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More Exercises for the Sute. 3. What is the amount of 16% yards, 17$ yds, and 3f yards ? 9. 374.
4. Add together and 4. A. 1385.
SUBTRACTION OF FRACTIONS.
1 XLV. 1. William, having of an orange, gave & to Thomas; how much had he left? How much does from ieave ?
2. Harry had of a dollar, and Rufus $; what part of a dol'ar has Rufus more than Harry? How much does from $ Leave ?
3. How much does 18 from 1 % leave ?
From the foregoing examples, it appears that fractions may be subtracted by subtracting their numerators, as well as added, and for the same reason.
1. Bought 203 yards of cloth, and sold 154 yards; how much remained unsold? OPERATION.
In this example, we cannot taks 3 and 4, reduced to a com from 1, but, by borrowing 1 mon denominator, make (unit), which is 12, we can pro
and 12; then, ceed thus 14 and I are 1, from 204 = 20
which taking i., or 9 parts from 151 = 152
20 parts, leaves 11 parts, that is,
1}; then, carrying 1 (unit, for that 410 yds., Ans. which I borrowed) to 15, makes 16;
then, 16 from 20 leaves 4, which, joined with+1, makes 411, Ans.
2. From take . % and , reduced to a common denomina tor, give }8 and 3%; then, from 18 leares 30, Ans.
From these illustrations we derive the following
RULE. 1. What is the rule? A. Prepare the fractions as in addition, then, the difference of the numerators, written over the denom inator, will give the aifference required.
More Exercises for the Slate. 2. From 1 take
A. 2 3. From 15 take ti.
A. 96 4. From tá take .
A. 412. 5. From } take .
A. 18. 6. From 11 take f.
A. 16 7. From of $ take .
A. . 8. From 1 of % take 1 of 2. 9. From 194 take of 19. A.466
DIVISION OF FRACTIONS.
1 XLVI. To divide a Whole Number by a Fraction.
I est you may be surprised, sometimes, to find in the following 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; half (!) 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 £, and 6 dollars are 6 times, that is, ; then for 3 parts are contained in 4, or 24 parts, as many times as 3 is contained in 24; that is, 8 times.
A. 8 yards
Til 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. I How do you proceed to divide a whole number by a frac. tion? A. Multiply the dividend by the denominator of the dividing fraction, and divide the product by the numerator.
Exercises for the Slate. 2. At Po of a dollar a bushel, how many bushels of ryo can I have for 80 dollars ? OPERATION.
In this examplo, we 80 dividend.
see more fully illustrated 16 denominator. the fact that division is
the opposite of multipli480
cation; for, to multiply 80
80 by 16, we should
multiply by the numerNumer. 5) 1280
ator, and dividc by the
denominator, 1 XXXIX Quotient, 256 bushels, Ans.
3. If a family consume & of a quarter of flour in one week, now many weeks will 48 quarters last the same family?
A. 128 weeks. 4. If
you borrow of your neighbour ab of a bushel of mea] at one time, how many times would it take you to borrow 96 dushels ? A, 960 times.
5. How many yards of cloth, at ļ of a dollar a yard, may be bought for 200 dollars ? A. 1000 yards.
6. How many times is in contained in 720 ? A. 140.
7. How many times is of contained in 300: Reduce 8f to an improper fraction. A. 36.
8. Divide 620 by 811 4. 75%.
9. Divide 84 by 18. A. 160. 10. Divide 92 by 41. A. 20% 11. Divide 100 by 2 .A. 36+1 12. Divide 86 by 157. 9. 57624 12. How many rods in 20 yards? A. 40 rods.
14 How many sq. rods in 1210 sq. yards? A. 40 sq. roda 15. How many barrels in 1260 gallons A. 40 barrels. 1 XLVII. To divide one Fraction by another. 1. At 1 of a cent an apple, how many apples may be bought for of a cent? How many times I in? How many times in ?
2. William gavo of a dollar for one orange; how many oranges, at that rate, can he buy for f of a dollar'? How many for of a dollar ? For ķ? For 24? For 2? For xo?
A. 2 yards.
Hence we see that fractions, having a common denominator, may be divided by dividing their numerators, as well as subtracted 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 ?
In this example, as common denoninator, thus :
the common denomi.
nator is not used, it is } *
plain that we need not find it, but only multi
ply the numerators by Then A is contained in ias many the same numbers as times as 4 is contained in 9,=21.
before. This will be found to consist in
multiplying the numerator of the divisor into the denominator of the dividend, and the denominato of the divisor into the numerator of the divi. dend. But it will be found to be moru 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, mul divis :r, become i and if;
tiplicd by $, the divisor, thus, then, idi==2; yards,
X1, gives is the divisor. as before, Ans. From these illustrations we derive the following
RULE. 1. Horo do you proceed to divide one ; action by another
. I invert the divisor, then multiply the per terms logethe for a new numerator, and tho lower for a vis denominator
Note. Mixed numbers must be reduced to improper fractions, and compound to simple terms.
Proof. It would be well for the pupil to prove cach result as in Simple Multiplication, by multiplying the divisor and quotient together, to obtain the dividond.
More Exercises for the Slate. 2. Ats of a dollar a peck, how many pecks of salt may be bought for } of a dollar ? A. 48 pecks.
3. Divide by jt. A. =2 4. Divide by zł. A. 144=214. 5. Divide 1% by it. A. 336. 6. Divide 94 by of g. A. 37. 7. How many times is contained in ? .4. 11. 8. How many times is to contained in 1? 2134. 9. What number multiplied by 5 will make 13? A. 215.
REDUCTION OF FRACTIONS. It will be recollected!, that in Reduction (1 XXIX.) włrole 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. 1 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 qrs.? Is 3 qrs.? 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 yds.? 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 grs ? 13. What part of 1s is 1 d.; Is 2 d.? Is 3d.?