ART. 26. To reduce an improper fraction to a whole or mixed number.

Ex. 1. Reduce 42 to a mixed number.

49÷5-9 Ans.

Explanation.-Since 5 fifths make 1, there will be as many ones in 49 fifths as 5 is contained times in 49, which is 94.

ART. 27. To reduce a whole or mixed number to an im

proper fraction.

Ex. 1. Reduce 5 to an improper fraction.

Explanation.-Since there are 4 fourths in 1, in 5 there are 5 times 4 fourths-20 fourths, and 20 fourths +3 fourths= 23 fourths. Ans. 23.

RULE.

Multiply the whole number by the denominator of the fraction, to the product add the numerator, and under the result place the denominator.

24729

5. Reduce 1236 to an improper fraction. Ans, 247

6. Reduce 5 to an improper fraction.

20

7. Reduce 23% to an improper fraction. 8. Reduce 13313 to an improper fraction. 9. Reduce 563 to an improper fraction. 10. Reduce 80061 to an improper fraction.

11. Reduce 24 to fourths.
12. Reduce 35 to twentieths.

13. Reduce 312 to twelfths.
14. Reduce 19 to twenty-fifths,

15. Reduce 1008 to ninths.

20

Ans, 1912.

Ans. .

Ans. 7.

ART. 28. To reduce compound fractions to simple ones,

Ex. 1. Reduce of to a simple fraction.

of

Explanation of }

and if of is, of

4 x 52

5 × 5

3 Ans. is, and of is 5 times or, 13

is 4 times, or = Ans. This is in effect multiplying the numerators together and also the denominators.

RULE.

Multiply the numerators together for the numerator of the simple fraction, and the denominators together for its denominator.

Note. If there are whole or mixed numbers, first reduce them to improper fractions.

Examples.

2. Reduce of of to a simple fraction.

3. Reduce 31 of 2 of to a simple fraction. 4. Reduce 2 of 1 of 3 to a simple fraction.

15

6. Reduce of 1 of 3 of 2 to a simple fraction. Ans. †.

7. Reduce of 6 to a simple fraction.

8. Reduce of 2 of 3 to a simple fraction,

CANCELLATION.

ART. 29. The above operations may be abbreviated by indicating the multiplications to be performed, and then cancelling the factors common to both terms, as shown in the following examples.

Ex. 1. Reduce of of of 21 to a simple fraction.

2. Reduce of of of 1⁄2 of; of 1 to a simple fraction.

Note.-1 remains as a factor in the numerator.

3. Reduce of 41 of of 4. Reduce

99

to a simple fraction. of 213 of 75 of

to a simple fraction.

Ans. 75.

Ans. 33.

5. Reduce of of of 1% of 12 to a simple fraction.

6. Reduce of 31 of 1 of 11 of 1 of 71⁄2 to a simple fraction. Remark. The principle of cancellation may often be used with great advantage. Whenever, to obtain a certain result, several multiplications and divisions are to be performed, indicate the operations and cancel the factors common to the multipliers and divisors.

7. Divide the product of 24, 163, 8, 331 by 12, 163, and 663.

8. Multiply 48, 32, 5280 and 27 together, and divide the result by 16, 264, 54 and 6.

Ans. 160.

9. How many cords of wood in a pile 144 feet long, 12 feet high and 3 feet wide?

10. Multiply 9, 8, 18, 45, 36, 90, 81 together and divide the result by 72, 180, 27, 24, 4 and 18. Ans. 25.

ART. 30. To reduce fractions to a common denominator. Ex. 1. Reduce,,, and to equivalent fractions having a common denominator.

13 15 20

247 24

5 5

ΤΣ

247

15

16

Solution. First Method.-It is evident upon a little inspection that each of the fractions can be changed to twentyfourths. According to Art. 24, 3=!!, !=!1, }=}}, } = !!, and. Hence, §, §, and are respectively equal to 1, 1, 1, 1 and 14, fractions having a common denominator. Second Method. The least common multiple of 4, 8, 6, 3 and 12 (denominators) found by Art. 22, is 24, which, being divided by 4, 8, 6, 3 and 12 respectively, give the multipliers by which both terms of their respective fractions are to be multiplied. 3×6

2X8

3х6
4× 6

18 5X315 5 X 4
247
8 X 3
6×4

247

120

247

16, and 7×2

3 X 8

249

12×2

RULE FOR SECOND METHOD.-Find the least common multiple of the denominators. Then divide the least common multiple by the denominator of each fraction and multiply both of its terms by the quotient.

Note. The first method is the one generally used. In ordinary examples, the common denominator can be seen at a glance.

Examples.

Reduce the following fractions to equivalent fractions having a common denominator.

ADDITION OF COMMON FRACTIONS.

ART. 31. Ex. 1. What is the sum of,, and ?
3 + 2 + 3 + } =

10+18+18+21=1=3} Ans.

Ex. 2. Add 3 of 3, of and 2}.

223

Reduce the fractions to a common denominator; then add their numerators, and under their sum place the common denominator.

Notes.-1. First reduce mixed numbers to improper fractions, and compound fractions to simple ones.

2. The integers may be set aside and subsequently added to the sum of the fractions.