6. How much is 2 times ģ? How does the result compare with 3? How is it obtained from 3?

7. In multiplying, a fraction what part of the fraction do we multiply?

8. Multiply by 2. by 3. Í by 7.

9. Express 2 times in smallest terms. How is this result obtained from the fraction g?

10. In what other way then may we multiply a fraction? 11. How much is 3 times 5 ? 4 times j ? 6 times iz? 12. How much is 5 times $? 6 times ? 9 times ? 13. How much is 4 times ? 3 times iz? 5 times } ? 14. How much is 4 times ž? 7 times 4? 9 times 277?

181. PRINCIPLE.— Multiplying the numerator or dividing the denominator of a fraction by any number, multiplies the fraction by that number.

WRITTEN EXERCISES.

PROCESS.

Or,

1. Multiply 4 by 6.

ANALYSIS.–6 times 13 twen18 X 6=13*1==31

ty-fourths are 78 twenty-fourths,
or 31. Or,

Since dividing the denomi

nator multiplies the fraction 1 3 X 6=2417=*=31 (Prin.), 6 times 34 are ¥, or 31.

RULE.—Multiply the numerator or divide the denominator by
the integer.
Multiply:
Multiply:

Multiply:
2. î by 5. 7. Hi by 7. 12. by 9.
3. 14 by 7. 8. 15 by 6.

13. 16 by 13.
4. 5 by 5.
9. 76 by 8.

14. 35 by 14.
5. by 3. 10. 31 by 3. 15. by 18.
6. 11 by 17.
11. 1.4 by 7.

16. jg by 75.

17. What is the value of a load of 17 bushels of apples at

S} a bushel ?

PROCESS.

18. If a boy earns $s per day, how much can he earn in 9 days? 19. At $7 a barrel what will 7 barrels of flour cost?

ANALYSIS.-In multiplying a mixed number, we mul$7 tiply the fractional part and integer separately and add

7 the results.
51

Thus, 7 times $1=$4 =$51. 7 times $7 are $49, and 49

the sum of $19 and $51 is $541.

We may reduce the mixed number to an improper 541

fraction before multiplying.

20. If a man travel 213 miles per day, how far can lie travel in 4 days?

21. What will 13 yards of cloth cost at $61 a yard?

22. If a steamship sails 17} miles an hour, how far can she sail in 9 hours ?

CASE II.

182. To multiply an integer by a fraction.

1. Henry had 6 rabbits and sold of them to James. How many did he sell?

2. Jane had 10 cherries and gave of them to her sister. How many did she give to her sister?

3. How much is of $18? of $18?

4. How much is t of 7 apples? of 9 bushels? 1 of 5 ounces? 1 of 3 lemons?

5. How much is of 52 of 5? of 5?
6. How much is 4 of 77 of 82 of 12?
7. What is of 36? of 32? 3 of 54?
8. How much is of 35 tons? 4 of 49 horses? of 80?

183. PRINCIPLE.—Multiplying by a fraction is taking such ( part of a number as is indicated by the fraction.

PROCESS.

1. Multiply 75 by ?

ANALYSIS.—To multiply 75 hy is

to find of 75. = 3 times } } of 75 Xf=1843 -45 75 is 15, and =3 times 15, or 45.

Or,
Since =} of 3, } of 75=š of 3 times 75, or 25X3 = 45.

15

RULE.—Multiply the integer by the numerator of the multiplier, and divide the product by the denominator.

When possible use cancellation.

PROCESS,

14. A man owned a mill worth $7850. How much money should he receive for of it?

15. A sportsman shot 48 birds one day, and 6 as many the next. How many did he shoot in both days? 16. Multiply 46 by 53.

ANALYSIS.—In multiplying 46 0r, 5}=8

by a mixed number we multi46 Xxx=1288 ply by the integer and fraction 278

separately and add the products. 1238=257Thus, i of 46 is 135, or 273. 5 230

times 46=230, and 273 + 230 257

= 2577, the product. Or, we

may reduce the mixed number to an improper fraction and multiply.

17. Multiply 36 by 5}; 68; 7; 819; 1015.

CASE III.

184. To multiply a fraction by a fraction.

1. How much is 1 of 4 fifths of a yard? How much is 1 of 3 ninths of a yard? of 8 twelfths of a yard?

2. If James has g of a dollar, and Anna has į as much, how much has Anna? If Henry has as much as James, how much has he? How much is į of g? of z?

3. A man who owned of a steamship sold of his share. What part of the vessel did he sell? How much is } of 5 ?

4. How much is of 112 of 1? 6 of 43? 1 of 1?

5. If l of a yard be divided into 2 equal parts, what part of a yard will each part be? How much is } of 1 yard? 1 of 1 yard?

6. At $1 a bushel what will į of a bushel of oats cost? How much is of į? 1 of į? 1 of 1? of 1? of į?

7. What is the value of L of }? of ?? of }? of }?

WRITTEN EXERCISES.

PROCESS.

1. Multiply by

ANALYSIS.—To multiply 4 by is to

find of $, or 3 times } of 4. } of 4 4x=4*3=1

*5, and are 3 times or X3=j3. RULE.- Reduce all integers and mixed numbers to improper fractions.

Multiply the numerators together for the numerator of the product, and the denominators together for its denominator. X 1. When possible use cancellation.

2. The word of between fractions is equivalent to the sign of multiplication. Such expressions are sometimes called compound fractions. Thus, of } is equal to X.

3. Integers may be expressed in the form of fractions by writing 1 as a denominator. Thus, 4 may be written as 1.

PROCESS.

23. Multiply 1 of 1ž by of 1'. of 4.

ANALYSIS.—All the mixed X3X6X12X1=, or 11

numbers must be changed to

improper fractions, and all whole numbers expressed in the fractional form. Multiplying and cancelling we have X, or 11.

24. Multiply of 4 of 5 by ii of of 31.
25. Multiply of 12 of 8 by of 10 of 15.
26. Multiply 3.1 times 4 by 4 times of 7.
27. Multiply 54 times of 18 by 5 of 3 times } of 4.

28. What will be the cost of % of a yard of cloth at $j a yard?

29. 162 feet make a rod. How many feet are there in 54 rods?

30. A man who owned of a mill, sold of his share. What part of the mill did he sell?

31. How many yards of cloth are there in 124 pieces of cloth, each piece containing 423 yards?

32. At $167 per ton how much can be realized from the sale of 44 tons of hay?