. Then he would have sold 47 yards of black, 94 yards of blue, and 141 yards of mixed. 39. In a certain school, of the pupils learn navilearn algebra, and all the

gation, learn geometry,

rest, a, learn arithmetic. How many pupils are there

or 60 x

60 x

15 x + 12 x + 10 x + 60 a;
37 x + 60 a, by addition;

23 x = 60 a, by transposition;

Instead of a, in the question, use any number that can be divided by 23 without a remainder, and the answer will be a whole number.

Substitute, for instance, 46 in the place of a.

ANS. 120 pupils.

The last two examples are intended to exhibit the

manner in which questions may be prepared, whose answers shall be free from fractions.

40. Three men, A, B and C, trade in company, and gain $1350. Now, if A put into the joint stock $7 as often as B put in $6, and B put in $6 as often as C put in $5, what is each man's share of the gain?

ANS. A, $525; B, $450; C, $375

To generalize this question, we will suppose A put in a dollars as often as B put in b dollars; and that B put in b dollars as often as C put in c dollars; that is, we will use the letters a, b, c, instead of the numbers 7,6, 5. Let the amount gained be represented by g A's share of the gain.

Let x

b x

сх

Then x +*+* = g, by the question.

a

a