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OPERATION

Ex. 1. Add is, , 1z, and 11 together. Ans. 29 2.

These fractions istitis + 1} = = 1 = 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, : 2, the sum required. 2. What is the sum of ], is, it, and }; ?

Ans. 2117.

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 6 5 tors.

2012 X 13 156 Sum of numerators, 6 3 1

2141, Ans. Least com. denom., 2 40 The given fractions not expressing the same kind of fractional unit, we reduce them to their least common denominator, 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

2216, 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.

obtain it

EXAMPLES.

Ans. 314. 3. Add , ft, 14, 14, and 16 together.

Ans. 21%. 4. Add , 21, 21, , and 3 together. 5. What is the sum of 4, 87, 49, and 37 ? 6. What is the sum of 4, 45, 46, and 147 ? Ans. 174. 7. What is the sum of 11, 431, 491, and ?

Ans. 241. 8. Add 1, }, 11, and š together.

Ans. 23. 9. Add 1, , , and į together.

Ans. 2488 10. Add is, is, fd, and } together.

Ans. 9213.

Ans. 1. 11. Add B 1:56, and į together. 12. Add 16, 17, 18, and together.

Ang. 378 13. Add }, }, }, }, , and together.

Ans. 188 14. Add , #, and 53 together.

Ans. 6301 15. Add t, 22, and 91 together. 16. Add , , and 4d together.

Ans. 63 17. Add 4, 73, and 8f together.

Ans. 1726. 18. Add , 31, and 5$ together. 19. Add 63, 73, and 43 together.

Ans. 1836. 20. What is the sum of 175, 143, and 137 ? 21. What is the sum of 167, 87, 9, 31, and 17 ?

Ans. 4045 22. What is the sum of 37119, 61438, and 814?

Ans. 106827 23. Add 4 of 1811, and 11 of of 6 11 together.

Ans. 12693 24. Add of 18, and 4 of 11 of 7 4 together.

Ans. ਨੂੰਹ

OPERATION

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

We first find the Sum of the denominators, 5 +4= 9 sum of the denomProduct of the denominátors, 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, jo, 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

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 47 127, 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. 10

OPERATION.

130, Ans.

EXAMPLES.

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3. Add ik to £, £ to , } to }, } to }, } to š, } to $, što . 4. Add I to 1, i to je t'i to 1, i to }, is to đ, ili to 7. 5. Add to to }, i to }, t'o to , to to }, i'o to , To to +. 6. Add {to }, to }, {to , {to }, i to § 1 to 7, 4 to g. 7. Add } to }, } to 4, š to $, š to š, š to }, } to }, } to . 8. Add 7 to 3, 4 to }, 7 to 1, 7 to }, 7 to 5, 4 to $, 7 to š. 9. Add f to }, } to }, } to 4, to }, } to d, f to 7, f to go 10. Add to fr, to 1s, fto , to , to foto 11. Add to 4, to $, to tt, to , to ą, to , i to 4. 12. Add ; to , fto , to , to , & to & éto 13. Add to $,$to , &to , i to , fi to , i to fg. 14. Add to far, i to i, i to , i to 15is to 17, is to . 15. Add it to 11, io to 14 10 to 16 to 17, 1% to 98.

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SUBTRACTION OF COMMON FRACTIONS.

230. SUBTRACTION of Fractions is the process of finding the difference between two fractions.

NOTE. — When the fractions express different fractional units, they require to be brought to those of the same kind before the subtraction can be performed.

To subtract one fraction from another.

Ans. Tਡ

OPERATION.

5 12

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Ans. 44.

OPERATION.

Ex. 1. From 11 take rze

- }. The fractions both being twelfths, having 12 ii

for a common denominator, we subtract the

less numerator from the greater, and write the difference, 6, over the common denominator, 12. Thus, we have

as the difference required. 2. From 11 take #.

The given frac7 7 common denominator.

tions not express

ing the same kind 11 7 X10 : 70 new numerators.

of fractional unit, 711 x 4= 44

we reduce them 26 dif. of numerators.

to a common de

nominator, and 77 common denominator.

thus make the

fractional parts all of the same kind. We next find the difference of the new numerators, which we write over the common denominator, and obtain 46, the answer required.

Rule. - Reduce the fractions, if necessary, to a common, or the least common denominator. Write the difference of the numerators over their common denominator.

NOTE. ~ If the minuend or subtrahend, or both, are compound fractions, they must be reduced to simple ones.

36

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EXAMPLES.

Ans. 17 3. Subtract fit from 14.

Ans. jog 4. Subtract A from ts. 5. From take 387

Ans. • 6. From 1 take 15.

Ans. ag. 7. From take 18. 8. From , take 14.

Ans. I 9. Subtract 24 from to

Ans. 36 10. Subtract 16 from 144

Ans. do 11. Subtract 100 from 1o.

Ans. 269 12. Subtract 3277 from 5-96 1728

172go Ans.

13 13. Subtract 178, from 1000 14. From 14 taken

Ans. 17. 15. From To take iz

Ans. 134 16. From 17 take is:

Ans. 17. From 1; take jg.

Ans. 3931 18. From if take it. 19. From 1 take io.

Ans. 136 20. From 1} take zo

Ans. 21. From 1, take to

Ans. 36 22. From taket

Ans. 19% 23. From 74 take 4 of 9. 24. What is the value of of 8 - of 5 ?

Ans. I. 25. What is the value of 1 of 3

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1 of 2 ?

Ans. 3.

231. To subtract a proper fraction or a mixed number from a whole number. Ex. 1. From 7 take 32.

Since we have no fraction from which to subtract From 7 the s, we must add 1, or its equal, $, to the minuend,

leaves . We write the below Take 3g the line, and carry 1 to the's in the subtrahend, and Rem. 3} subtract as in subtraction of simple whole numbers.

The result will be obtained, if we

OPERATION.

Subtract the number denoting the numerator from that denoting the denominator, under the remainder write the denominator, and adding one to the whole number in the subtrahend, subtract the sum from the minuend.

NOTE. — When the subtrahend is a mixed number, we may reduce it to an improper fraction, and change the whole number in the rninuend to a fraction having the same denominator, and then proceed as in Art. 230.

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FIRST OPERATION.

4 43

SECOND OPERATION,

232. To subtract one mixed number from another mixed number. Ex. 1. From 88 take 44.

Ans. 333. We first reduce the fractional parts to a comFrom 87

815 mon denominator, and obtain as their equivalents Take 4

and h. Now, since we cannot take it from

f, we add 1, equal to 3, to the in the minRem. 33; uend, and obtain .. From taking it, we

have left it, which we write below the line, and carry 1 to the 4 in the subtrahend, and subtract from the 8 above as in subtraction of simple whole numbers.

In this operation, we reduce From 84 5229

the mixed numbers to improper Take 44

fractions, and these fractions to

a common denominator. We Rem.

127 3 then subtract the less fraction

from the greater, and, reducing the remainder to a mixed number, obtain 3}}, as before. Hence, in performing like examples, we may

Reduce the fractional parts, if necessary, to a common denominator, and subtract the fractional parts and the whole numbers separately. Increase the fractional part of the minuend, when otherwise it would be less than the subtrahend, before subtracting, by as many parts as it takes to make a unit of the fraction (Art. 208), and carry 1 to the whole number of the subtrahend before subtracting it. Org

Reduce the mixed numbers to improper fractions, then to a common denominator, and subtract the less fraction from the greater.

295
35

= 168

35

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