11. The numerator and denominator of a fraction, taken together, are called the terms of the fraction.

12. A fraction is said to be in its lowest terms, when no number greater than 1, or unity, will divide the terms of the fraction without a remainder.

EXAMPLES.

1. If I divide an apple into 8 parts, by what fraction will one of those parts be expressed? 2 of those parts? 3-4-5-6-7-8? Ans. 1, 2, 3, 4, §, §, 7, §.

2. If one be divided by 4, what will be the quotient? if by 6, what? if by 7? if by 8? if by 9? if by 10? if by 11? if by 12? 1 Ans. 1, 4, 4, 4, 4, 1o, TT, 12.

2

if

3. If 2 be divided by 4, what will be the quotient? by 6? by 7? by 8? by 13? Ans. 4, 4, 4, 3, 13: 4. If a bushel be divided equally among 4 persons, what part of a bushel does each receive?

5. If 2 bushels of apples be divided equally among 4 persons, what will each receive?

6. If a bushel of corn be divided into 4 parts, what are those parts called? if into 5 parts? into 6? into 7?-8? 9?-10?

7. If I give away 6 quarts of nuts, what part of a peck is it? if 7? if 8? if 9?

8. How many of the four last questions are proper fractions? Are any improper, and which are they?

9. If I divide an apple into halves, and give away of , what part do I give away? and what is the fraction by which it is expressed in the question?

10. How is the quotient of 4, divided by 3, expressed? Questions similar to the foregoing, should be multiplied and proposed to the scholar, until his ideas of fractions are clear and correct.

To reduce Improper Fractions to Mixed Numbers, and Mixed Numbers to Improper Fractions.

IMPROPER FRACTIONS.

67

MIXED NUMBERS.

1. Change $7 to a whole 2. Change 16 to an imor mixed number.

4)67

improper fraction.

162

As the denominator of a fraction denotes the number

163

4

of parts into which the unit

67

The scholar will perceive, that the mixed number, 163, was the quotient in the last question, of 64 divided by 4; and

is divided, it is evident, that let it be remembered that a 67 contains as many units, or mixed number is the quotient wholes, as four is contained of a division, whose divisor times in 67, which we find by is the denominator of the trial, to be 16 times and of fraction; therefore in reduca time.

Hence the Rule

ing a mixed number to an improper fraction, we only find

To reduce an improper the dividend.-(See p. 40.) fraction to a whole, or mixed

number.

Hence the Rule

To reduce a mixed num

Divide the numerator by ber to an improper fraction. the denominator, and the Multiply the whole number quotient will be the whole by the denominator of the number; the remainder, if fraction, and to the product any, written over the denom-add the numerator; under inator, must be placed at the result, place the denomithe right hand of the quo-nator of the fraction.

tient.

EXAMPLES.

EXAMPLES.

3. Change 42 to a whole 4. Change 5 to an imor mixed number. proper fraction. 6. In 65, how many ninths? 8. In 10 weeks, how many 15ths?

5. In 59, how many wholes? 7. In 159 of a week, how many weeks?

9. Change 2233 to a whole or mixed number?

11. In 3247 of a day, how many days?

39

13. In 49364 of a year, how many years?

15. In of a cent, how many half dimes?

17. Change 229 to a whole or mixed number.

15

10. Change 1597 to an improper fraction. 12. In 2021 many 16ths?

days, how

14. In 12653 years, how many 39ths?

16. Change 33 half dimes to the fraction of a cent?

18. Change 57 to an improper fraction?

19. In 504 of a minute, 20. In 72 minutes, how how many minutes?

many 7ths?

To reduce a Fraction to its lowest terms.

1. Reduce to its lowest terms.

If of a bushel of apples were divided equally between two persons, it is evident that one person would receive g

QUESTIONS. 1. What is the rule for reducing an improper fraction to a whole or mixed number? 2. For reducing a mixed number to an improper fraction?

of a bushel. By dividing the numerator by 2, the fraction is diminished. If we divide the denominator by 2, it becomes, and is therefore increased; but, if we divide both the numerator and denominator of by 2 it becomes ; if we divide the terms of the fraction again by 2 it becomes , which is equal to or, for in either case the numerator is one half of the denominator. Hence it appears that the value of a fraction is not affected by dividing or multiplying both the numerator and denominator by the same number. To reduce a fraction to its lowest terms, we have this RULE.

Divide both the numerator and denominator by any number that will divide both without a remainder; and so continue to do, until no number greater than 1 will divide them.

2. Reduce to its lowest terms.

2

4. Reduce 72, 771, 188, 24, to their lowest terms. Were the greatest number known which would divide the terms of the fraction, a simple division would at once reduce the fraction; but, as this is not the case, the greatest divisor may be found by the following

RULE.

