DIVISION. 124. Division is (1) the process of finding how many times one number is contained in another number; or (2), it is finding one of the equal parts of a number. Note.—The word number as used above stands for measured magnitude. 125 The dividend is the number (of things) to be divided. Note.—Since in multiplication the multiplicand and product must always be considered concrete (see foot-note, p. 181), then in division, the dividend, and either the divisor or the quotient, must be so regarded. 126. The divisor is the number by which we divide. Note—The word number as used in Art. 126 may stand for measured magnitude or for pure number, according to the aspect of the division problem. In the problem 324 -f- 6, if we desire to find how many times 6 is contained in 324, the 6 stands for measured magnitude—a number of things. But if we desire to find one sixth of 324, then the 6 is pure number, and is the ratio of the dividend to the required quotient. 127. The quotient is the number obtained by dividing. Note If the divisor is pure number, the quotient represents measured magnitude. If the divisor represents measured magnitude, the quotient is pure number. 128. The sign +, which is read divided by, indicates that the number before the sign is a dividend and the number following the sign a divisor. See notes 7 and 8, page 445. 1. In example No. 1, we are required to find 2. In example No. 2, we are required to find 3. In example No. 3, we are required to find 4. In example No. 4, we are required to find 5. In example No. 5, we are required to find 6. In example No. 6, we are required to find Note.—Let it be observed that all the examples given on this page, indeed all division problems, may be regarded as requirements to find how many times one number of things is contained in another number of like things. Referring to example No. 2 given above: If one were required to find one fifth of 1565 silver dollars, he might first take 5 dollars from the 1565 dollars, and put one of the dollars taken in each of five places. He might then take another five dollars from the number of dollars to be divided, and put one dollar with each of the dollars first taken. In this manner he would continue to distribute fives of dollars until all the dollars had been placed in the five piles. He would then count the dollars in each pile. Observe, then, that one fifth of 1565 dollars is as many dollars as $5 is contained times in $1565. It is contained 313 times; hence one fifth of 1565 dollars is 313 dollars. It is not deemed advisable to attempt such an explanation as the foregoing with young pupils; but the more mature and thoughtful pupils may now learn that it is possible to solve all division problems by one thought process—finding how many times one number of things is contained in another number of like things. * Fill the blank with the words, how many limes five dollars are contained in $1565. fFill the blank with the words, one fifth of $1565. Division—Simple Numbers. 130. Find the quotient of 576 divided by 4. "Short Division." Explanation No. 1. ,4\576 One fourth of 5 hundred is 1 hundred with a remain —-— der of 1 hundred; 1 hundred equals 10 tens; 10 tens plus 7 tens are 17 tens. One fourth of 17 tens is 4 tens with a remainder of 1 ten; 1 ten equals 10 units; 10 units plus 6 units are 16 units. One fourth of 16 units is 4 units. Hence one fourth of 576 is 144. Explanation No. 2. Four is contained in 5 hundred, 1 hundred times, with a remainder of 1 hundred; 1 hundred equals 10 tens; 10 tens and 7 tens are 17 tens. Four is contained in 17 tens, 4 tens (40) times with a remainder of 1 ten; 1 ten equals 10 units; 10 units and 6 units are 16 units. Four is contained in 16 units 4 times. Hence 4 is contained in 576, 144 times. 131. Find the quotient of 8675 divided by 25. "Long Division." Explanation. 2518675^347 Twenty-five is contained in 86 hundred, 3 ye hundred times with a remainder of 11 hundred; 11 hundred equal 110 tens; 110 tens plus 7 tens equal 117 tens. Twenty-five is contained in 117 tens 4 tens (40) times with a remainder 175 of 17 tens; 17 tens equal 170 units; 170 units 175 plus 5 units equal 175 units. Twenty-five is contained in 175 units 7 times. Hence 25 is contained in 8675, 347 times. 132. Problems. 1. 93492+49 5. 5904 + 328 2. 92169 + 77 6. 7693 + 157 3. 72855 + 45 7. 8190 + 546 4. 34694 + 38 8. 12960 + 864 (a) Find the sum of the eight quotients. 117 Division—Decimals. 133. Find the quotient of 785.65 divided by .5. Operation. Explanation. .5)785.6v5 First place a separatrix (v) after that figure in 1 -71 „ the dividend that is of the same denomination as the right-hand figure of the divisor—in this case after the figure 6. Then divide, writing the decimal point in the quotient when, in the process of division, the separatrix is reached— in this case after the figure 1. It was required to find how many times 5 tenths are contained in 7856 tenths. 5 tenths are contained in 7856 tenths 1571 times. There are yet 15 hundredths to be divided. 5 tenths are contained in 15 tenths 3 times; in 15 hundredths 3 tenths of a time. Note.—By holding the thought for a moment upon that part of the dividend which corresponds in denomination to the divisor, the place of the decimal point becomes apparent. 5 apples are contained in 7856 apples 1571 times. 134. Solve and explain the following problems with special reference to the placing of the decimal point: 1. Divide 340 by .8 .8)340.0v 2. Divide 468.5 by .25 .25)468.50v 3. Divide 38.250 by 12.5 12.5)38.2v50 4. Divide 87 by 2.5 2.5)87.0v 5. Divide 546 by .75 .75)546.00v 6. Divide .576 by 2.4 2.4).5v76 7. 86 -^ .'375 = 8. 94.5 -^ ,8 = 9. 75 +- .15= 10. 125 + .5 = 11. 12.5 -*- .05= 12. 1.25 -*- .5 = (a) Find the sum of the twelve quotients. -United States Money. by $.27. Explanation. This means, find how many times 27 cents are contained in 75465 cents. 27 cents are contained in 75465 cents, 2795 times. Problem. At 27^ a bushel, how many bushels of oats can be bought for $754.65? As many bushels can be bought as $.27 is contained times in $754.65. It is contained 2795 times. by 27. Explanation. This means, find one 27th of $754.65. One 27th of $754.65 is $27.95. Note One might find 1 27th of $754.65 by finding how many times $27 is contained in $754.65. Problem. If 27 acres of land are worth $754,65, how much is one acre worth? |