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3. A compound fraction is the fraction of a fraction, coupled by the word, of, thus; of, of of §.

4. A mixed number is composed of a whole number and a fraction joined together, thus; 84, 163, 193, &c.

A whole number may be expressed as a fraction by drawing a line under it and putting 1 for a denominator; thus, 6, and 12=Y, &c.

PROBLEM I.

To reduce Fractions to their lowest terms.

The numerator and denominator together are called terms of a fraction; and may often be changed without altering the value of a fraction; thus, take an orange, or any thing, and divide it into 2 equal parts, and 1 of these parts will be of the orange again, if we divide it into 4 equal parts, it is evident that 2 of those parts (2) will be just of the orange and if we divide it into 8 equal parts, 4 of these parts () will be just equal to of the orange and the fractions,, and, are all equal in value, but expressed in different terms. Hence the terms of fractions may be changed without altering the value of the fraction; for if we multiply both the terms of the fraction by 2 it becomes 2, which is equal to : again, if we divide the terms by 2, the fraction will be, which is expressed in its lowest terms possible.

RULE.

i. Divide the terms of the fraction by any number that will

divide them both without a remainder.

2. Divide these quotients again in the same manner, and so on, until no number greater than 1 will divide them:

EXAMPLES.

1. Reduce to its lowest terms.

Thus 9), and 5)15=3, the answer.

2. Reduce 132 to its lowest terms.

3. Reduce

4. Reduce

264

432 to its lowest terms.

576

147 to its lowest terms.

Ans.

Ans.

Ans.

168

49 to its lowest terms.

Ans.

343

Ans:

192

7. Reduce

77 to its lowest terms.

Ans.

5. Reduce
6. Reduce 16 to its lowest terms.

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To change a Whole or Mixed Number to an Improper

Fraction.

RULE.

Multiply the whole number by the denominator of the fraction and to the product add the numerator; this sum written over the denominator will form the fraction required.

EXAMPLES.

1. In 273 dollars how many fourths of a dollar?

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$14 fourths of a dollar, and 27 dollars=27 times 4 or 108 fourths, and 3 fourths added to 108 fourths make 111 fourths=111 the Ans.

2. In 36ğ dollars, how many eighths of a dollar?

3. Reduce 45 to ninths.

Ans. 293.

Ans. 405.

4. Reduce 85 to an improper fraction, that is, reduce it to sixths.

5. Reduce 33 to an improper fraction.

Ans. 53.

Ans. 100.

6. Reduce 28 to a fraction having 12 for a denominator,

that is, reduce it to twelfths.

7. Reduce 45 to fifths.

95

8. Reduce 6125 to an improper fraction.

140

9. Reduce 84 to an improper fraction.

10. What improper fraction is equal to 5618 ?

Ans. 336

12.

Ans.239. Ans. 8635 Ans. 933

140

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Ans. 2259.

11. What improper fraction is equal to 148?

12. What improper fraction is equal to 225?

Ans. 1489.

Ans. 1893.

PROBLEM III.

To change an Improper Fraction to a Whole or Mixed

Number.

RULE.

Divide the numerator by the denominator, and the quo tient will be the value of the fraction.

EXAMPLES.

1. In 4 of a dollar, how many dollars?

Operation.

$1=6)45

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of a dollar are equal to 1 dollar, and 6 is contained in 45, 7 times and of another time; therefore the answer is 7 dollars 7 dollars.

2. Find the value of 100 of a cent. 3. Find the value of 249 of a cwt. 4. Reduce 53 to a mixed number. 5. Reduce 405 to a whole number.

9

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Ans. 33 cts. Ans. 49 cwt. Ans. 8.

Ans. 45.

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To Multiply a Whole Number by a Fraction.

RULE.

I. Divide the whole number by the denominator of the fraction, (when it can be done without a remainder,) and multiply the quotient by the numerator; or,

II. Multiply the whole number by the numerator of the fraction and divide the product by the denominator.

EXAMPLES.

1. What is the product of 48 multiplied by ??

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By this example we see, there are two ways of multiplying a whole number by a fraction, and that both methods produce the same result. Thus, by the first method, we get

of 48, and this repeated 3 times is evidently equal to 2, for 3 times of any number is equal to 2 of that number. By the second method, we repeat 48, 3 times, and then take of that product, which is the same as 3 times of 48. 2. At 25 dollars per acre, what is the cost of 1 of an acre of land? Ans. 237 dolls. 3. If a ship sail 246 miles a day, how far will she sail

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16

Ans. 1914 miles.
Ans. $1537,963.

Ans. 150.

Ans. 450

7. The interest of $750 for 1 year, is $45; what is the interest on the same sun for 5 months, or

of a year? Ans. $18,75.

Note. If the multiplier of any sum be greater than a unit or 1, the multiplicand will be increased as many times as the multiplier is greater than a unit; that is, the multiplicand will be taken as many times as the multiplier contains units. But when the multiplier is a fraction or part of a unit, the product will be only a part of the multipli*cand. Hence in multiplying by a proper fraction, the product is always less than the multiplicand, as will be seen by the preceding examples.

PROBLEM V.

To Multiply a Fraction by a Whole Number.

RULE.

Multiply the whole number and the numerator of the fraction together, and write the product over the denominator; and if it produce an improper fraction, change it to a whole or mixed number, by Prob. 3.

EXAMPLES.

1. If a man spend of a dollar a day, how much will he spend in 11 days?

If he spend & in 1 day, he will spend 11 times 5-55 in 11 days, and 55 of a dollar =91 dollars, the answer.

2. If 1 yard of cloth cost of a dollar, what will 15 yards cost?

Ans. $9.

3. If a bushel of oats cost of a dollar, what will 23 bushels cost?

Ans. $91

4. A certain lot contains of an acre of land; how much land would 37 such lots contain? Ans. 27 acres.

5. If a bushel of potatoes cost of a dollar, what will 56 bushels cost? Ans. $16.

Note. The process of multiplying a fraction by a whole number, may be shortened, thus: Divide the denominator of the fraction by the whole number, (when it can be done without a remainder,) and over the quotient write the

numerator.

6. If a pound of sugar cost of a dollar, what will 20lb. cost.

Divide the denominator, 100, by 20, and the quotient 5, is a new denominator; then write the numerator over it, and it becomes of a dollar=2} dollars, the answer. 7. If a pound of nails cost of a dollar, what will 11lb. cost? Ans. $1. 8. If a pound of butter cost of a dollar, what will 5lb. cost?

6 66

20

Ans. $3.

9. At of a dollar per pound, what will 11lb. raisins

3

come to ?

Ans. $3

PROBLEM VI.

To divide a Whole Number by a Fraction.

RULE.

Multiply the whole number by the denominator of the fraction, and divide the product by the numerator.

EXAMPLES.

1. How many times is of a dollar contained in $9 ? 1 dollar is 4, and 9 dollars is 9 times as many; 9x4=; and is contained in 36 as many times as 3 is contained in 35.

2. How many times is & contained in 16?

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Ans. 12

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