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Analysis.—6tons is of 10 Operation. tons. Now, if 1 ton costs 75 dol- dolls. 75, price of 1 ton. lars, 10 tons will cost 10 times as
10 much, or 750 dollars; and f of 3)750
of 10 tons 750 dollars, (6*=j of 10,) are
250 500 dollars, which is the answer
dolls. 500, of 6 tons PROOF.--75 dolls. x6 =500 dolls., the same as above. 26. Multiply 320 by 63. 28. Multiply 837 by 63. 27. Multiply 277 by 663. 29. Multiply 645 by 6663. 30. What will 124 acres of land cost, at 46 dollars per acre ?
Analysis.--127 acres is of 100 Operation. acres; now since 1 acre costs 46 dol- dolls. 46, price of 1 A. lars, 100 acres will cost 100 times as
100 much, or 4600 dollars. But we wished 8)4600
• 100 A. to find the cost of only 125 acres, which dolls. 575 12 A. is } of 100 acres. Therefore } of the cost of 100 acres, will obviously be the cost of 122 acres.
PROOF.-46 dolls. X121=575 dolls., the same as before.
Note.-In like manner, if the multiplier is 371, 62), or 871, we may multiply by 100, and f, /, or of the product will be the answer. Hence,
224. To multiply a whole number by 121, 371, 621, or 87.
Annex two ciphers to the multiplicand, then take }, }, }, or }, of the number thus produced, as the case may be, and the result will be the answer required.
OBs. The reason of this contraction may be seen from the fact that 124 is , 371 is , 62is , and 87} is 3 of 100. 31. Multiply 275 by 37%.
Ans. 103127. 32. Multiply 381 by 12. 34. Multiply 643 by 627. 33. Multiply 425 by 371. 35. Multiply 748 by 877. 225. To multiply a whole number by 19, 163, 1663, &c.
Annex as many ciphers to the multiplicand as there are integrul figures in the multiplier, then of the number thus produced will be the product required.
Obs. The reason of this contraction is evident from the fact that 11 is } of 10; 16; is of 100; 1663 is # of 1000, &c.
36. What will 16 j bales of Swiss muslin cost, at 735 dollars
per bale ?
Solution.-Annexing two ciphers to 735 dolls., it becomes 73500 dolls.; and 73500-6=12250 dolls. Ans.
37. Multiply 767 by 1$. 39. Multiply 489 by 164. 38. Multiply 245 by 163. 40. Multiply 563 by 1664. Note.-Specific rules might be added for multiplying by 15, 115, 1114, 85, 83, 833}, 6}, &c., but they will naturally be suggested to the inquisitive mind, from the contractions already given.
226. Dividing a fraction by a whole number.
Ex. 1. If 4 yards of calico cost of a dollar, what will 1 yard past?
Analysis.--1 is 1 fourth of 4; therefore 1 yard will cost 1 fourth part as much as 4 yards. And 1 fourth of 8 ninths of a dollar, is 2 ninths. Ans. of a dollar.
Operation. We divide the numerator of the fraction by 4= Ans. 4, and the quotient 2, placed over the denomi
nator, forms the answer required. 2. If 5 bushels of apples cost 7 of a dollar, what will i bushel sost?
Since we cannot divide the numer11
ator by the divisor 5, without a re12-5=1275, or
mainder, we multiply the denominator by it, which, in effect, divides the fraction. (Art. 188.)
PROOF.-7. dolls. X 5=#f dolls., the same as above. Hence,
227. To divide a fraction by a whole number.
Divide the numerator by the whole number, when it can be done without a remainder ; but when this cannot be done, multiply the denominator by the whole number. 3. What is the quotient of 16 divided by 5 ? First Method.
15 15 3 20:5=20 Ans.
20X5100 4. Divide by 9.
7. Divide 4 by 12. 5. Divide $1 by 7.
8. Divide 1325 by 25. 6. Divide 94 by 16.
9. Divide 155 by 29.
CASE II. 228. Dividing a fraction by a fraction.
10. At } of a dollar a basket, how many baskets of peaches can you buy for of a dollar ?
Analysis.-Since of a dollar will buy 1 basket, of a dollar will buy as many baskets as } is contained times in ; and } is contained in , 4 times. Ans. 4 baskets.
11. At ș of a dollar per yard, how many yards of cloth can be bought for of a dollar ?
Obs. 1. Reasoning as before, ; of a dollar will buy as many yards, as ; is contained times in 3. But since the fractions have different denominators, it is plain we cannot divide one numerator by the other, as we did in the last example. This difficulty may be remedied by reducing the fractions to a common denominator. (Art. 200.)
