42. Rule.-Reduce 1 to a fraction having the given denominator. Multiply the given decimal by the fraction so obtained, and the result will be the fraction required. UNITED STATES MONEY. 44. The sign for dollars is $. read 25 dollars. It is read dollars. $25 is Since there are 100 cents in a dollar, one cent is 1 onehundredth of a dollar; the first two figures of a decimal part of a dollar represent cents. Since a mill is of a cent, or Too of a dollar, the third figure represents mills. Thus, $25.16 is read twenty-five dollars and sixteen cents; $25.168 is read twenty-five dollars, sixteen cents, and eight mills. 45. To change dollars to cents, move the decimal point two places to the right; to change dollars to mills, move the decimal point three places to the right. Thus, to change $143.75 to cents, we have $143.75 = 14,375 cents, or 143,750 mills. The decimal point is always understood as following the unit figure, whether written or not; hence, to change $100 to cents, write it thus, $100.; to move the decimal point two places to the right, it is necessary to annex two ciphers, thus, 10000; in other words, $100 10,000 cents. Moving the decimal point two places to the right is evidently the same thing as multiplying by 100, since it changes the unit figure from the first order to the third order. Thus, in 143.75, 3 is a figure of the first order; but, in 14375., 3 is a figure of the third order; and, since all the other figures have also been advanced two orders, the number has been multiplied by 100. 46. To change cents to dollars, move the decimal point two places to the left; or, to change mills to dollars, move the decimal point three places to the left. Thus, to change 143,750 mills to dollars, move the decimal point (understood to follow the 0) three places to the left, obtaining $143.750 = $143.75. Similarly, 14,375 cents $143.75, and 10,000 cents = $100.00. = ALIQUOT PARTS. 47. An aliquot part of a number is a number that will divide it without a remainder. For example, 15 is an aliquot part of 60, because 15 is an exact divisor of 60. For the same reason, 5, 6, 10, 12, etc. are also aliquot parts of 60. 48. About the only case of any importance to the business man is that concerning the aliquot parts of 100, which are given in the following table: The aliquot parts given in the above table should be carefully memorized; in many cases calculations may be shortened by using them. Thus, in order to multiply any number by one of the numbers in the above table: 49. Rule. Move the decimal point of the multiplicand two places to the right, and divide the multiplicand by the denominator of the fraction opposite the multiplier in the table. Then multiply the result by the numerator of the fraction. EXAMPLE.-Multiply 478.4 by 25. SOLUTION.-Applying the rule, the fraction opposite 25 is. Hence, 478.4 X 25 47,840 4 X 1 11,960. Ans. = The result may be proved to be true by actual multiplication. 50. The reasoning on which the rule is based is this: The last example required the multiplication of 478.4 by 25. = But 25 is of 100; that is, 25 = 100. Hence, in multiplying by 25, 100 can be used as a multiplier, and the operation becomes 478.4 X 100 47840 = 11,960. Since, in order to multiply by 100, it is necessary only to move the decimal point two places to the right, the correctness of the rule is evident. EXAMPLE.-Multiply (a) 50.64 by 163; (b) 1,894 by 37. 8 51. It matters not where the decimal point is placed in the number denoting the aliquot part, the principle can still be applied with a slight modification. Thus, suppose it is required to multiply 72 by 625. Now, 621, or 62.5, is of 100; that is, 62.5 100 x = 500. = It is an axiom in mathematics that if equals be multiplied or divided by equals, the result will be equal; that is, if 4 = 4, and both 4's be multiplied or divided by the same number, the results will be equal. Thus, multiplying by 10, 40 40; dividing by 10, .4.4. Hence, if in the expression 62.5 = 500, both numbers are multiplied by 10, the result is 625 5000, or 625 = 1,000 X. Therefore, to multiply 72 by 625, move the decimal point three places to the right, divide by 8 and multiply by 5. (It makes no difference whether we divide by the denominator or multiply by the numerator first.) Then, 72 × 625 72000 × 5: 45,000. Had it been required to multiply 72 by .0625, we note that 62.5 100 x, and that moving the decimal point in 62.5 three places to the left will give .0625. Hence, moving the decimal point in the other number (the 100) three places to the left, we have .0625 5 .5 = = = whence, 72 x .0625 = 8 8 = 4.5. 8 ог To multiply any number by 5, multiply by 10 and divide by 2. 52. To divide by one of the aliquot parts of 100, simply reverse the rule. Thus: Rule.-Move the decimal point two places to the left, multiply by the denominator of the equivalent fraction in the table, and divide by the numerator. 53. To divide by a number having all the figures of the aliquot part, but with the decimal point in a different place, proceed in the same manner as in Art. 52. Thus, to divide 45,000 by 625, we have 62.5 100 ; hence, 625 = 1,000 × = 5000, and 45,000 ÷ 625 = 8 = 45,000 ÷ 5000 8 Such 54. The student will find the principle of aliquot parts extremely convenient for accurate and rapid work. numbers as 12, 16, 25, 331, 37, 621, 75, and 87 are of very frequent occurrence in business accounts, and the method can be readily employed in such cases. If these numbers are given as .12, .163, etc., as is usually the case, the example becomes even easier, as they are then equivalent to,, etc. EXAMPLE.-What will be the cost of 48 yards of carpeting at $1.33 per yard? 1. Find the cost of 12 dozen hats at $4.12 per dozen. Ans. $49.50. 2. Find the cost of 24 boxes of note paper at 163 cents per box. 3. Find the cost of 75 books at 25 cents each. Ans. $4. Ans. $18.75. 4. Find the cost of 30.19 hundredweight of bran at 62 cents per hundredweight. Ans. $18.867. Ans. $81. Ans. $4.35. 5. Find the cost of 36 pairs of shoes at $2.25 per pair. |