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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?

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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.

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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 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 7 of a day?

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

6. Multiply 750 by

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 =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. $972. 4. A certain lot contains of an acre of land; how much land would 37 such lots contain? Ans. 273 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.

6

59

Divide the denominator, 100, by 20, and the quotient 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?

Ans. $3.

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

come to ?

PROBLEM VI.

Ans. $3.

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, and 9 dollars is 9 times as many; 9x4=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

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Ans. 12.

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

or 12÷3= Ans. 18. Ans. 48.

Ans. 40.

4. How many times can I have in 27 ? 5. How many times is 17 contained in 34 ? 6. How many men can I divide 75 dollars among, so as to give each of a dollar?

Ans. 100 men.

examples, that The reason of number, 12 for

Note. It will be seen by the 6 preceding the quotient is greater than the dividend. this is as follows. If we divide a whole 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.

8. What part of 19 acres is 5 acres? 9. 18 inches is what part of 56 inches? 10. What part of £1 is 12s. 9d. 3qrs. ? Operation.

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12s. 9d. 3qrs.

12

153

4

Numerator 615qrs.

£1=20s.

12

240
4

Denominator 960qrs.=815=44.

11. What part of a shilling is 44d.?

Ans.

Ans.

Ans. 9

12. What part of a pound Troy is 7 oz. 4pwt.
13. What part of 1cwt. is 2qr. 16lb.?
14. What part of a yard is 3qr. 3na. ?
15. What part of a hogshead is 35gal. 2qts.?
16. What part of a furlong is 6rd. 3yds. 2feet?
17. What part of an acre is 3 roods 21rds. ?
18. Reduce 54 gallons to the fraction of a hogshead.

Ans. 15 16. Ans. 126 1 Ans. Ans. 1.

Ans. 9.

19. Reduce 6fur. 26rds. 11ft. to the fraction of a mile.

20. What part of 1 year is 7 weeks 1 day? 21. What part of a yard is 2qr. 2na.?

Ans. . Ans..

Ans. yd.

(Reduce 2qr. 2na. to nails, then reduce the nails to thirds, adding in the 2 thirds, for the numerator; then reduce 1 yard to thirds of nails, for the denominator.)

22. What part of a day is 11 hours 59 minutes?

Ans. 4661 10080' 23. Reduce 4 shillings 6 pence to the fraction of a dollar or 6 shillings.

24. What part of the year had transpired the October, 1836, including that day?

25. What part of a dollar at 8 shillings, is 2

pence ?

PROBLEM VIII.

Ans. $3. 26th day of Ans.. shillings 8 Ans. $.

To find the Value of a Fraction in Whole Numbers of less

Denominations.

RULE.

1. Multiply the numerator by the parts in the next lower denomination, and divide the product by the denominator.

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