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7. What is the value of It of 15of 570 of 100 ?

Ans. 575814 8. What is of of 11? 9. What is the value of 11 of it off of $71?

Ans. $ 1.75. 10. What is the value of of i, of is of 3 gallons ?

Ans. gal. 11. What part of a ship is į of şof it? 12. What is the value of 4 of zo of 1 of 14 of $ 34 ?

Ans. $ 6.75.

A COMMON DENOMINATOR.

224. Fractions have a common denominator when all their denominators are alike.

225. A common denominator of two or more fractions is a common multiple of their denominators; and their least common denominator is the least common multiple of their denominators.

226. To reduce fractions to a common denominator.

Ex. 1. Reduce , i' , and 15 to other fractions of equal value, having a common denominator.

FIRST OPERATION.

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1056
1536

7 X 12 X16=13 4 4 new numerator. } 1346 5 X 8 X16= 640

14436 Ans. 11x 8 x 12 =105 6

16 8 X 12 X16= 15 3 6 common denominator.

We first multiply the numerator of } by the denominators 12 and 16, and obtain 1344 for a new numerator. We next multiply the numerator of is by the denominators 8 and 16, and obtain 640 for a new numerator; and then we multiply the numerator of il by the denominators 8 and 12, and obtain 1056 for a new numerator. Finally, we multiply all the denominators together for a common denominator, and write it under the several numerators, as in the operation.

By this process, since the numerator and denominator of each fraction are multiplied by the same numbers, their relation to each other is not changed, and the value of the fraction remains the same. (Art. 217.)

SECOND OPERATION.

66

66

48 least common denominator. 8 12 16 8 6 X 7=42, new numerator. } 3 12 4 X 5 = 20,

Is=% Ans. 16 3 X 11= 33, “

it= 16 X 3 48, least common multiple, and least common denomina .

tor.

Having first obtained the least common multiple of all the denominators of the given fractions, we assume this to be their least common denominator. We then take such a part of this number, 48, as is expressed by each of the fractions separately for their respective new numerators. Thus, to get a new numerator for }, we take of 48, the least common denominator, by dividing it by 8, and multiplying the quotient 6 by 7. We proceed in like manner with each of the fractions, and write the numerators thus obtained over the least common denominator. In this process the value of each fraction remains unchanged, as 'both terms are multiplied by the same number. (Art. 217.)

The method used in the second operation, it will be perceived, expresses the fractions of the result in lower terms than that used in the first. On this account it is often to be preferred to the other.

RULE. Find the least common multiple of the denominators for the LEAST COMMON denominator.

Divide the least common denominator by the denominator of each of the given fractions, and multiply the quotients by their respective numerators, for the new numerators. Or,

Multiply each numerator by all the denominators except its own, for the new numerators; and all the denominators together for A COMMON denominator.

NOTE 1. Compound fractions must be reduced to simple ones, whole and mixed numbers to improper fractions, before finding a common denominator, and all to their lowest terms, before finding the least common denominator.

NOTE 2. Fractions may sometimes be reduced to a common denominator most readily by multiplying both terms of one or more of them by such a number as will make all the denominators alike. Thus ) and may be brought to a common denominator simply by multiplying both terms of the } by 2, and changing in that way its form to .

Note 3. — Fractions may often be reduced to lower terms, without destroying their common denominator, by dividing all their numerators and denominators by a common divisor.

EXAMPLES Reduce the following fractions to their least common denominator :2. Reduce , , }, and Ps.

Ans. 1, 42, 43, 49 3. Reduce f1, 15, 18, and A. Ans. 1936, 7324, 126, 126

140 165

12

4. Reduce +, Pyt, and age
5. Reduce tf, 4, it, and to Ans. 43, 44, 43, 41.
6. Reduce 11, 11, 4, and . Ans. 348, 36, 37, 385.
7. Reduce }, }, , and .

Ans. 48, 48, 46, . 8. Reduce 4, 5, T, and TT: Ans. $4, 95, 96, 93 9. Reduce , š, , and 34. 10. Reduce , , 4, and 41. Ans. 79, , 138185. 11. Reduce 4, , 1, and J. Ans. 1, 15, 1, . 12. Reduce 8, 12, 13, and Ans. 334, 353, 389, 2 13. Reduce 38, 3o, , and zo. Ans. 270, 660, , 600* 14. Reduce }, 12, 16, and zu Ans. 218, 218, 2it, ** 15. Reduce ș, 7, 8, and 54. Reduce the following fractions to a common denominator :

16. Reduce ş, , and iš to fractions having a common denominator.

120, 120, 125. 17. Reduce 4, 5, and %

Ans. 260, 560. 189. 18. Reduce it, 4, and is Ans. FT) 1, 1901 19. Reduce , t, and 73. 20. Reduce 14, 4, and Ms. Ans. 199, 93, 92105 21. Reduce , tr, and 1175. Ans. 398, 40, 4636 22. Reduce 1, §, 4, and 8. Ans. 41, 42, 41, 23. Reduce , 11, and of 7%. Ans. 174, 1999', 1993. 24. Reduce }, }, 5, and 17. 25. Reduce 11, of 6, and 211. Ans. 118, 126, 228 26. Reduce , 11, 13, 4, and .

