24

ANALYSIS. In order that fractions may be added, they must be reduced to equivalent fractions having a common =25 denominator. (Art 242.) Reducing the fractions thus, divide the sum of their numerators by the common denominator. Hence,

18+25+20=28=138.

To add two or more fractions.

265. Reduce the fractions to equivalent fractions having a common denominator; find the sum of the numerators ; divide this sum by the common denominator.

NOTE I. It is usual to reduce the fractions to their least common denominator.

NOTE II. Results should be reduced to their simplest forms.

Find the sums of the following fractions :

SOLUTION. The difierence of 8 elevenths minus 5 elevenths, is 3 elevenths. 5 .

2. Find the difference of and .

12

SOLUTION.=25, and 2=24. 25-24-36.

267. To find the difference of two fractions :

Reduce the fractions to equivalent fractions having a common denominator; find the difference of their numerators; divide this difference by the common denominator.

57=514=514

81=8,9=727 ANALYSIS. Reduce the fractions to equiva lent fractions having a common denominator. Since is less than 14, take 1=18, from the 8 of the minuend, and add it to the, then from 218 77 take 514.

First perform the operations indicated in the parentheses.

29. 59-(83+5+18).

SOLUTION. 5 times 2 thirds are 10 thirds=31; 235=10=3}. The product is always the same denomination as the multiplicand. (Art. 109 ) 2. Multiply by 5.

SOLUTION By the preceding method, 45=28=1=1}. But instead of taking 5 times as many fractional units, that is 20 fifteenths, the same result is obtained by taking the same number of fractional units,

269. To multiply a fraction by an integer :

I. Multiply the numerator of the fraction by the integer, and divide this product by the denominator.

If the integer is a factor of the denominator,

II. Divide the denominator of the fraction by the integer; then divide the numerator of the fraction by this quotient.

If the denominator and the integer have a common factor, or factors,

III. Cancel the factor or factors, common to the denominator of the fraction and the integer; find the product of the numerator and the remaining factor of the integer, and divide this product by the remaining factor of the denominator.

These three rules are all included in this general 270. RULE. Multiply the numerator of the fraction, or, Divide the denominator, by the integer.

5

by 15; by 14; by 9.

8. Multiply by 8; by 10;

9. Multiply by 10; by 15; by 18; by 20; by 36.

7 24

10. Multiply by 5; by 8; by 12; by 24; by 30. 11. Multiply by 4; by 5; by 30; by 35; by 40.

8

15

12. Multiply 71 4; by 31 by 3; 81 by 6.

PROCESS. 7×4=(7×4=28)+(3×4=2)=30 13. Multiply 163 by 6; 331 by 3; 121 by 8.

14. Multiply 371 by 2; 61 by 8; 83 by 8.

70) 246

338

510

ANALYSIS. Multiply 41 by 6, and divide the product by 70; this 388 is the product of X6. To this add the product of 85×6, reducing the lowest terms.

to

21x3=64. Observe that the result is the same by Case I. §×9= 27-64. Hence,

272. To multiply an integer by a fraction:

Multiply the numerator by the integer, and divide the product by the denominator of the fraction.

Since the product of two factors is the same, which ever is used as the multiplier. Cases I and II are identical, so far as the process is concerned. But since the denomination of the product is always that of the multiplicand, case II is required.