PROBLEM 8.

146. To reduce a complex fraction to a simple one.

Ex. 1. The complex fraction equals what simple fraction?

욕

The operation required is only to divide a fraction by a frac

tion; thus, 1÷÷4=1×3=3. Hence,

= =

RULE. First, if necessary, reduce the numerator and denomi nator of the complex fraction each to a simple fraction; then divide the fractional numerator by the fractional denominator (Art. 145).

NOTE. A complex fraction may also be made simple by multiplying each term of the complex fraction by the least common multiple of their denominators; thus, in Ex. 1, the least common multiple of the two denominators, 4 and 7, is 28, whose factors are 4 and 7. Multiplying the numerator,, by 4, gives 3 (Art. 142, a), and multiplying 3 by 7, the other factor of the multiple, gives 21 for the numerator of the reduced fraction. In like manner, multiplying the denominator, 4, by 7, and that product by 4, gives 20 for the denominator of the reduced fraction.

146. Rule for reducing a complex fraction to a simple one? Reason? An other mode?

† = 4÷6 = 1⁄2, Ans., by Art. 143, Rule 2; or,

X

7

8

B

9

9

15. Reduce

of of of 2

to its simplest form. Ans. 1.

PROBLEM 9.

147. To reduce fractions that have not a common denominator to equivalent fractions that have a common denominator.

Ex. 1. Reduce and to equivalent fractions having a common denominator. Ans. and {.

213

5

OPERATION.

X

བབ

14

7 21

3 15

X
7 =

3 21

Multiplying both terms of each fraction by the denominator of the other fraction will not alter the value of either fraction (Art. 84, a), but it will necessarily make the denominators alike, for each new denominator is the product of the two given denominators.

Similar reasoning applies, however many fractions are to be reduced. Hence,

RULE 1. Multiply all the denominators together for a common denominator, and multiply each numerator into the continued product of all the denominators, except its own, for new numer

ators.

147. Common denominator, how found by Rule 1? How the numerators? Explanation?

2. Reduce 3, 4, and to equivalent fractions having a com

84

60

Ans. 8, 198, and f.

140

Ans. 1, 18, and 389. Ans. 18, 158, 188, and 118. 11: Reduce, 8, 7, and 11. 12. Reduce, 4, 3, and 71. 13. Reduce , 1, 4, and 14. 14. Reduce, 27, 34, and 25. 15. Reduce 13, 17, 2, and 1.

3. Reduce 3, 2, and 3. 4. Reduce, 3, and 4. 5. Reduce,, †, and §. 6. Reduce,, and . 7. Reduce, †, and §. 8. Reduce, and. 9. Reduce,, and . 10. Reduce, 1, and. (a) The foregoing rule will always give a common denominator, but not always the least integral common denominator; this, however, may always be effected by

23

RULE 2. Reduce each fraction, if necessary, to its lowest terms (Art. 141). Find the least common multiple of the denominators (Art. 127) for a common denominator. Divide this multiple by each given denominator, and multiply the several quotients by the respective numerators for new numerators.

NOTE 1. Each of these rules is founded on the principle that multiplying both terms of a fraction by the same number does not alter its value. 16. Reduce, §, and 72.

147. Rule for finding the least common denominator? Rule for finding the numerators? Principle?

17. Reduce o, §, 1, and 2.

89

18. Reduce, fe, 40, and go.

19. Reduce, fe, 13, and 13.

69

Ans. 28, 38, 33, 38.

NOTE 2. The first clause of Rule 2 is omitted by many authors, but its necessity is apparent from the following example:

20. Reduce, 1, and to equivalent fractions having the least common denominator.

Disregarding the first clause of the rule, we find 72 to be the least common multiple of the denominators, and the fractions §, , and, reduce to 4, 4, and ; but, regarding the first clause, we have,, and = 4,, and = f,, and which have a common denominator less than 72.

21. Reduce, §, 12, and . 22. Reduce §, 18, 1, and 1. -23. Reduce, 13, 12, and 18. 24. Reduce 18, 18, 18, and 18.

Ans. 18, 18, 18, and 18.

NOTE 3. In this and the following problems, each fraction should be in its simplest form before applying the rule.

REMARK. The numerators, as well as the denominators, of fractions, may be made alike by reduction; thus, and are equal in value to 18 and 1; also ‡ and f = 14 and 13; also 4, 11, and = 41, 44, and 44; etc. The process is simple, but of little practical importance, and therefore seldom presented in Arithmetic.

147. May the numerators of fractions be made alike? How?

PROBLEM 10.

148. To reduce a fraction of a higher denomination to a fraction of a lower denomination.

Ex. 1. Reduce of a penny, to the fraction of a farthing.

As 1 penny is equal to 4 farthings, so any fraction of a penny will be 4 times as great a fraction of a farthing; .. d. 4 times qr.qr., Ans.

2. Reduce of a shilling to the fraction of a farthing.

As 1s. is equal to 12d., so

d.4 times qr.#qr., Ans.

s. 12 times 4d.=d., and

=

Hence,

RULE. Multiply the fraction by such numbers as are necessary to reduce the given to the required denomination.

3. Reduces. to the fraction of a farthing.

×

s. (=3d. X 12)=3d. (=3qr. × 4)=2&qr., Ans. ; or, 7 X 12 X 4 7 X 12 X 4 28

= qr., Ans., as before.

3

NOTE.1. The sign of multiplication, in these examples, is written only between the numbers which are given before the canceling is begun; thus, in Ex. 3, no sign is written between 36 and 3, for they are not to be multiplied together, but the 3 is obtained by canceling 12 in 36. So in Ex. 4, the 12 comes from canceling 20 in 240, and the 3 from canceling 4 in 12.

4. Reduce of a ton to the fraction of a dram.

7 X 20 X 4 X 25 X 16 X 16
12 3

240

44800

dr., Ans.

3

NOTE 2. In the first statement of Ex. 5, the 163, in the numerator, is equal to 33, and, in the second statement, the 33 is retained in the numera tor as a factor in the dividend, and the 2 is put in the denominator as a fac tor in the divisor.

148. Rule for reducing a fraction from a higher to a lower denomination? Explanation? How is Ex. 5 solved?