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?

273

Operation.

+4-fourths in 1 dollar. 108-fourths in 27 dollars. +3=fourths in 3.

111-fourths Ans. 11.

=

$1-4 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 36 dollars, how many eighths of a dollar?

3. Reduce 45 to ninths.

Ans. 223.

8

Ans. 405 405.

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.

61140

8. Reduce 6125 to an improper fraction. 9. Reduce 84 to an improper fraction. 10. What improper fraction is equal to 5618?

11. What improper fraction is equal to 148?

12. What improper fraction is equal to 225?

Ans. 1189.

Ans. 1898,

PROBLEM III.

To change an Improper Fraction to a Whole or Mixed

Number.

RULE.

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

EXAMPLES.

1. In 45 of a dollar, how

many dollars?

of a dollar are equal to 1 dollar, and 6 is contained in 45,7 times and of another time; therefore the answer is 73 dollars 7 dollars.

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 ??

By this example we see, there are two ways of multiply ing 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 3, for 3 times of any number is equal to 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 15 of an acre of land? Ans. 23 dolls. 3. If a ship sail 246 miles a day, how far will she sail in of a day?

4. How much is of $1845,56 ? 5. Multiply 400 by 3.

6. Multiply 750 by 3.

Ans. 191 miles.
Ans. $1537,962.

Ans. 150.

Ans. 450

7. The interest of $750 for 1 year, is $45; what is the interest on the same sum 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 multiplicand. 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 2. If 1 yard of cloth cost yards cost?

9 dollars, the answer.
of a dollar, what will 15
Ans. $9.

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

Ans. $97

of an acre of land; how Ans. 27 acres.

4. A certain lot contains much land would 37 such lots contain?

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.

11

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

100,

Divide the denominator, 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? 9. At 2 of a dollar per pound, what will 11lb. raisins 'come to ?

20

Ans. $3.

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.

of a dollar contained in $9?

1. How many times is 1 dollar is, and 9 dollars is 9 times as many; 9×4=38; and is contained in 36 as many times as 3 is contained in 35.

2. How many times is contained in 16?

Thus, 16

X6 denominator.

Numerator=5)96=sixths in 16
Ans. 19}.

Ans 12

3. How many times is contained in 12 how many?

4. How many times & can I have in 27 ?
5. How many times is contained in 34 ?

or 12÷="

Ans. 18. Ans. 483.

Ans. 40.

6. How many men can I divide 75 dollars among, so as to give each of a dollar? Ans. 100 men. Note. It will be seen by the 6 preceding examples, that the quotient is greater than the dividend. The reason of this is as follows. If we divide a whole number, 12 for example by 2, the quotient will be 6, which is equal to half the dividend; and if we divide it by 1, the quotient will be 12, for 1 is contained in any number twice as often as 2. Again, if we divide by 2, the quotient will be increased, for is contained in any number twice as often as 1; thus, 12 is 24 halves, and is contained in 24, 24 times. Hence when the divisor is less than a unit, it will be contained in the dividend a greater number of times. Therefore dividing a whole number by any proper fraction, the quotient will always exceed the dividend.

PROBLEM VII.

To Reduce any given Quantity to a Fraction of a higher Denomination of the same kind.

RULE.

1. Reduce the given quantity to the lowest denomination mentioned, for a numerator.

2. Reduce 1 of the higher denomination to the same name, for a denominator.

EXAMPLES.

1. What part of 5 yards is 3 yards?

Thus, lyd. is of 5yds., and 3 yards are 3 times as much;

3 times is, the answer.

2. What part of 7lb. is 4lb.?

Ans. 4.

3. What part of 17 cents is 9 cents?

Ans. 17.

4. What part of 18 dollars is 4 dollars?

Ans. 3.

Note. Reduce all the fractions to their lowest terms.

5. What part of £15 is £6?

Ans.

6. What part of 25 rods is 15 rods?

Ans.

7. What part of 63 gallons is 9 gallons?

Ans.