First Operation. and j reduced to a common denominator, become it and (Art. 200.) Now 4+4=*; and 1=1. Ans. 1+ yards.
OBs. 2. It will be perceived that no use is made of the common denominator, after it is obtained. If, therefore, we invert the divisor, and then multiply the 'Wo fractions together, we shall have the same result as before.
Second Operation. *x* (divisor inverted)=it, or 16 yards, the same as above.
QUEST.-227. How is a fraction divided by a whole number?
229. Hence, to divide a fraction by a fraction.
I. If the given fractions have a common denominator, divide the numerator of the dividend by the numerator of the divisor.
II. When the fractions have not a common denominator, invert the divisor, and proceed as in multiplication of fractions. (Art. 219.)
Obs. 1. When two fractions have a common denominator, it is plain one numerator can be divided by the other, as well as one whole number by another; for, the parts of the two fractions are of the same denomination.
2. When the fractions do not have a common denominator, the reason that inverting the divisor and proceeding as in multiplication, will produce the true answer, is because this process, in effect, reduces the two fractions to a common denominator, and then the numerator of the dividend is divided by the numerator of the divisor. Thus, reducing the two fractions to a common denominator, we multiply the numerator of the dividend by the denominator of the divisor, and the numerator of the divisor by the denominator of the dividend; (Art. 200;) and, then dividing the former product by the latter, we have the same combination of the same numbers as in the rule above, which will consequently produce the same result.
We do not multiply the two denominators together for a common denominator; for, in dividing, no use is made of a common denominator when found, therefore it is unnecessary to obtain it. (Art. 228. Obs. 2.)
The object of inverling the divisor is simply for convenience in multiplying.
3. Compound fractions occurring in the divisor or dividend, must be reduced to simple ones, and mixed numbers to improper fractions.
230. The principle of dividing a fraction by a fraction may also be illustrated in the following manner. Thus, in the last example,
Dividing the dividend by 2, the quo- Operation. tient is 16. (Art. 188.) But it is required 중=
=2=76 to divide it by 1 third of two; consequently Hox3=4 the Zo is 3 times too small for the true And it=lt. Ans. quotient; therefore multiplying to by 3, will give the quotient required ; and 7X3=it, or 17.
Note.-By examination the learner will perceive that this process is precisely
QUEST.—229. How is one fraction divided by another when they have a common denoninator ? How, when they have not common denominators ? Obs. When the fractions have a common denominator, how does it appear that dividing one numerator by the other will give the true answer ? When the fractions have not a common denominator, how does it appear that inverting the divisor and proceeding as in multiplicatiou will give the true answer? What is the object of inverting the divisor? How proceed when the divisor or dividend are compound fractions, or mixed numbers ?
the same in effect as the preceding; for, in both cases the denominator of the dividend is multiplied by the numerator of the divisor, and the numerator of the dividend, by the denominator of the divisor. 12. Divide of by 23.
Ans. bå, or 13. Divide 83 by 37.
Ans. *, or 221. 14. Divide the by .
16. Divide 551 by 164. 15. Divide so by
17. Divide 464 by 689. 231. The process of dividing fractions may often be contracted by canceling equal factors in the divisor and dividend; (Art. 146 ;) or, after the divisor is inverted, by canceling factors which are common to the numerators and denominators. (Art. 191 )
18. Divide of 4 of it by off of +. Operation.
For convenience we arrange the numerators, (which answer to dividends,) on the
right of a perpendicular line, and the de11 2
nominators, (which answer to divisors,) on 45
the left; then canceling the factors, 2, 3, 4,
and 7, which are common to both sides, 17
(Art. 151,) we multiply the remaining fac1115=Ans. tors in the numerators together, and those
remaining in the denominators, as in the rule above. Hence,
232. To divide fractions by CANCELATION.
Having inverted the divisor, cancel all the factors common both to the numerators and denominators, and the product of those remaining on the right of the line placed over the product of those remaining on the left, will be the answer required.
OBs. 1. Before arranging the terms of the divisor for cancelation, it is always necessary to invert them, or suppose them to be inverted.
2. The reason of this contraction is evident from the principle, that if the numerator and denominator of a fraction are both divided by the same number, the value of the fraction is not altered. (Arts. 148, 191.) 19. Divide 189 by 63.
QUEST.-232. How divide fractions by cancelation ? How arrange the terms of the given fractions ? Obs. What must be done to the divisor before arranging its terms? How does it appear that this contraction will give the true answer ?