Ans. 14814, 128, 12910900, 1907. 27. Reduce 1, tỉ7, and 1728

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1784 31 552, 178131552, 17873 1 552

Ans. 80

630 630 630

3 36 42

37088064

72 28 13

ADDITION OF COMMON FRACTIONS.

227. Addition of fractions is the process of finding the value of two or more fractions in one sum.

NOTE. — Only units of the same kind, whether integral or fractional, can be collected into one sum; if, therefore, the fractions to be added do not express the same fractional unit, they require to be brought to the same, by being reduced to a common or the least common denominator.

228. To add together two or more fractions.

OPERATION.

Ex. 1. Add 1, 3, 72, and 14 together. Ans. 19 = 2

These fractions is + + Io + t = f g = lal = 24. all being twelfths,

that is, having 12 for a common denominator, we add their numerators together, and write their sum, 26, over the common denominator, 12.

Thus we obtain {, which, being reduced, 27, the sum required. 2. What is the sum of 3, , 1d, and 38 ?

Ans. 21 t.

OPERATION

8 12

16 | 20 12 40 least common denominator. 3 4

8 30 x

7 210 12 20 X

5 100 new numera20 X 4 X 3 = 240 16 15 X 11 1 65 tors.

20 1 2 x 13 1 5 6 Sum of numerators, 6 3 1

2150, Ans. Least com. denom., 2 40 The given fractions not expressing the same kind of fractional unit, we reduce them to their least common dunominator, and thus make the fractional parts all of the same kind. The fractions now all expressing two-hundred-fortieths, we add their numerators, and write the result, 631, over the least common denominator, 240, and obtain 94 = 2435, the answer required.

RULE. -- Reduce the fractions, if necessary, to a common, or the least common denominator, and write the sum of the numerators over their common denominator.

Note 1. – Mixed numbers must be reduced to improper fractions, and compound fractions to simple fractions, and each fraction to its lowest terms, before attempting to obtain the common denominator.

NOTE 2. — In adding mixed numbers, the fractional parts may be added separately, and their sum added to the amount of the whole numbers.

EXAMPLES

Ans. 319. 3. Add , ft, it, 14, and 19 together. 4. Add , 11, 13, and if together.

Ans. 23%. 5. What is the sum of 47, 47, 49, and 37 ? 6. What is the sum of 14, 4445, and 147 ? Ans. 141. 7. What is the sum of $11, 481 167, and 11?

Ans. 2441 8. Add 1, 5, 11, and s together.

Ans. 231 9. Add 11, 22, , and ļ together.

Ans. 2133 10. Add to, to, and together.

Ans. 1. 11. Add 7, 3, 1996, and 1 together.

Ans. 318. 12. Add 16, 17, 18, and % together. 13. Add , }, , 5, 1, and together.

Ans. 1886 14. Add I, 4, and 5% together.

Ans. 6301 15. Add , , and 931 together. 16. Add }, , and 44 together.

Ans. 61 17. Add 4, 73, and 84 together.

Ans. 1725. 18. Add 7, 34, and 54 together.

Ans. 9113. 19. Add 63,7%, and 4g together.

Ans. 18351 20. What is the sum of 175, 141, and 134 ? 21. What is the sum of 16%, 87, 93, 31, and 17 ?

Ans. 4046 22. What is the sum of 37113, 61418, and 814 ?

Ans. 106887 23. Add 4 of 1811, and 14 of of 6,3 together.

Ans. 1289 24. Add gof 18, and 4 of 11 of 7 together.

OPERATION.

229. To add any two fractions, whose numerators are alike. Ex. 1. Add to .

Ans. •

We first find the Sum of the denominators, 5 + 4 9

sum of the denomProduct of the denominators, 4 X 5

20 inators, which is

9, and then their product, which is 20; and the 9 being written as a numerator of a fraction, and the 20 as its denominator, the result, zo, is the answer required. The reason of the operation is, that the process reduces the fractions to a common denominator, and then adds their numerators. Hence, to add two fractions whose numerators are a unit,

Write the sum of the given denominators over their product.

2. Add i to š: Sum of the denominators X

by one of the numerators, (4 + 5) X 3 27 Product of the denominators, 4 x 5

= 130, Ans.

20 By multiplying the sum of the denominators by one of the numerators for a new numerator, and the denominators together for a new denominator, we reduce the fractions to a common denominator, and add their numerators, and thus obtain ž7 17o, the answer required. Hence, to add fractions whose numerators are alike, and greater than a unit,

Write the product of the sum of the given denominators by one of the numerators over the product of the denominators.

Ans. 120

OPERATION.

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