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remainder, and continuing to divide the last divisor by the last remainder until nothing remains. The last divisor will be the greatest common divisor sought.

Ex. 3. Find the greatest common divisor of the two numbers 63 and 81. 63)81(1

PROOF. 63

9)6307 18)63(3

63 54

98109 Gr. com. div. 9)18(2

81 18 4. Find the greatest common divisor of 315 and 405.

Ans. 45. 5. What is the greatest common divisor of the two numbers 2205 and 2835 ?

Ans. 315. Note. § 122. If it be required to find the greatest common divisor of more than two numbers, find first the greatest common divisor of two of them, then of that common divisor and one of the remaining numbers, and so on, for all the numbers : the last common divisor will be the greatest common divisor of all the numbers.

Ex. 6. What is the greatest common divisor of 246, 372 and 522 ?

Ans. 6. 7. What is the greatest common divisor of 492, 744 and 1044 ?

Ans. 12. QUESTIONS. * 111. How many kinds of Vulgar Fractions are there?

are thoy? What is a proper fraction ? Is its value

greater or less than 1 ? What is an improper fraction? Why is it called improper! When is its value equal to 1? What is a simple fraction? What is a compound fraction ? What is a mixed number !

0 112. What are the numerator and denominator of a fraction, taken together, called ? How many terms has every fraction ?

0 113. How may a whole number be expressed fraction. ally ? Does this alter its value ?

$ 114. What does the denominator of a fraction show ? What does its numerator show ? Repeat Prop. I. 6 115. Repeat Prop. II.

116. Repeat Prop. III $117. Repeat Prop. IV. $ 118. Repeat Prop. V. $ 119. Repeat Prop. VI.

$ 120. What is the common divisor of two or more num. bers ? What is the greatest common divisor ?

$ 121. How is the greatest common divisor found?

§ 122. How do you find the greatest common divisor of three or more numbers ?

REDUCTION OF VULGAR FRACTIONS. § 123. Reduction of Vulgar Fractions is the meth. od of changing the forms of the fractions without altering their values.

Ø 124. A fraction is said to be in its lowest terms, when there is no number greater than 1 that will divide the numerator and denominator without a remainder

CASE I, § 125. To reduce an improper fraction to its equivalent whole or mixed number.

RULE. Divide the numerator by the denominator, the qu: tient will be the whole number, and the remainder there be one, placed over the given denaminator, form the fractional part.

Ex. 1. Reduce and to their equivalent whole or mixed numbers. 4)84

9)67 Ans. 21

Ans. 7

It was shown in $ 59, that the value of every fraction is equal to the quotient arising from dividing the numerator by the denominator : hence the value of the fraction is not changed by the reduction. 2. Reduce you to a whole or mixed number.

Ans. 121. 3. In y of yards of cloth, how many yards?

Ans. 24yd. 4. In id of bushels, how many

bushels ?

Ans. 5fbu. 5. If I give } of an apple to each one of 15 children, how many apples do I give ?

Ans. 5.

CASE II. 126. To reduce a mixed number to its equivalent improper fraction.

RULE. Multiply the whole number by the denominator of the fraction ; to the product add the numerator, and place the sum over the given denominator.

Ex. 1. Reduce 44 to its equivalent improper fraction. Here 4*5=20: then 20+4=24; hence a is the equivalent fraction.

Ans. ** This rule is the reverse of Case I. In the example 44 we have the integer number 4 and the fraction : Now 1 whole thing is equal to 5 fifths, and 4 whole things are equal to 20 fifths; to which, add

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4 fifths and we have y, equal to the mixed number 44 2. Reduce 47} to its equivalent improper fraction.

Ans. 231 3. Reduce 67637, 8744, 690 , 367187, to their equivalent improper fractions.

Ans. 2413, 299, 41893, 2471?.

CASE III 6 127. To reduce a fraction to its lowest terms.

RULE. Divide the numerator and denominator by any number that will divide them both without a remain. der, and then divide the quotients arising in the same way until there is no number greater than 1 that will divide them without a remainder.

Or, find the greatest common divisor of the nume. rator and denominator and divide them by it. The value of the fraction will not be altered by reduction. Ø 119. Ex: 1. Reduce 1% to its lowest terms.

1st method. 5) 707)14 2 5175=235 which is the lowest term of the fraction, since no number greater than 1 will divide the numerator and denominator without a remainder.

2d method, by the common divisor.

70)175(2
140

35) 70 2

Ans Greatest com. div. 35,70(2 35)1755

70 2. Reduce f to its lowest terms.

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CASE IV. $ 128. To reduce a whole number to an equivalent fraction having a given denominator.

RULE. Multiply the whole number by the given denominator, and set the product over the said denominator.

Ex. 1. Reduce 6 to a fraction whose denominator shall be 4.

Here 6x4=24; therefore is the required fraetion. It is plain that the fraction will in all cases be equal to the whole number, since it may be reduced to the whole number by Case I.

2. Reduce 15 to a fraction whose denominator shall be 9.

Ans. ?{. 3. Reduce 139 to a fraction whose denominator shall be 175.

Ans. 21935.

CASE V. $ 129. To reduce a compound fraction to its equivalent simple one. Let us take the fraction of .

fraction is equal to 3xį of . But off

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