PROOF.-6 dolls. X320 dolls., the same as before.

Note.-In like manner, when the multiplier is 33, 333}, &c., if we multiply by 100, 1000, &c., of the product will be the answer. Hence,

222. To multiply a whole number by 31, 331, 3331, &c. Annex as many ciphers to the multiplicand as there are 3s in the integral part of the multiplier; then take of the number thus produced, and the result will be the answer required.

OBS. 1. The reason of this contraction is evident from the principle that annexing a cipher to a number multiplies it by 10, annexing two ciphers multiplies by 100, &c. (Art. 98.) But 3 is of 10; 33 is of 100, &c.; therefore annexing as many ciphers to the multiplicand, as there are 3s in the integral part of the multiplier, gives a product 3 times too large; consequently of this product must be the true answer.

2. When the multiplicand is a mixed number, and the multiplier is 31, 331 &c., it is evident we may multiply by 10, 100, &c., as the case may be, and of the number thus produced will be the answer required.

18. Multiply 158 by 331. 19. Multiply 148 by 31. 20. Multiply 256 by 331. 21. Multiply 1728 by 333.

Ans. 52663.

22. Multiply 297 by 3334.

23. Multiply 5611⁄2 by 31. 24. Multiply 4262 by 331.

223. To multiply a whole number by 6%, 66, 666, &c.

Annex as many ciphers to the multiplicand as there are 6s in the integral part of the multiplier; then take of the number thus produced, and the result will be the answer required.

OBS. The reason of this contraction is manifest from the fact that 63 is 3 of 10; 663 of 100, &c.

25. What will 63 tons of iron cost, at 75 dollars per ton?

QUEST.--222. How may a whole number be multiplied by 31, 331, &c.? 223 How miay a whole number be multiplied by 63, 663, &c.

PROOF.-75 dolls. ×6=500 dolls., the same as above.

26. Multiply 320 by 63.

27. Multiply 277 by 663.

28. Multiply 837 by 64.

29. Multiply 645 by 6662.

30. What will 12 acres of land cost, at 46 dollars per acre?

Operation. dolls. 46, price of 1 A.

100

8)4600 " 100 A.

Analysis.-12 acres is of 100 acres; now since 1 acre costs 46 dollars, 100 acres will cost 100 times as much, or 4600 dollars. But we wished to find the cost of only 12 acres, which is of 100 acres. Therefore of the cost of 100 acres, will obviously be the cost of 12 acres.

dolls. 575

66

12.A.

PROOF.--46 dolls. X124=575 dolls., the same as before. Note. In like manner, if the multiplier is 37, 621, or 871, we may multiply by 100, and %, §, or 1⁄2 of the product will be the answer. Hence,

224 To multiply a whole number by 121, 371, 621, or 871. Annex two ciphers to the multiplicand, then take,,, or 7, 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 121 is 1, 37 is 1, 621 is §, and 871 is of 100.

31. Multiply 275 by 371⁄2.

32. Multiply 381 by 121. 33. Multiply 425 by 371.

Ans. 10312.

34. Multiply 643 by 621.
35. Multiply 748 by 871.

225. To multiply a whole number by 13, 163, 1663, &c.

Annex as many ciphers to the multiplicand as there are integral 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 13 is of 10; 163 is of 100; 1663 is of 1000, &c.

36. What will 16 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 13. 38. Multiply 245 by 16%.

39 Multiply 489 by 163. 40. Multiply 563 by 166%.

Note.--Specific rules might be added for multiplying by 11, 111, 1111, 81, 831, 8331, 61, &c., but they will naturally be suggested to the inquisitive mind from the contractions already given.

DIVISION OF FRACTIONS.

CASE I.

226. Dividing a fraction by a whole number.

Ex. 1. If 4 yards of calico cost of a dollar, what will 1 yard cost?

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.

43 Ans.

We divide the numerator of the fraction by 4, and the quotient 2, placed over the donominator, forms the answer required.

2. If 5 bushels of apples cost 11 of a dollar, what will 1 bushel cost?

tor by it, which, in effect, divides the fraction. (Art. 188.)

1

PROOF. dolls. X5-1 dolls., the same as above. Hence,

6

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 15 divided by 5?

228. Dividing a fraction by a fraction.

10. At of a dollar a basket, how many baskets of peaches can you buy for 4 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, 4 times. Ans: 4 baskets.

11. At 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 } of a dollar is contained in . 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.)

and

First Operation.

reduced to a common denominator, become 21 and 1. (Art. 200.) Now 24÷14-21; and 21%. Ans. 1

16

16

6

5

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 two fractions together, we shall have the same result as before.

Second Operation:

5.

*x (divisor inverted), or 1 yards, the same as above,

16

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 #he 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 cominon 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 inverting 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

will give the quotient required; and X3, or 1. 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 denominator? How, when they have not common denominators? Obs. When the fractions have a common denominator, how does it appear that dividing any 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 multiplication 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 ?