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124. Let us consider the converse of the preceding case; for instance, suppose it were required to reduce the mixed quantity 5 to an improper fraction. Since unity is supposed to be divided into 7 equal parts, every unit=7-sevenths; therefore, 5 units=5 × 7-sevenths, or 35-sevenths, and 2-sevenths make altogether 37-sevenths. Therefore, 5=37.

Hence, to reduce a mixed quantity to an improper fraction, multiply the integer by the denominator of the fraction, and to the product add the numerator of the fraction; the sum is the new numerator, and the denominator of the fraction will be its denominator.

125. Exercises: Express the following mixed quantities as improper fractions:-33, 74, 163, 147, 268, 3511, 4817, 1242}. 126. RECAPITULATION.-EXERCISES.

1. Express that an object is divided into 25 equal parts, and 17 of them taken away.

2. Exhibit the quantity represented by an object broken into 134 equal parts, and 21 of them taken away.

79

3. Write down the following fractions in words:-1, 7, 8, 17, 31. 4. What is the meaning of, 4, 3, 12, 11?

5. Write down in figures the following expressions -five-sixths, one-thirteenth, seven-tenths, thirteen-fifteenths, three-fifths, four-ninths, three-nineteenths, fifteen-twenty-eighths.

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oor? Why?

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7. Which is the greater of,, I, 1‍2, 16 ?

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8. Express in figures that a workman has done but threequarters of his work.

9. Two workmen are in a factory, one works five-sevenths of the time, and the other five-sixths. Express in figures the time each worked, and also mention which worked the longer time.

10. Two men work in the same manufactory, the first performs one-fifth less than his daily work, and the second one-fifth more. How much does each perform?

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11. What name is given to quantities of this form: 15, 11 21, 76, 100, 134 ? and express them in mixed quantities. 12. Which quantity is twice as great as ??

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three times
five times

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13. The treble of is required; the quadruple of the double of?

14. A mason builds, in one day, of a wall. How much will he do in 7 days?

15. A man goes, in one day, of his journey. How much will he accomplish in 8 days?

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17. How much is one-quarter of one quarter?

19. In 5 days,

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of a work is performed; how much is done in one day? In nine days, &; how much in one day? In seven days,; how much in one day?

20. A and B divide a sum of money equally between them; A divides his share equally between his three children, and B divides his equally between his four children. What part of the whole money does each of the children of A and B receive?

21. A person received of an inheritance, which was trebled in trade; his son quadrupled that fortune; but the grandson, failing in business, lost one-seventh of his inheritance. What part of the whole inheritance was left?

22. Required, the greatest common measure of 28 and 98; of 345 and 2415; of 22893 and 79245.

23. Reduce to their lowest terms, if possible,

720 1664

18 317
639

8739

504 12969

24. A shopkeeper was asked for the 31 of a yard of cloth, but not understanding the question, what must be done?

24. A boy who was offered the part of an orange, asked that the quantity be put in a simpler form. How is this to be done?

ADDITION

127. If to of a yard

The sum of § and 3 is 7.

OF FRACTIONS.

be added, the sum is of a yard.

Find the sum of ++ +}=}; } + } ==1}
Find the sum of s+++}}· 13+13=} }; } } +15

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Therefore, when fractions have the same denominator, add all the numerators together, and place their sum over the denominator, the result will be the sum of the fractions, which must be reduced to a whole or mixed quantity, when possible.

128. How much are +?

These fractions, not having the same denominator, cannot be put together without transforming them to fractions of the same denominator, for instance :

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Hence,

+3+3=5. We observe that

sixths by multiplying the terms by 3:

formed by multiplying the terms by 2:

is transformed into

3 x 1

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3 × 2

6

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129. The sum of 3, 4, and § is required.

Here the fractions can all be transformed into twelfths :

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Therefore, ++} = A +++8=}}=2135=24.

To transform into twelfths the terms must be multiplied by 4.

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Therefore, when the fractions proposed have not the same denominator, reduce them to a common denominator, and then proceed as before.

130. We have seen that fractions cannot be added together without being of the same kind, or of the same denominator. Let us now determine a method which will enable us to reduce fractions to a common denominator. The common denominator of two or more fractions is such a number as will contain the several proposed denominators, without remainders; thus, in our last example, 12 is a common denominator of 3, 4, 6; so is 24, 36, &c., any multiple of 12; but 12 is the least, and it will be preferable to find that one, because we shall thereby shorten the operations.

131. What is the least common denominator of §, t, I'm, 11, 28?

First, let it be understood that to resolve a number into its prime factors is to divide it by 2, and the quotient by 2, and this second quotient again by 2. When the quotient can no longer be divided by 2, we must try by 3, then by 5, by 7, &c., until the quotient is a prime number. The several divisors and the last quotient are the prime factors of the numbers.

Let the above denominators be resolved into their prime factors.

6=7x3
8=2x2x1

18=4×3×3

24=2×2×2×z
45=3x3x5.

Now, we observe that the two of the first line, the three twos of the second, and the two of the third, are contained in the three twos of the fourth line, for which reason they are struck out; the three of the first line, the two threes of the third, and the three of the fourth are contained in the two threes of the fifth line, then they are also struck out. The product of the remaining

factors will evidently be the least common denominator, or 2×2×2×3×3×5=360.

132. In practice, the operation is carried on in this manner:—

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The several divisors, and the last quotient 5, are the factors found above. But, by inspection, we easily see that 6 and 8 are contained in 24; and, therefore, may be neglected, and the is thus abridged :—

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and the least common denominator is 2 × 3 × 3 ×4x5=360, as before.

133. To transform the proposed fractions into others, having 360 for their denominator, it is merely necessary to multiply the terms of every fraction by the quotient resulting from dividing 360 by its denominator.

Now, 6 is contained in 360, 60 times; therefore, &

60 × 5

-388

60 × 6

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