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4. A compound fraction is a fraction of a fraction, connected by the word of; as į of of g of 1. 5. A mixed number is an integer with a fraction ; as

76,53 6. A compound mixed fraction is one whose numerator or

denominator, or both, is a mixed number; as ? 7. The greatest common measure of two or more numbers is the largest number, that will divide them without a remainder. 8. The least common multiple of two or more numbers is the least number, that may be divided by them without a remainder. 9. A fraction is in its lowest terms, when no number but

a unit will measure both its terms. 10. A prime number is that which can be measured only

by itself or a unit ; as 7, 11, and 19. 11. A perfect number is equal to the sum of all its ali

quot parts ; as 6, 28, 496, &c. 12. A fraction is equal to the number of times the nu

merator will contain the denominator. 13. The value of a fraction depends on the proportion, which the numerator bears to the denominator.

I. To find the greatest common measure of two or more numbers ; that is, to find the greatest number that will divide two or more numbers.

1. What is the common measure of 84 and 132 ; that is, what is the largest number, that will divide both of these numbers without a remainder ?

Ans. 12.

OPERATION.

As 12 will divide 36, it is ev84) 132 (1

ident it will also divide 48, 84

which is equal to 12+36. It 48) 84(1

will also divide 84 ; because 48

84 is equal to 36 +48 ; for, as 36) 48(1

12 will divide each of these 36 numbers, it is evident it will

divide their sum. For the same 12) 3 6 (3 36 reason, it will also divide 132,

which is equal to 84+48. We therefore find, that 12 is the largest number, that will di

vide 48 and 132 without a remainder. It is, therefore, its greatest common measure. Hence the following

RULE.

Divide the greater number by the less, and if there be a remainder, divide the last divisor by it, and so continue dividing the last divisor by the last remainder, until nothing remains, and the last divisor is the greatest common meas

ure.

If there be more than two numbers, find the greatest common measure of two of them, and then of that common measure and the other numbers. If it should happen, that 1 is the common measure, the numbers are prime to each other, and are incommensurable. 2. What is the greatest common measure of 85 and 95 ?

Ans. 5. 3. What is the greatest common measure of 72 and 168 ?

Ans. 24. 4. What is the greatest common measure of 119 and 121 ?

Ans. ). 5. What is the largest number that will divide 324 and 586 ?

Ans. 2. 6. What is the largest number that will divide 582 and 684 ?

Ans. 6. 7. What is the greatest common measure of 32 and 172 ?

Ans. 4. 8. What is the largest number that will divide 84 and 1728 ?

Ans. 12. 9. What is the greatest common measure of 16, 20, and 26 ?

Ans. 2. 10. What is the greatest common measure of 12, 18, 24, and 30 ?

Ans. 6.

II. To reduce fractions to their lowest terms. Note. A fraction is said to be in its lowest terms, when no num. ber but a unit will divide its numerator and denominator. 1. Reduce is to its lowest terms.

We find by the last Rule, that 5 is 5) Ps= } Ans. the largest number, that will divide

both the numerator and denominator of the fraction; and having divided them both by it, we

OPERATION.

find the result to be }, and that is equal to its is evident from the fact, that the ratio of 5 to 15 is equal to the ratio of 1 to 3. And, as the value of a fraction depends on the ratio, which the numerator bears to the denominator, if their ratios are equal, the fractions are also equal. Q. e. d. Hence the following

RULE.

Divide the numerator and denominator by any number that will divide them both without a remainder; and so continue until no number will divide them but unity. Or, divide the numerator and denominator by the greatest common

measure.

174

Ans. t.

Ans. 2. Reduces to its lowest terms. 3. Reduce 36 to its lowest terms.

Ans. . 4. Reduce je to its lowest terms.

Ans. š. 5. Reduce 96 to its lowest terms.

Ans. 4. 6. Reduce 107 to its lowest terms. 7. Reduce i to its lowest terms.

Ans. 12. 8. Reduce 8 3867 to its lowest terms.

Ans. 7. 9. Reduce ild to its lowest terms. Ans. 34 10. Whàt is the lowest expression of ff? Ans. 375.

III. To reduce mixed numbers to improper fractions.

MENTAL OPERATIONS.

1. In 3 dollars how many halves ? How many thirds ? 2. In 7 apples how many tenths ? How many twelfths ? 3. In 87 dollars how many sevenths ? 4. In 3] oranges how many fourths ? 5. In 941 gallons how many elevenths ? 6. In 7 quarts how many fifths of quarts ?

We analyze this question by saying, as 7 there are 5 fifths in one quart, there will

5 be 5 times as many fifths as quarts ; there35 fore, in seven quarts and three fifths, there

3 will be 38 fifths, which should be expressed 38

thus, 18. And this fraction, by definition

2d, on page 76, is an improper fraction. 5

Hence the following

OPERATION

RULE.

Multiply the whole number by the denominator of the fraction, and to the product add the numerator, and place their sum over the denominator of the fraction. 7. Reduce 8} to an improper fraction.

Ans. 41 8. Reduce 151 to an improper fraction.

Ans. 183. 9. In 187 how many ninths ?.

Ans. 162, 10. In 16111'y how many one hundred and seventeenths ?

Ans. 18848. 11. Change 43114 to an improper fraction Ans. 242. 12. What improper fraction will express 27195 ?

Ans. 3 13. Change 11liti to an improper fraction ?

Ans. 14112. IV. To change improper fractions to integers or whole numbers.

MENTAL OPERATIONS.

1. How many dollars in 4 halves ? In 5 halves ? In 6

halves ? In 7 halves ? In 12 halves ? In 19 halves ? 2. How many dollars in 5 quarters? In 9 quarters ? 3. How many dollars in 10 eighths ? In 20 eighths ?

FOR THE SLATE.

OPERATION.

4. How many dollars in it dollars ?

Ans. 216 This question may be analyzed by say16) 37 (216 ing, as 16 sixteenths make one dollar,

32 there will be as many dollars in 37 six5 teenths as 37 contains 16, which is 26 times,

$ 276. This answer is called a mixed number by definition 5th, page 77. Hence we see the propriety of the following

RULE.

Divide the numerator by the denominator, and if there be a remainder, place it over the denominator at the right hand of the integer. 5. Change Y to a mixed number.

Ans. 1087

Ans. 1896

6. Change Y to a mixed number. Ans. 1011t: 7. Change 174 to a mixed number. 8. Reduce you to a mixed number. Ans. 1424. 9. Reduce 373 to a whole number.

Ans. 1. 10. Change 567 to a whole number. Ans. 567. 11. What is the value of 375 ?

Ans. 12. What is the value of 345?

Ans. 13. Change 125 to an improper fraction.

Ans. 125. V. To change or reduce compound fractions to simple fractions.

MENTAL OPERATIONS.

1. What part of an orange is a { of a half ?
2. What part of an apple is a 1 of a half ?
3. What part of a bushel is a 1 of a peck ?
4. What part of a quart is a { of a pint ?

FOR THE SLATE.

Ans.

OPERATION.

5. What is of 1?

This question may be analyzed X=Ans. by saying, if 4 of an apple be

divided into 5 equal parts, that one of these parts is 3's of an apple ; and, if of it be 5's, it is evident, that of 11 will be 7 times as much. 7 times it's is 35 ; and, if { of 11 be 35. of 11 will be 4 times as much. 4 times 35 is s.

We therefore induce the following

RULE.

Change mixed numbers and whole numbers, if there be any, to improper fractions; then multiply all the numerators together for a new numerator, and all the denominators together for a new denominator ; the fraction should then be reduced to its lowest terms. 6. What is şof of ¢ ?

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