4 units annexed to 280 units make 284 units. 59 is contained 4 times in 284, with a remainder of 48, a number smaller than the divisor, hence the division is completed, and we have found that 59 is contained 364 times in 21524, with a remainder 48. Observe carefully the following analysis of the process in the preceding example: From these illustrations we obtain the following 130. RULE.-I. Find how many times the divisor is contained in the least number of orders at the left of the dividend that will contain it, and write the result for the first figure of the quotient. II. Multiply the divisor by this quotient figure, and subtract the result from the part of the dividend that was used; to the remainder annex the next lower order of the dividend for a new partial dividend and divide as before. Proceed in this manner with each order of the dividend. III. If there be at last a remainder, place it after the quotient, with the divisor underneath. PROOF.-Multiply the divisor by the quotient and add the remainder, if any, to the product. This result will be equal to the dividend, when the division has been performed correctly. 132. Additional examples for practice should be taken from Arithmetical Table No. 2, page 58, as follows • Dividend four figures, Divisor two. 1. Take the dividends in order from columns A, B, C, D ; B, C, D, E; C, D, E, F; D, E, F, G; E, F, G, H; F, G, H, I; G, H, I, J. 2. Take as divisors in each set the figures immediately above the dividend, in the two right-hand columns of those used. Dividend six figures, Divisor three. 1. Take the dividends in order from columns A, B, C, D, E, F; B, C, D, E, F, G; C, D, E, F, G, H; D, E, F, G, H, I; E, F, G, H, I, J. 2. Take the divisors as before from the three right-hand columns of those used for dividend. WRITTEN EXAMPLES. 133. 1. A hogshead of molasses contains 63 gallons; how many hogsheads in 16002 gallons? SOLUTION.-As one hogshead contains 63 gallons, 16002 gallons will make as many hogsheads as 63 is contained times in 16002. 16002 ÷ 63 =254. Hence there are 254 hogsheads in 16002 gallons. 2. A man wishes to carry to market 2623 bushels of potatoes; if he carries 61 bushels at a load, how many loads will they make ? Ans. 43 loads. 3. An army contractor furnished horses, at $72 each, to the amount of $1131264; how many did he furnish? Ans. 15712. 4. A certain township contains 192000 acres ; how many square miles in the town, there being 640 acres in a square mile? Ans. 300 miles. rate of $43 an acre; 5. A man paid $1548 for a farm at the how many acres did the farm contain? 6. How many acres of land at $100 an for $26700? Ans. 36 acres. acre, can be bought Ans. 267 acres. 7. A certain product is 43964 and one of the factors is 58; what is the other factor? Ans. 758. 8. At what yearly salary will a man earn 20400 dollars in 17 years? Ans. $1200. 9. If light travels 192000 miles in a second, in how many soconds will it travel 691200000 miles? Ans. 3600. 10. Henry Pendexter divided $47400 into 3 equal parts, one of which he gave to his wife; the rest, after paying a debt of $3280, he divided equally among 4 children; what did each child receive? Ans. $7080. 11. A piano maker expended in one year for material $20644, and for labor $4925, paying each week the same amount; what was his weekly expense? 12. A western farmer raised in one year 13475 bushels of wheat; the average yield was 49 bushels per acre; how many acres did he have sown? DIVISION BY FACTORS. PREPARATORY STEPS. 134. STEP I.-Any number may be expressed in terms of one of its factors by taking another factor as the Unit. (14.) Thus, 124+4+4; hence, 12 may be expressed as 3 fours, the four being the unit of the number 3. Write the following numbers: 1. Express 12 as 2's; as 3's; as 4's; as 6's. 2. Express 36 as 3's; as 9's; as 18's; as 12's; as 6's. 3. Express 45 as 5's; as 3's; as 9's; as 15's. 4. Express 42 and 24 each as 6's. 5. Express 45 and 225 each as 9's; as 5's; as 3's. 135. STEP II.— When a number is made into three or more factors, any two or more of them may be regarded as the unit of the number expressed by the remaining factors. For example, 24 = 3 × 4 × 2. This may be expressed thus, 243 (4 twos). Here the 3 expresses the number of 4 twos; hence, (4 twos) is regarded as the unit of the number 3. Write the following: 1. Express 12 as (3 twos); as (2 twos); as (2 threes). 2. Express 30 as (3 twos); as (2 fives); as (3 fives). 3. Express 42 and 126 each as (2 sevens); as (7 threes). 4. Express 75, 225, and 375 each as (5 threes). 5. Express 66, 198, and 264 each as (11 threes) and as (2 elevens). 136. STEP III.-When the same factor is made the unit of both the dividend and divisor, the division is performed as if the numbers were concrete. Thus, 6012 may be expressed, 20 threes÷ 4 threes, and the division performed in the same manner as in 6 feet ÷ 3 feet. 4 threes are contained 5 times in 20 threes; hence, 12 is contained 5 times in 60. The division may be performed in this way when the factors are connected by the sign of multiplication; thus, 60 12 = (20 × 3) (4 x 3). We can regard as before the 3 as the unit of both dividend and divisor, and hence say, 4 threes are contained 5 times in 20 threes. Perform the division in each of the following examples, without performing the multiplication indicated: 137. PROB. III.-To divide by using the factors of the divisor. Ex. 1. Divide 315 by 35. 9 EXPLANATION.-1. The divisor 35=7 fives. 2. Dividing the 315 by 5, we find that 31563 fives. (138.) 3. The 63 fives contain 9 times 7 fives; hence 315 contains 9 times 7 fives or 9 times 35. Ex. 2. Divide 359 by 24. 4 (3 twos) | 59 (3 twos) and 2 twos remaining Quotient, 14 and 3 (3 twos) remaining True remainder, EXPLANATION.-1. The divisor 24 = 4 × 3 × 2 = = 1 = 18 23 2. Dividing 359 by 2, we find that 359 179 troos and 1 unit remaining. 3. Dividing 179 twos by 3 twos, we find that 179 twos = 59 (3 twos) and 2 twos remaining. 4. Dividing 59 (3 twos) by 4 (3 twos), we find that 59 (3 twos) contain 4 (3 twos) 14 times and 3 (3 twos) remaining. Hence 359, which is equal to 59 (3 twos) and 2 twos + 1, contains 4 (3 twos), or 24, 14 times, and 3 (3 twos) + 2 twos + 1, or 23, remaining. |