214. To reduce fractions to equivalent fractions having a common denominator.

1. How many fourths in 1? In ?

2. How many ninths in 1? In ? In ?

3. Express,, and, each as twelfths.

4. Change and to fractions of the same denominator. 5. What is a multiple of 4? Of 6? Of 8? Of 9? 6. What is a common multiple of 3 and 4? Of 4 and 5 ? 7. What is the least common multiple of 3, 4, and 6? 8. What is the least common multiple of the denominators of,, and §? Of }, }, and { ?

9. Reduce and to eighteenths. To twenty-sevenths. 10. Name some fractions that can be changed to 16ths. 11. Name four fractions that can be changed to 24ths.

DEFINITIONS AND PRINCIPLES.

215. A Common Denominator is a denominator common to two or more fractions.

216. The Least Common Denominator of two or more fractions is the least denominator to which they can all be reduced.

Since all higher terms of a fraction are multiples of its corresponding lowest terms (207, Note), hence the following

217. PRINCIPLES.-1. A common denominator of two or more fractions is a common multiple of their denominators.

2. The least common denominator of two or more fractions is the least common multiple of their denominators.

218. 1. Reduce,, and to equivalent fractions having a common denominator.

OPERATION.

2×3×5=30 } = 3x0 =15

XX

5

× 2 × 3

ANALYSIS.-Multiply each denominator by the other two, and the product, 30, is a common de nominator of the three. (PRIN. 1.)

But since the value of the fractions is not to be changed, each numerator must be multiplied 18 by the same multiplier as its denominator. Hence, multiplying the terms of by 3 and 5, the result is 15; of, by 2 and 5, the result is and 3, the result is 18. Or,

8; and of % by 2

To find the numerators, take such part of the common denominator 30, as the given fraction is part of 1. Thus, of 30 is 15, etc.

Reduce to fractions having a common denominator

8. Change,, and to equivalent fractions having the least common denominator.

RULE.-1. To reduce two or more fractions to equivalent fractions having a common denominator.

Multiply the terms of each fraction by the denominators

of all the other fractions.

2. To reduce them to their least common denominator. I. Find the least common multiple of the denominators of the given fractions for their least common denominator. II. Divide this common denominator by the denominator of each of the given fractions, and multiply its numerator by the quotient. The products are the new numerators, Mixed numbers must first be reduced to improper fractions. Reduce to fractions having the least common denomi nator.

219. 1. What is the sum of and ? Of and †?

2. How many times 1 is the sum of 4, 4, and 4? 3. Sold of an acre of land to one man, to another, and to a third. How much was sold to all?

How are fractions added that have a common denominator?

4. Mary paid $ for some ribbon, and $ for a pair of gloves. How much did she pay for both?

is equal to, and

ANALYSIS.-She paid the sum of $2 and $5. is equal to ; and 19 are 1%, or 1. Hence she paid $177.

5. A man having of a ton of coal, bought of a ton more. How much had he then ?

How are fractions added that have different denominators?

6. Henry gave $ for a book, $1 for a slate, and $1 for a bottle of ink. What did he pay for all?

7. What is the sum of,, and &? Of, t, and t?

8. Find the sum of §,, and .

9. Find the sum of,, and 1.

Oft,, and .

Of §, 1, and 17.

10. A farmer sold 31 tons of hay to one man, and 5} to another. How much did he sell to both?

ANALYSIS.-The sum of 3 tons and 5 tons. 5 and 3 are 8; and and are ğ, which added to 8 makes 8g tons.

11. A man bought 5 cords of wood at one time, and at another. How much did he buy in all?

How are mixed numbers added?

220. PRINCIPLE.-Fractions can be added only when

they have a common denominator, and when they express parts of like units.

WRITTEN EXERCISES.

221. 1. Find the sum of,, and .

merators, and write the sum, 75, over the common denominator 60,

and 38 14 is the required result.

2. What is the sum of 142, 257, and 7§?

RULE.-I. Reduce the given fractions to equivalent frac tions having the least common denominator, and write the sum of the numerators over the common denominator.

II. When there are mixed numbers or integers, add the fractions and integers separately, then add the results.

7. I+I+1+1=? 8. +78+21+ £= ? 9. 18+24+1}}=?

10.

+6+218+77=? 11. 18+1+73+60+J{=? 12. 1244+325+4019=?

13. Bought 3 pieces of cloth containing 1057, 864, and 58 yards respectively; how many yards in all ?

14. If it takes 5 yards of cloth for a coat, 34 yards for a pair of pantaloons, and 7 of a yard for a vest, how many yards does it take for all?

15. Four cheeses weighed respectively 465, 48, 49 and 57 pounds. What was their entire weight?

16. What number is that from which if 244 is taken, the remainder is 63?

17. A farm is divided into 4 fields: the first contains 29 acres, the second 50 acres, the third 414 acres, and the fourth 692 acres. How many acres in the farm?