Divide the denominator by the numerator, and if there be no remainder, the numerator will be that divisor; but if there be a remainder, divide the last divisor by the last remainder, and thus proceed until there be no remainder; and the last divisor will be the greatest common measure sought.

5. Reduce 48 to its lowest terms.

Operation. 16)76(4

64

12)16(1
12

4)12(3
12

We find by trial that 16 is contained in 76, 4 times and 12 remainder ; hence 16 is not the common divisor; dividing the last divisor 16, by the remainder 12, we have the quotient 1, and 4 remainder therefore 12 is not the common divisor. Had 12 divided 16 without a remainder,

it is evident, that it would have been a divisor common to

both terms of the fraction, because 76=16×4+12. It is plain, if 12 would divide 16 without a remainder, it would also divide 4 times 16+12 without a remainder. Again, we find that 12, the last divisor, will contain 4, the last remainder, 3 times and no remainder. 4, is therefore the greatest common divisor of the terms of the fraction 6, and is the answer in its lowest terms, for no number will divide 4 and 19 without a remainder.

19

6. Reduce 182 to its lowest terms.

7. Reduce 468 to its lowest terms.

1184

32256

1296

Ans., 13. Ans., 117 Ans., 14 Ans.,

296

8

8. Reduce 6912 to its lowest terms. 9. Reduce 36 to its lowest terms. If it be required to find the greatest common measure of more than two numbers, find the greatest common measure of two of them as before; then, of that common measure, and of one of the other numbers, and so on thro' the whole. The common measure last found, will be the one sought.

10. What is the greatest common measure of 48 and 192? Answer, 48. Answer,

11. Reduce to its lowest terms. 48

192

12. What is the greatest common measure of 35—42— 63! Answer, 7.

GENERAL RULE FOR THE MULTIPLICATION AND DIVISION OF FRACTIONS BY WHOLE NUMBERS -WHOLE NUMBERS BY FRACTIONS--FRACTIONS BY FRACTIONS.

-

Draw a perpendicular line, and place all those figures which are to be multiplied together for a numerator, or dividend, on the right of the line, and those figures which are to be multiplied together for a denominator, or divisor, on the left hand of the line. (See p. 43.) It will appear obvious to the scholar by the analysis under each of the following sections, that the numerators of fractions to be multiplied or divided, must be placed upon the right of the line, and their denominators on the left; and also that the numerators of fractions, by which the division is to be made, on the left. The question thus stated, equals on

QUESTIONS. 3. What is the rule for reducing a fraction to its lowest terms? 4. Were the greatest number known which would divide the terms of the fraction, how might you proceed? 5. When this is not the case, how may the greatest divisor be found? 6. How is the common measure of more than two numbers found? 7. What is the rule for the multiplication and division of fractions, &c. by cancelling?

each side of the line may be crossed, as cancelling each other. (See p. 43.) When no two numbers remain, one on each side of the line, capable of being divided by any one figure, (See p. 90.) multiply the figures on the right of the line for a numerator, or dividend, and those on the left for a denominator, or divisor, and the result will be the answer in the lowest terms of the fraction.

Multiplication of Fractions by Whole Numbers.

1. If a man receive of a dollar for one day's work, what will he receive for two days' work? It is evident, if a man receive of a dollar for 1 day's work, that he would receive for two day's work twice as much or =. If I multiply the numerator by 2, I give him, or 2 quarters of a dollar; if I divide the denominator by 2, I give him or quarters. Hence, to multiply the numerator of a fraction is the same in effect, as to divide the denominator. If the numerator of be multiplied by 2, it becomes =1; if the denominator be divided by 2, it becomes 1. Therefore

To multiply a Fraction by a Whole Number-Multiply the numerator, or divide the denominator; and the result will be the answer required.

2. If a pound of lead cost of a dollar, how much will 8 pounds cost?

Operation.

2 161
21=

the

As, the cost of 1 pound, is to be multiplied by 8, number of pounds; therefore, place the numerator of, the multiplicand, and 8, the multiplier, on the right of the line; since 8, on the right is contained in 16, on the left, twice, cross the 8 and 16, and write 2 on the left of 16; we now have 1 on the right to be divided by 2 on the left, which is, the answer.

3. What will 8 bushels of apples cost at a dollar per bushel? Answer, $4,00.

QUESTIONS. 8. When the question is stated, what is the method of procedure? 9. When no two numbers are left, one on each side of the line, capable of being divided by any one figure, what is to be done? 10. How do you multiply a fraction by a whole number? 11. Why, in example 2, are 8 and the numerator of the fraction placed on the right of the line?