226. To divide an integer or a fraction by a fraction.

a. How many thirds are there in 2? how many fourths? eighths? tenths?

b. 3 are how many times ? ? ? ? 12? 26? Illustrative Example. How many yards of cloth at of a dollar a yard can be bought for $6?

d. How many steps of a foot high will be required to ascend 8 feet? 12 feet?

e. How many pounds of tea at 7 of a dollar a pound can be bought for $5?

f. A heaped bushel is § of an even bushel. How many bushels of apples, heaped measure, can be put into barrels that hold just 7 bushels of shelled beans, even measure? g. A dealer bought $9 worth of hats at each.

How many hats did he buy?

of a dollar

h. If the dealer had bought his hats at $1 each, how many could he have bought?

i. 6 are how many times ? ? ? j. How many are 8 ÷ 3? 6 ÷ ?

10

11

12

? ? 1?
9 ÷ 11?

4 ÷ 10
??

Dividing by is equivalent to multiplying by what?

To divide an integer by a fraction, how must the integer be changed? What change must be made in a mixed number used as a divisor?

k. Divide by 1; 2 by 2; 38 by ; 23 by

How is the division of a fraction by a fraction performed when both fractions have a common denominator?

m. How many pillow slips of a yard long can be made from 12 yards of cloth? From 8 yards?

n. How many paces, each 2 o. When coffee cups hold cups can be filled with 54 pints

228. Illustrative Example.

feet, are there in a rod?

of a pint each, how many of coffee? With a gallon?

How many yards of cloth

at of a dollar a yard can be bought for of a dollar?

229. From the above illustrations may be derived the following

Rules.

1. To divide a fraction by an integer: Divide the numerator or multiply the denominator by the integer.

2. To divide an integer or a fraction by a fraction: Change the dividend and divisor to fractions having a common denominator, and then divide the numerator of the dividend by the numerator of the divisor. Or,

Invert the expression of the divisor and proceed as in mul tiplication of fractions.

*For other explanations of division of fractions, see Supplement, Art. 10.

271. How many towels 24 yards, and what part of a 272. How many sheets 2 yards in length can be cut from 42 yards, and how many yards of cloth will be left? 273. In what time, paying $14 per week, will a debt of $12 be paid?

of a yard long can be cut from yard will be left?

274. How many widths of cartridge paper of a yard wide will reach around a room 12 feet by 15 feet?

275. How many yards of sheeting at 161⁄2 cents a yard, will pay for 23 bushels of pease at 35 cents a peck?

276. 71⁄2 marks of Germany equal $1.781. What is the value of 1 mark?

277. At $23 per thousand feet for lumber, how many thousand feet can be obtained for $186,4?

278. If $13.51 were exchanged for francs at 193

cents

279. If $5691 worth of flour was sold at $41 a barrel, what was the number of barrels sold?

280. An agent sells flour, receiving 4 cents on a dollar of the amount of the sales. If he receives $25.92, what is the amount of the sales?

281. If telegraph poles are set one to every 5 rods, how many are set in 31⁄2 miles?

282. How many railroad ties will be required for 300 feet of a single track railroad which is supported at the ends and throughout by ties 21 feet apart?

231. Illustrative Example. What is the quotient of 20?

20 51 by 5. (Art. 125.) Multiplying both terms by 2, we have 40 divided by 11, which equals 3.

denotes that 20 is to be divided

Perform the operations indicated below:

Ans. 3.

Expressions like those above are called complex fractions.

How do you multiply a fraction by an integer ? a mixed number by an integer? Explain, by an example, the method of multiplying an integer by a fraction. Multiply a fraction by a fraction; explain and give the rule. How do you multiply a mixed number by a mixed number or by a fraction? How can you simplify compound fractions?

How do you divide a fraction by an integer? a mixed number by an integer? an integer by a fraction? Dividing by is equivalent to multiplying by what? Multiplying by 4 is equivalent to dividing by what? Explain by an example the method of dividing a fraction by a fraction, and give the rule. How can you simplify the expres sions called complex fractions.

SECTION IX.

MISCELLANEOUS AND REVIEWS.

232. To find what part one number is of another.

a. What part of 5 is 1? 2? 3? 4? Of 7 is 4? 5? b. One is what part of 3? 5? 10? 20? 100?

In comparing 1 with any number to see what part it is of that number, what do you take as the numerator? as the denominator ?

c. What part of 30 days is 1 day? of 32 quarts is 1 quart? d. What part of 1 dozen is 6? 3? 4? 9? 8? 2? 10? e. What part of 1 score is 10? 4? 5? 15? 16? 8? 12? f. If a cistern can be drained in 12 minutes, what part of it can be drained in 5 minutes?

g. A firm failing can pay but 30 cents on a dollar of what they owe. What part of their debts can they pay?

h. Goods which cost 50 cents per yard sold for 75 cents. What part of the cost was the gain?

i. What part of the cost is the loss, when brooms which cost 30 cents sell for 24 cents?

j. Fred and Frank buy a quire of paper together; Fred uses 8 sheets, Frank uses 16. What part of the cost should

each pay?

233. From the foregoing illustrations may be derived. the following

Rule.

To find what part one number is of another: Make the number which is the part the numerator of a fraction, and the number with which it is compared the denominator.