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2. Harry had of u imilar. w .. E : IN I. dollar has Rotiis more nan s i IV DUC IES in

leave?
3. How much does m ean?
4. How much does n eare:
5. How much does from EXTE:
6. How much does smiette?

From the foregoing erampies, moen sat Surus mar be subtracted by szokong tiez IETZUN, 32 and for the same reason.

1. Boaght 2 pais ef eiach, und siz 1373 eis; bor za remained insold:

OPERATION. 1 In this example, we canno!

and , reduced to a ese- take som e bybermon denominator, make it moving 1 , isti, and ; then,

we za rezectia 13 and 20%= 2012

T are i se siseh taking 151 =15

2, 9 pass from 20 parts,

I leares 11 perts, that is. H; 48 yards, Ans. thencarrying 1 (unit, for that

which I borrowed) to 15, makes 16; then, 16 from 20 leaves 4, which, joined with it, makes 413, Ans.

2. From take land, reduced to a cominon denomi dator, give 1% and B; then, from leaves 30, Ans. From these illustrations we derive the following

RULE.
Q. What is the rule ?

A. Prepare the fractions as in addition, then the difference of the numerators written over the denominator, will give the difference required.

More Exercises for the Slate
2. From 1 take .
3. From 1} take ti.

A. 1965.
4. From lá take 6.
5 From } take

A. 18

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1 XLVI. TO DIVIDE A WHOLE NUMBER BY A

FRACTION. Lest you may be surprised, sometimes, to find in the fol. lowing examples a quotient very considerably larger than the dividend, it may here be remarked, by way of illustration, that 4 is contained in 12, 3 times, 2 in. 12, 6 times, 1 in 12, 12 times; and a half (1) is evidently contained twice as many times as 1 whole, that is, 24 times. Hence, when the divisor is 1 (unit), the quotient will be the same as the dividend; when the divisor is more than 1 (unit), the quotient will be less than the dividend; and when the divisor is less than 1 (unit), the quotient will be more than the dividend.

1. At of a dollar a yard, how many yards of cloth can you buy for 6 dollars ? 1 dollar is 4, and 6 dollars are 6 times 4, that is, 44 ; then, &, or 3 parts, are contained in 2, or 24 parts, as many times as 3 is contained in 24, that is, 8 times. A. 8 yards.

In the foregoing example, the 6 was first brought into 4ths, or quarters, by multiplying it by the denominator of the divisor, thereby reducing it to parts of equal size with the divisor; hence we derive the following

RULE. Q. How do you proceed to divide a whole number by a fraction ?

A. Multiply the dividend by the denominator of the dividing fraction, and divide the product by the numerator.

Exercises for the State. 2. At of a dollar a bushel, how many bushels of rye can I have for 80 dollars ?

OPERATION.

In this example, 80 dividend. we see more fully

illustrated the fact 16 denominator.

that division is the 480

opposite of multi

plication; for, to 80

multiply 80 by Lot Numerator, 5) 1280

we should multi ply by the numera

tor, and divide by Quotient, 256 bushels, Ans.

the denominator ;

1 XXXIX. 3. If a family consume f of a quarter of flour in one week, how many weeks will 48 quarters last the same family?

A. 128 weeks. 4. If you borrow of your neighbor Th of a bushel of meal at one time, how many times would it take you to borrow 96 bushels ? A. 960 times.

5. How many yards of cloth, at 3 of a dollar a yard, may be bought for 200 dollars ? A. 1000 yards.

6. How many times is 36 contained in 720 ? A. 140.

7. How many times is 84 contained in 300? Reduce 84 10 an improper fraction. A. 36. 8. Divide 620 by 811.

A. 75%. 9. Divide 84 by 19.8.

A. 160. 10. Divide 99 oy 41.

A. 20%. 11. Divide 160 by 2

A. 36-41 12. Divide 86 by 157.

A. 57 13. How many rods in 220 yards ? A. 40 rods. 14. How many sq. rods in 1210 sq. yards ? A. 40 sq. rods. 15. How many barrels in 1260 gallons ? A. 40 barrels.

