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find the result to be t, and that is equal to 1's is evident from the fact, that the ratio of 5 to 15 is equal to the ratio of 1 to 3. And, as the value of a fraction depends on the ratio, which the numerator bears to the denominator, if their ratios are equal, the fractions are also equal. Q. e. d. Hence the following

RULE.

Divide the numerator and denominator by any number that will divide them both without a remainder; and so continue until no number will divide them but unity. Or, divide the numerator and denominator by the greatest common

measure.

Ans. } 2. Reduces to its lowest terms.

Ans. . 3. Reduce to its lowest terms.

Ans. $. 4. Reduce je to its lowest terms.

Ans.. 5. Reduce 96 to its lowest terms.

Ans. 1. 6. Reduce 101 to its lowest terms,

Ans. ing. 7. Reduce to its lowest terms.

Ans. 7. 8. Reduce to its lowest terms.

Ans. Ziit 9. Reduce 789 to its lowest terms. 10. What is the lowest expression of t? Ans. 175.

386

III. To reduce mixed numbers to improper fractions.

MENTAL OPERATIONS.

OPERATION.

1. In 3 dollars how many halves ? How many thirds ? 2. In 7 apples how many tenths ? How many twelfths ? 3. In 83 dollars how many sevenths ? 4. In 3 oranges how many fourths ? 5. In 971 gallons how many elevenths ? 6. In 7 quarts how many fifths of quarts ?

We analyze this question by saying, as 7 there are 5 fifths in one quart, there will

5 be 5 times as many fifths as quarts ; there35

fore, in seven quarts and three fifths, there 3

will be 38 fifths, which should be expressed

thus, 3. And this fraction, by definition 38

2d, on page 76, is an improper fraction. 5

Hence the following

RULE.

Multiply the whole number by the denominator of the fraction, and to the product add the numerator, and place their sum over the denominator of the fraction. 7. Redace 8 și to an improper fraction.

Ans. 41. 8. Reduce 1575 to an improper fraction.

Ans. 483. 9. In 187 how many ninths ?

Ans. 169. 10. In 16111'how many one hundred and seventeenths ?

Ans. 18148, 11. Change 43114 to an improper fraction Ans. 3142 12. What improper fraction will express 2715?

Ans. 362 13. Change 11lstī to an improper fraction ?

Ans. 14112.

IV. To change improper fractions to integers or whole numbers.

MENTAL OPERATIONS.

1. How many dollars in 4 halves ? In 5 halves ? In 6

halves ? In 7 halves ? In 12 halves ? In 19 halves ? 2. How many dollars in 5 quarters? In 9 quarters ? 3. How many dollars in 10 eighths ? In 20 eighths ?

FOR THE SLATE.

OPERATION.

4. How many dollars in it dollars ?

Ans. 216 This question may be analyzed by say16) 37 (276 ing, as 16 sixteenths make one dollar, 32

there will be as many dollars in 37 six5 teenths as 37 contains 16, which is 276 times, =

$ 276. This answer is called a mixed number by definition 5th, page 77. Hence we see the propriety of the following

RULE.

Divide the numerator by the denominator, and if there be a remainder, place it over the denominator at the right hand of the integer. 5. Change 44 to a mixed number.

Ans. 10%

6. Change 111 to a mixed number.
7. Change 1792 to a mixed number,
8. Reduce 1000 to a mixed number.
9. Reduce 37% to a whole number.
10. Change 567 to a whole number.
11. What is the value of 375 ?
12. What is the value of 345?
13. Change 125 to an improper fraction.

Ans. 10111:
Ans. 188
Ans. 1429.

Ans. 1.
Ans. 567.
Ans.
Ans.
Ans. 125

V. To change or reduce compound fractions to simple fractions.

MENTAL OPERATIONS.

1. What part of an orange is a { of a half? 2. What part of an apple is a { of a half ? 3. What part of a bushel is a of a peck ? 4. What part of a quart is a f of a pint ?

FOR THE SLATE.

Ans.

OPERATION.

5. What is of 7 ?

This question may be analyzed A XH= Ans. by saying, if It of an apple be

divided into 5 equal parts, that one of these parts is 's of an apple ; and, if } of be o's, it is evident, that of 11 will be 7 times as much. 7 times it is 375 ; and, if t of IT be 35 of 11 will be 4 times as much. 4 times 35 is s.

We therefore induce the following

RULE.

Change mixed numbers and whole numbers, if there be any, to improper fractions; then multiply all the numerators together for a new numerator, and all the denominators together for a new denominator ; the fraction should then be reduced to its lowest terms.

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7. What is of of 7?

OPERATION.

* X 1 X 1 = 4* = 58 Ans. 8. What is of of off ?

Ans. _756

1928 = 1476 . 9. Change 17 of of of 20 of 7 to a simple fraction.

Ans. 22. Note ). If there be numbers in the numerators and denominators, that be alike, an equal number of the same value may be cancelled. 10. Reduce of 4 of ☆ of 11 to a simple fraction.

STATEMENT.

CANCELLED.

3 X 4 X 5 X 7 3 X4 X5 X 7 3

Ans. 4 X 5 X 7 XII 4 X 5 X 7 X 11 11 In performing this question, we perceive that there is a 4 and 5 and 7 among the numerators, and also the same numbers among the denominators ; these we cancel before we commence the operation. 11. Required the value of of t of 17 of 37 of 54.

STATEMENT.

3 x 4 x 11 x 17 x 23
5 x 11 x 17 x 23 x 4

CANCELLED.

3 X 4 X11 X 17 X 23 3

Ans. 5 X 11 X 17 X 23 X 4 5 12. Reduce ț of g of ii of ý of to a simple fraction.

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1 X 8 X 9 X 5 X 3 1 X 8 X 9 X 5 X 3 3

Ans. 5 X 9 X 11 X 8 X 7 $ X 9 X 11 X 8 X 7 77 13. Reduce of of 7 of % of 44 to a simple fraction.

Ans. NOTE 2. When there are any two numbers, one in the numerators and the other in the denominators, which may be divided by a num. ber without a remainder, the quotients arising from such division may be used in the operation of the question instead of the original numbers.

14. Reduce té of g of 11 to a simple fraction.

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In performing this question, we find that the 15 among the numerators and the 9 among the denominators may be divided by 3, and that the quotients will be 5 and 3. We write the 5 above the 15, and the 3 below the 9. We also find an 8

among

the numerators, and a 16 among the denominators, which may be divided by 8, and that the quotients will be 1 and 2. We write the l over the 8, and the 2 under the 16. We then multiply the 5, and 1, and 7 together for a new numerator, and the 2, and 3, and 11 together for a new denominator. That the result will be the same by this process as by the other, is evident from the fact, that the multiples of any number have the same ratio to each other, as the numbers themselves.

This cancelling principle, when well understood, will often facilitate the operations of many questions, when the divisors and dividends have a common denominator.

15. Reduce ii off of z3 of 95 to a whole number.

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3 11 8 X 22X15X77 8 X 22 X 15 X77 3

=3 Ans. 11 X 35 X 22 X 8 11 X 35 X 22 X 8 1

5 16. Divide the continued product of 18, 24, 27, and 30, by the continued product of 20, 21, 9, and 10.

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17. Divide the continued product of 20, 19, 18, 17, 16,

15, 14, 13, 12, and 11, by the continued product of 10, 9, 8, 7, 6, 5, 4, 3, 2, and 1.

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