CHAPTER IX ALIQUOT PARTS, BUSINESS FRACTIONS, SHORT METHODS IN MULTIPLICATION 64. Aliquot parts. The aliquot parts of a number are the parts that are contained in the number an integral number of times. Thus, 50, 25, and 6 are aliquot parts of 100. The knowledge and use of these parts not only facilitates the work, but makes for accuracy and rapidity in all business computations. Thus, in finding by the usual method, the cost of 24 yd. of cloth at 124, 12 is instantly recognized as of $1.00. 8 yd. would cost $1.00; therefore 24 yd. would cost as many dollars as 8 is contained times in the number. The quantity (24) is divided by 8 and the result obtained is $3.00. (1) How many cents in $1? In $? In $2? In $? 1. Cost, price, and quantity being the elements of a business transaction, the following use of aliquot parts in the solution of business problems is given: (1) To find the cost, when the price of 1 article is 50¢, consider the quantity as dollars, and divide by 2. (2) To find the cost, when the price of 1 article is 25¢, consider the quantity as dollars, and divide by 4. (3) To find the cost, when the price of 1 article is 75¢, from the quantity considered as dollars, taken of itself. ORAL EXERCISE (1) Using method given in Sec. 64 state the cost of: a. 64 yd. cloth @ 50¢; at 25¢; at 75¢. b. 128 yd. cloth @ 50¢; at 25¢; at 75¢. C. 160 neckties @ 50¢; at 25¢; at 75¢. d. 5600 pairs of stockings @ 50¢; at 25¢; at 75¢. e. 360 tennis balls @ 50¢; at 25¢; at 75¢. (2) Compare the cost of 64 yd. cloth at 50¢ a yard with the cost of 50 yd. cloth at 64¢. (3) Twenty-five cents is of $1.00. Therefore, 4)$1.00 of $.25 is $, or $.12. 2)$ .25 $ .12 $.12 $.25 $.37 $.50 $.62 $.75 $.87 +12+12+12+12+12+12+121/ $.25 $.37 $.50 $.62 $.75 $.87 $1.00 (4) How many eighths of a dollar is $.12+$.121, or $.25? $.50? $.87? $.75? $.621? $1.00? (5) Count by eighths of a dollar to $1.25, thus: $.12} = $}; $.25 $3, or $1; $.374=$3, etc. From the above exercise it will be readily seen that: a. To find the cost, when the price of 1 article is $.12, consider the quantity as dollars, and divide by 8. 1 article is $.621, of it. b. To find the cost, when the price of 1 article is $.371⁄2, consider the quantity as dollars, and find 3 of it. c. To find the cost, when the price of consider the quantity as dollars, and find d. To find the cost, when the price of consider the quantity as dollars, and find quantity, considered as dollars, take of itself. ORAL EXERCISE (1) State the cost of: 1 article is $.87, of it, or, from the a. 96 yd. cloth @ 371⁄2¢; 50¢; 75; 6216; 871⁄2¢; 121¢. b. 288 yd. cloth @ 12; 25¢; 371⁄2¢; 50¢; 621¢; 75¢. C. 840 yd. cloth @ 12; 256; 37; 50; 62; 75; 87. d. 1080 yd. cloth @ 12; 25; 37; 50¢; 62; 75; 87. e. 3264 handkerchiefs @ 12; 25; 50; 37; 75; 62. 871⁄2¢. f. Compare the cost of 84 yd. cloth @ 371⁄2¢, with the cost of 37 yd. cloth @ 84¢. (2) What part of $1 is $.33? of $.333, or $.16? What part of one dollar is $.331+$.333, or 3)$1.00 (4) How many sixths of a dollar is $.334? $.50? $.66? $.16? $.83? $1.00? (5) Count by thirds of a dollar to $1.33, thus: $.33=$; $.663=$, etc. (6) Count by sixths of a dollar to $1.33, thus: $.163=$1; $.331=$2 or $1; $.50= $3 or $12, etc. From the above it will be readily seen that: a. To find the cost when the price of 1 article is $.163; divide the quantity, considered as dollars, by 6. b. To find the cost when the price of 1 article is $.83; from the quantity, considered as dollars, take of itself. c. To find the cost, when the price of 1 article is $.333; consider the quantity as dollars, and divide by 3. d. To find the cost, when the price of 1 article is $.66; from the quantity considered as dollars, take of itself. ORAL EXERCISE (1) State the cost of: a. 72 yd. cloth @ 163; at 331; at 50¢; at 121; at 371⁄2¢; at 75¢; at 62; at 664; at 831. b. 240 pairs stockings @ 12; at 25¢; at 331¢; at 371⁄2¢; at 50¢; at 621; at 663; at 163; at 754; at 831¢. c. 1440 pieces of braid @ 12; at 163; at 25¢; at 331¢; at 37; at 50; at 621; at 662; at 75¢; at 831; at 871⁄2¢. d. 1080 baskets @ 12; at 163; at 25¢; at 331⁄2¢; at 371⁄2¢; at 50¢; at 62; at 663; at 75¢; at 833; at 87. e. 3600 collars @ 12; at 16; at 25¢; at 331¢; at 371; at 50¢; at 621; at 663; at 75¢; at 871; at 831¢. (2) a. How many cents in of $1? In $3? In $3? In $? b. What aliquot part of $1 is 20¢? 40¢? 60¢? 80¢? c. State a short way of finding the cost, when the price of 1 article is 20; 40; 60; 80¢. (3) What is the cost of: a. 200 yd. cloth @ 20¢? At 40%? At 60%? At 80¢? b. 450 yd. lace @ 20¢? At 40¢? At 60%? At 80¢? 1 c. 800 dozen buttons @ 20¢? At 40¢? At 60¢? At 80¢? (4) a. Since $.163 is of a dollar, what part of a 2)$.163 dollar is of $.163, or $.083? $.08 b. How many twelfths of a dollar is $.083+$.08 or $.163? $.163+$.081 or $.25? $.334? $.50? $.66? $.75? $.831? c. State a short way of finding the cost, when the price of 1 article is 81¢. (5) a. How many times is 15¢ contained in $1.00? b. 63 is what part of $1.00? c. State a short way of finding the cost of goods, when the price of 1 article is 63¢. (6) State the cost of: 96 yd. tape @ 81¢ 90 yd. tape @ 6¢ 288 spools silk @ 8¢ 3060 doz. buttons @6 (7) a. How much more than $1.00 is $1.12? What fractional part of 100 is 12? 1 ILLUSTRATIVE EXAMPLE: Find the cost of 56 yd. of silk @ $1.12. Solution: At $1 a yard, the cost would be $56.00, but $1.12 is more than $1. Adding of $56 to itself, the result is $63.00, the required cost. b. State a short way of finding the cost when the price of 1 article is $1.163; $1.25; $1.331; $.66; $1.75; $1.87; $1.83}. (8) State the cost of: a. 36 yd. @ 50; 75; $1.25; $1.75; $1.163 b. 75 yd. @ 331; $1.663; $1.833; $.063/ C. 160 yd. @ $1.25; $1.75; $1.121; $1.871⁄2 d. 1080 handkerchiefs @ $.121; $.063; $.083; $.663; $.833 e. 432 yd. velvet @ $1.12; $1.163; $1.081; $1.75 1. Memorize parts of $1 given in the above table, so that in whatever form they occur, they will be familiar to you. EXERCISE (1) Find the total cost of the following (without copying): 56 yd. table damask @ $1.25 48 yd. linen finish damask @ 33 160 yd. heavy damask @ 87¢ |