I XLVII. TO DIVIDE ONE FRACTION BY ANOTHER.

1. At & of a cent an apple, how many apples may be bought for of a cent? How many times 4 in { ? How many times in ?

2. William gave of a dollar for one orange; how many oranges, at that rate, an he buy for f of a dollar ? How many for f of a dollar ? For ? Fer 44 ? For 47 ? For MO?

Hence we see that fractions, having a common denominator, may be divided by dividing their numerators, as well as sua tracted and added, and for the same reason.

1. At of a dollar a yard, how many yards of cloth may be bought for of a dollar? OPERATION

In this example, as Reducing the fractions f and to a the common denomi. common denominator, thus :

nator is not used. it is plain that we need not find it, but only mul. tiply the numerators

by the same numbers Then, is contained in as many | as before. This will times as 4 is contained in 9, =21. I be found to consist in

A. 24 yards. multiplying the nu.

merator of the divisor into the denominator of the dividend, and the denominator of the divisor into the numerator of the dividend. But it will be found to be more convenient, in practice, to invert the divisor, then multiply the upper terms together for a numerator, and the lower terms for a denominator; thus, taking the last example,

and , by inverting the Proof. , the quotient, divisor, become į and }; then, | multiplied by }, the divisor, **= =24 yards, as be- thus, i }, gives =, the fore, Ans.

| divisor.
From these illustrations we derive the fo lowing

RULE.
Q. How do you proceed to divide one fraction by another ?

A. I invert the divisor, then multiply the upper terms together for a new numerator, and the lower for a new denominator.

Note.--Mixed numbers must be reduced to improper fractions, and compound to simp.e terms.

PROOF. It would be well for the pupil to prove each result, as in Simple Multiplication, by multiplying the divisor and quotient together, to obtain the dividend.

More Exercises for the Slate. 2. At of a dollar a peck, how many pecks of salt may be bought for } of a dollar? A. 48 pecks.

3. Divide by it. A. 2.
4. Divide by z5. A. 159=217.

5. Divide 14 by 41. A. 376.
6. Divide 91 by 1 of $. A. 37.
7. How many times is contained in 1. A. 1}.
8. How many times is it contained in fi ? A. 2154.
9. What number multiplied by , will make 195? A. 24.

REDUCTION OF FRACTIONS.

It will be recollected, that in Reduction (1 XXIX.) whole · numbers were brought from higher to lower denominations

by multiplication, and from lower to higher denominations by division; hence, fractions of one denomination may be reduced to another after the same manner, and by the same rules. I XLVIII. TO REDUCE WHOLE NUMBERS TO THE

FRACTION OF A GREATER DENOMINATION. 1. What part of 2 miles is 1 mile? 2. What part of 4 miles is 1 mile? Is 2 miles? Is 3 miles? 3. What part of 1 yd. is 1 gr.? Is 2 grs.? Is 3 grs. ? 4. What part of 8 gallons is 1 gallon? Is 3 gallons ? 5. What part of 9 oz. is 1 oz. ? Is 2 oz. ? Is 5 oz. ? 6. What part of 7 yds. is 1 yd.? Is 6 yde.? Is 7 yds.? 7. What part of $21 is $17? Is $11? Is $13? 8. What part of 271 inches is 11 in.? Is 251 in. ? 9. What part of 1 month is 1 day? Is 2 days ? 10. What part of 1 hour is 11 minutes ? Is 21 minutes ? 11. What part of 19 cents is 11 cents ? Is 3 cents ? 12. What part of 1 d. is 1 farthing? Is 2 qrs.? Is 3 qrs.? 13. What part of 1 s. is 1 d.? Is 2 d.? Is 3 d. ?

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