WRITTEN EXERCISES.

1. Divide by 5, and also by 6.

SOLUTION.-Dividing the numerator off by 5, OPERATION.

we have 5 equals (Prin. 2).

SOLUTION 2D.-Multiplying the denominator of H by 6, we have f÷6 equals 18, or (Prin. 3).

2. Divide 822§ by 4.

SOLUTION.-Dividing 822 by 4, we have 205 and a remainder of 2; 2 equals 2, which, added to §, equals ; divided by 4 equals ; hence the quotient is 20517.

H÷5=A

OPERATION.

}÷6=18=√

OPERATION.

4)822

2057

Rule.-Divide the numerator or multiply the denominator of the dividend by the divisor.

NOTE.-Reduce a mixed number to a fraction; or divide the integer, unite the remainder with the fraction, and divide the result.

MENTAL EXERCISES.

1. How many times is contained in 3?

SOLUTION.-One is contained in 3, 3 times; and if 1 is contained in 8, 8 times, is contained in 3, 4 times 3 times, which are 12 times; and is contained in 3, § of 12 times, or 4 times. Therefore, etc.

10. In dividing 12 by 4, by what do we multiply 12, and by what do we divide the product?

11. How then may we divide a whole number by a fraction with out the analysis?

WRITTEN EXERCISES.

1. Divide 10 by the fraction §. SOLUTION.-10 divided by one equals 10, hence 10 divided by equals 6 times 10, and 10 divided by 5 sixths equals of 6 times 10, which is f times 10, which equals 12. Hence the

Rule I-Multiply the dividend by the denominator of the fraction, and divide the product by the numerator.

Rule II.-Invert the divisor and proceed as in multiplıcation of fractions.

MENTAL EXERCISES.

1. How many times is contained in §?

SOLUTION.-One is contained in §, § times, and if 1 is contained in §, times, is contained in §, 4 times times, which is 20, or 10 times; and is contained in §, of, or times. Therefore, etc.

10 In dividing § by we multiply § by what, and divide the pro

duct by what?

11. How then may we divide a fraction by a fraction without the analysis?

WRITTEN EXERCISES.

1. Divide by .

SOLUTION.- divided by one equals 7, hence divided by equals 6 times, and divided by 5 sixths equals of 6 times, which is times, which equals 3, or H. Therefore we have the following

Rule I.-Multiply the dividend by the denominator of the divisor, and divide by the numerator.

Rule II.-Invert the divisor and proceed as in multipli cation of fractions.

NOTE.-Reduce mixed numbers to simple fractions. When the divisor is a compound fraction, invert each term and multiply, cancelling when pos

sible.

1. If a yard of cloth cost of a dollar, how many yards can you buy for 12 dollars?

2. Mary gave $12 for silk at the rate of $21 a yard; how many yards did she buy?

3. How long will it require for a man to earn $33, at the rate of $2 a day?

4. If 2 barrels of apples cost $15, how many barrels can be bought for 12 dollars?

5. How many building lots of of an acre each, can we make out of 12 acres?

6. A lady distributed $29 equally among some poor people, giving them $5 apiece; how many did she aid?

7. A school girl bought 6 yards of ribbon worth 5 cents a yard; how many apples worth 13 cents each would cost the same?

8. Mrs. Bear exchanged 20 pounds of butter, at 15 cents a pound. for calico, worth 12 cents a yard; how many yards did she receive? 9. A school-boy shared 14 apples equally with his companions, giving to each 3 apples; required the number of his companions. 10. A grocer sold 8 bushels of potatoes, worth $ a bushel, and received for them eggs at the rate of $ a dozen; how many eggs did he get?

WRITTEN EXERCISES.

1. If a pound of brown sugar cost 7 cents, how many pounds can you buy for 25 cents? Ans. 31. SUGGESTION.-The analysis gives 151×19, which by cancelling and multiplying equals 3.

2. If a quart of vinegar cost 184 cents, how many quarts can be bought for 52 cents? Ans. 24.

3. If 7 yards of book muslin cost $57, how much will 5 Jards cost? Ans. $4.10. 4. If 15 yards of ribbon cost 45 cents, what will 6 yards cost? Ans. 17 cts.

5. If 21 pounds of tea cost $1888, what will 1 pound cost? Ans. $0.85.

6. If 22 barrels of sugar cost $308, what is the price per barrel? Ans. $137.

7. If 281 tons of Lehigh coal cost $183g, what is the price per ton?

8. Divide of of

9. Find the value of

Ans. $64.

by 8 of 14 of 21 of 51. Ans. 18. XXZ+H.

10. Find the value of (†×8)÷(11×11)

Ans. s.

Ans. 44.

Ans. 3.

11. Find the value of (—)XH, and the product divided

by (6-51).

Ans. 161.

12. Find the value of (+34)×93 and the product divided by (6-4). 13. Find the value of (771-44%)×(6§—5%) divided by (83+7-311).

Ans. 24.

14. If an errand boy earn $77 in a week, how long will it require him to earn $204? Ans. 24 weeks. 15. A lady distributed $231 among the poor, giving $283 to each person; how many were there? Ans. 8 persons.

16. Mr. B divided $416 equally among five persons; how much did each receive? Ans. $834.

17. By his father's will, Henry shared $9600 equally with his five brothers; how much did he receive? Ans. $1600.

18. The product of two numbers, diminished by 112, is 127, and one number is 15; required the other.

Ans. 16.

19. A boy shared 102 apples with his companions, giving to each 6 apples; required the number of companions.

Ans. 15.

20. A man divided 30 pounds of flour among the poor, giving to each 2 pounds; how many persons were there? Ans. 11 persons.

21. A seamstress bought a sewing-machine for $85.50, and paid $40 down; how much must she save each month, to pay for it in 7 months? Ans. $6.50. 22. A man's wages are $3 a day and his daily expenses are $13; how many days must he labor to enable him to buy a suit of clothes worth $46 ? Ans. 24 days.

23. Mr. Landis gained $228 on the sale of 67 acres of land; how much more would he have gained on each acre, if he had gained $382 on the whole? Ans. $22.40.

REDUCTION OF COMPLEX FRACTIONS.

183. The Reduction of Complex Fractions is the process of changing them to simple fractions.

NOTE.-A complex fraction is not really a fraction, according to the definition of a fraction. It is rather a complex fractional expression of one fraction divided by another.

Rule I.-Multiply the numerator of the complex fraction by its denominator inverted.

Rule II.-Multiply both terms of the complex fraction by the least common multiple of the denominators.