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 1.-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 footnote, p. 41), 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 = 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. 129. EXAMPLES IN DIVISION. No. 1. 313 No. 2. 5)81565 $313 No. 3. 123 No. 4. 123 bush. No. 5. No. 6. 3 ab + 4 ac – 6 a 2. In Example No. 2, we are required to find of 1565 dollars. † 3. In Example No. 3, we are required 4. In Example No. 4, we are required 5. In Example No. 5, we are required 6. In Example No. 6, we are required Note.—Let it be observed that all the examples given on this page, indeed 411 division problems, may be regarded as requirements to find how many times ont 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. But if this method is adopted great care must be taken both in understanding the conditions of the problems and in the interpretation of the results obtained. * Fill the blank with the words, how many times five dollars are contained. + Fill the blank with the words, one fifth. 4)576 Division-Simple Numbers. 130. Find the quotient of 576 divided by 4. "Short Division." Explanation No. 1. One fourth of 5 hundred is 1 hundred with a remain144 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. Fouris 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 tinies. 131. Find the quotient of 8675 divided by 25. “Long Division.” Explanation No. 1. 25) 8675(347 One twenty-fifth of 86 hundred is 3 hundred, 75 with a remainder of 11 hundred; íl hundred 117 equal 110 tens. 110 tens plus 7 tens equal 117 tens. One twenty-fifth of 117 tens is 4 tens, 100 with a remainder of 17 tens; 17 tens equal 170 175 units; 170 units plus 5 units equal 175 units. 175 One twenty-fifth of 175 units is 7 units. Hence one twenty-fifth of 8675 is 347. TO THE PUPIL.–Make another explanation of this process similar to Explanation No. 2, under Art. 130. 132. PROBLEMS. 1. 93492 49 6. 5904 — 328 2. 92169 - 77 7. 7693 · 157 3. 72855 – 45 8. 8190 ; 546 4. 34694 : 38 9. 12960 - 864 5. 54875 · 25 10. 10950 · 438 (a) Find the sum of the ten quotients. Division-Decimals. 133. Find the quotient of 785.6 divided by .5. Operation Explanation. .5)785.6'5 First place a separatrix (V) after that figure in the dividend that is of the same denomination as 1571.3 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 reachedin 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 at once. 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 7. 86 ; .375 .8)340.00 .25) 468,50 12.5)38.250 2.5)87.00 .75)546.00 2.4).5'76 8. 94.5 :.8 10. 125.5 (a) Find the sum of the ten quotients. Division-United States Money. 135. Divide $754.65 by $.27. Operation. Explanation. $.27)$754.65'(2795 This means, find how many times 27 54 cents are contained in 75465 cents. 27 cents are contained in 75465 cents, 2795 214 times. 189 PROBLEM. 256 At 274 a bushel, how many bushels of 243 As oats can be bought for $754.65 ? many bushels can be bought, as $.27 is 135 contained times in $754.65. It is con135 tained 2795 times Therefore, 2795 bushels can be bought. 136. Divide $754.65 by 27. Operation. Explanation. 27)$754. 65($27.95 This means, find one 27th of $754.65 54 One 27th of $754.65 is $27.95. NOTE.-One might find 1 27th of 214 $754.65 by finding how many times $27 189 is contained in $754.65. See p. 52, Note. 256 PROBLEM. 243 If 27 acres of land are worth $754 65, 135 how much is one acre worth? 135 137. Divide $754.65 by .27. Operation. Explanation. .27(8754 .65'($2795 This means find 100 27ths of $754.65. 54 One 27th of $754.65 is $27.95. 100 27ths of $754.65 is $2795. Nore.—In practice we find one 27th of 100 times $754. 65. 256 243 PROBLEM. If .27 of an acre of land is worth 135 $754.65, how much is 1 acre worth at the 135 sanie rate? |