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. 1. In Example No. 1, we are required to find in 1565 dollars.* 120 66 2. In Example No. 2, we are required to find 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, division problems, may be regarded as requirements to find how many t number of things is contained in another number of like things. Ref Example No. 2 given above: If one were required to find one fifth of 1 dollars, he might first take 5 dollars from the 1565 dollars, and put one o lars taken in each of five places. He might then take another five dollars number of dollars to be divided, and put one dollar with each of the do taken. In this manner he would continue to distribute fives of dollars the dollars had been placed in the five piles. He would then count the c each pile. Observe, then, that one fifth of 1565 dollars is as many dollar contained times in $1565. It is contained 313 times; hence one fifth of 150 is 313 dollars It is not deemed advisable to attempt such an explanation as the forego young pupils; but the more mature and thoughtful pupils may now learn possible to solve all division problems by one thought process-finding he times one number of things is contained in another number of like things this method is adopted great care must be taken both in understanding t tions of the problems and in the interpretation of the results obtained. * Fill the blank with the words, how many times five dollars are contain + Fill the blank with the words, one fifth. Division-Simple Numbers. 130. Find the quotient of 576 divided by 4. "Short Division." 4)576 144 Explanation No. 1. One fourth of 5 hundred is 1 hundred with a remainder 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." 25)8675(347 75 117 100 175 175 Explanation No. 1. One twenty-fifth of 86 hundred is 3 hundred, with a remainder of 11 hundred; 11 hundred equal 110 tens. 110 tens plus 7 tens equal 117 tens. One twenty-fifth of 117 tens is 4 tens, with a remainder of 17 tens; 17 tens equal 170 units; 170 units plus 5 units equal 175 units. 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. Division-Decimals. 133. Find the quotient of 785.6 divided by .5. Operation .5)785.6'5 1571.3 Explanation. First place a separatrix (♥) after that figure in 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 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 : Division-United States Money. 135. Divide $754.65 by $.27. Operation. $.27)$754.65' (2795 54 214 189 256 243 135 135 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 Therefore, 2795 bushels can be bought. |