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the quotient will be five times as much. Hence the quotient

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If be divided by c, the quotient will be ; if be divided by the dth part of c, the quotient will be d times as much.

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RULE TO DIVIDE A FRACTION BY A FRACTION.— -Multiply the numerator of the divisor by the denominator of the dividend, and the denominator of the divisor by the numerator of the dividend.

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13. If b shillings will buy ab yards of cloth; what will a c shillings buy?

Ans.

abac b

In the answer to the preceding question, we have abac

divided by b, which gives a a c, or a2c; for,

abac
b

is a ac multiplied by b, and that product divided by b, which gives aac upon the same principle, that five times 6, or 5 × 6 divided by 5 will make 6.

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1. Two men divide a dollars between them, so that one has m times as much as the other. What will represent what each has?

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3. Does the result of the addition in the second question, show that the answers to the first are correct?

4. If in the first question a = 20 and m values of x and of mx be?

= 4; what will the

5. State the 1st question, putting 20 instead of a and 4 instead of m.

6. The sum of two numbers is 40, and one is three times as many as the other. What are the numbers?

7. If you use 40 instead of a, and 3 instead of m; will the values of x and mx serve to find the answers to the sixth question? And if so, why?

8. Two men share 100 dollars between them, so that as often as one takes 1 the other takes 4. What will be each man's share?

9. What numbers must you use for a and m, so that the values of x and m x in the first question, will give the answer to the 8th?

10. Divide the number a into two such parts, that if one be multiplied by m and divided by n, it will be equal to the other.

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Multiply the numerator and denominator of the first fraction by n, that it may have the same denominator that the other has, and the sum of the two fractions will be

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12. Does the result of the additions in the 11th question

show that the answers to the 10th are correct?

13. If in the equations x =

we suppose n = 1; what do the equations become?

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n(n+m)

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These answers coincide with those found in the first question.

14. State the 10th question, using 1 instead of n, and observe if it coincides with the first.

15. There are two numbers, one of which is of the other; and the sum of the numbers is 40. What are the numbers?

16. State the 10th question, using 40 for a, 2 for m and 3 for n; and, observe if it coincides with the 15th question?

17. What numbers must you put in place of a, m and n, in the answers to the 10th question, in order to obtain the answers to the 15th question?

18. Two numbers make 36, and one is of the other. What are the numbers?

19. Two men divide 33 dollars between them, so that as often as one takes 5, the other takes 6. What is the share of each?

SECTION LXXVIII.

Generalization.

1. Divide the number a into three such parts, that the second may be m times, and the third n times as much as the

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1st. part,

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2. Add these three parts together, and observe if the result of the addition, shows the three parts found to be correct.

3. Divide the number 120 into three such parts, that the second shall be twice, and the third, three times as much as the first. What are the three parts?.

4. Three men share 50 dollars among them, so that as often as the first takes one dollar, the second takes 4, and the third What is the share of each?

5.

5. Three men have a dollars between them; the second has m times as many as the first, and the third, n times as many as the second. What will represent what each has?

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Verify the answers by adding the three parts together.

6. Read the 5th question, putting 90 in place of a, 2 in place of m, and 3 in place of n. What will the answer to the question then be?

7. Three men have a dollars; the second has m times as many as the first, and the third has n times as many as both the others. How many has each?

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8. Prove these to be the correct representatives of the parts,

by adding them together.

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