SECTION III.

To remove Denominators.

1. A man received $18 towards a debt, which was only of the sum due. What was the whole debt?

Let x denote the amount of the debt.

I

Then must be, and 2 must be of it.

3

3

It has already been shown, [See Sec. I.] that, if all the terms of an equation be either multiplied or divided by the same quantity, the equality of the members will not be affected; therefore, to remove the denominator of the first term,

Multiply by 3; 2 x = 54.

Divide by 2; x = 27.

ANS. $27.

We first multiply the whole equation by 3, the denominator of the fraction, and then find the value of · x as before. It has been shown, in a former chapter, [See Chap. VI. Sec. VI.] that a fraction is multiplied by its denominator, when that denominator is removed.

2. What number is that, and

Let x indicate the number.

of which are 44?

Then is half of it,

We first free the equations from fractions, by multiplying all the terms by the denominators, one at a time, and then find the value of x as before.

3. What number is that, which, being increased by of itself, becomes 146?

1, 2, and

Multiply by 7; 28 x + 14

Multiply by 2; 4 x + 2 x

x

+ 21x+8x+2x=4088.

Add the x's; 73 x = 4088.

Divide by 73; x = 56.

by dividing their denominators by 2; and we multi

ply the fractions

nominators.

8 x 7

and 27, by removing their de

To free an equation from fractions, we multiply all the terms by all the denominators, taking them in such be most convenient.

order

as may

4. Says A to B, "If to my age and

of my age be added, the sum will be 81; what is my age?"

Let x denote A's age.

3

Then x++= 81, by the question.

* 5. If of a ship cost $4000, what is the whole ship worth?

6. A man, driving his geese to market, was met by another, who said, "Good morrow, master, with your hundred geese." Said he, "I have not a hundred ; but if I had as many more, and half as many more, and two geese and a half, I should have a hundred." How many had he?

x + x + 1/ = 100 21.

7. A post, standing in a pond, is of its length under water, and above water, it being 14 feet from the top of the post to the bottom of the pond. What is the whole length of the post?

8. What number is that, whose sixth part exceeds its eighth part by 20?

*9. A man, having spent three fifths of his estate, Had $978 left. How much had he at first?

of it

10. A man spent of his life in England, in Scotland, and the remainder of it, which was 20 years, in the United States. What was his age?

11. What number is that, than of it by 21?

of which is greater

12. A man sold 75 bushels of wheat to two persons; to one, , and to the other, of all he had. How many bushels had he?

13. A man gave to three poor persons $6; to the first,, to the second, 4, and to the third, of all the money he had in his pocket. How much had he?

14. A and B divide $320 between them, of which B has three times and as much as A.

has each?

How much

15. A says to B, "Your age is twice and of my age, and the sum of our ages is 54 years."

the age of each?

What is

*16. If you divide $50 between two persons, giv ing one as much as the other, what will be the share of each?

17. A stranger in Boston spent, the first day, } of the money he brought with him; the second day, ; and the third day, ; when he had only $26 left. How much money did he bring?

18. A man, having invested of his property in bank stock, by which he lost of the sum invested, had stock worth $723 remaining. How much property had he at first?

19. A merchant retired from business when he had passed of his life, during of which period he had been engaged in trade, having commenced at the age of 21. At what age did he die?

20. The age of A is that of B, and the age of C is that of A, and the sum of all their ages is 120 years. What is the age of each?

21. A farmer wishes to mix 90 bushels of proven

der, consisting of barley, oats and corn, so that there may be as many bushels of oats as of corn, and as many bushels of barley as of oats. How many bushels of each sort must he use?

22. Three boys spent 98 cents for fruit. B spent as much as A, and C spent as much as B. What did each spend?

23. If you divide $75 between two men, in the proportion of 3 to 2, what will each man receive?

One man will receive 2 dollars as often as the other receives 3; that is, one will receive as much as the other.

* 24. Divide 84 into two numbers, which shall be to each other as 7 to 5.

25. What two numbers are to each other as 4 to 9, their sum being 91?

* 26. Three men trade in company, and gain $780. A put in $2 as often as B put in $3 and C put in $5. What part of the gain must each one receive?

27. Divide 234 into 4 numbers, which shall be to each other in the proportion of 5, 6, 7 and 8.

28. Divide 36 into three such parts, that of the