Had this remainder been 3 dollars, it is evident that each man would have had 3 times of a dollar, that is of a dollar, more; and in all cases where there is a remainder, we may obtain the true quotient by placing the divisor under the remainder, with a line between, as above. Thus Remainder 1 4 shows that the divisor 4 is contained in the remainder one fourth of a time. 19. How many cwt. of rice, at 4 dollars a cwt., may be bought for 947 dollars? Ans. 236 cwt. 20. How many cwt. of sugar, at 9 dollars per cwt., can be bought for 2944 dollars? Ans. 327. 21. How many barrels of pork, at 11 dollars a barrel, can be bought for 2478 dollars? Ans. 225 barrels. LONG DIVISION. When the divisor exceeds 12, we cannot conveniently perform the operation in the mind; we therefore set the quotient figures on the right hand of the dividend, and write down the whole computation at full length, and this is called long division. RULE. I. Find how many times the divisor is contained in the least number of the left hand figures of the dividend, that will contain it once, or more: place the figure expressing the number of times, to the right hand of the dividend for the first quotient figure. II. Multiply the divisor by this quotient figure, and place the product under that part of the dividend used, and subtract it therefrom. III. Bring down the next figure of the dividend to the right hand of the remainder, and divide this number as before. Thus proceed till you have brought down, and divided, all the figures of the dividend. Note. 1. If the product of the divisor by any quotient figure, be greater than that part of the dividend used, it shows that the quotient figure is too large, and must be diminished.—If the remainder at any time be equal to or greater than, the divisor, the quotient figure is too small, and must be increased. 2. When you have brought down a figure to the right hand of the remainder, if the number made up be less than the divisor, you must place a cipher in the quotient, and bring down the next figure of the dividend. Proof-The method of proof is the same as in Short Division. EXAMPLES. 1. How many times is 13 contained in 2785 ? Operation. Divisor. Dividend. Quotient. 13) 2785 (214 26 18 1 3 Remainder 55 52 3 Proof. Quotient 2 1 4 We find that the divisor (13) is contained in the two first figures of the dividend, 2 times. We place the fig ure (2) at the right hand of the dividend for the first quotient figure, and multiply the divisor (13) by it, and place the product (26) under the two first figures of the dividend, and subtract it therefrom, and 1 remains. To the right hand of this remainder bring down (8) the next figure of the dividend, which makes 18. We find that 13 is contained in 18, 1 time; placing 1 in the quotient, and 13 under the 18, subtract it therefrom, and 5 remain, to the right hand of which, we bring down (5) the next figure of the dividend, which makes 55. We then find that 13 is contained in 55, 4 times. Placing 4 in the quotient, multiply the divisor (13) by it, and place the product (52) under the 55, subtract it therefrom, and the remainder is 3. Thus we have brought down and divided all the figures of the dividend. Hence we find that 13 is contained in 2785, 214 times, and 3 remains. 642 214 2782 Remainder +3 2 7 8 5 Divid. 6. How many times is 13 contained in 29877? Ans. 2298 times. 7. How many times can you have 16 in 5520? 13 Ans. 345 times. 8. If a man travel 630 miles in 15 days, how many miles is that a day? Ans. 42 miles. 9. What is the quotient of 55081 divided by 17? Ans. 3240 10. 25 workmen received $3150, to be equally divided among them; how many dollars did each one receive? Ans. $126. 11. If the divisor be 29, and the dividend 72740, what is the quotient? 12. How often does 106215 contain 365 ? 13. Divide 6832 by 16. 14. Divide 28656 by 36. 15. Divide 37088 by 61. 16. Divide 410642 by 79. Ans. 25089 Ans. 291. Ans. 427: Ans. 796. Ans. 608. Ans. 5198: 17. Divide 1853219 by 91 Ans. 203651. Ans. 30766162 321. Ans. 8259305 Ans. 8049611707 MORE EXAMPLES FOR EXERCISE. 1. Divide 796976499 by 49654. 695. 57606 Remainder 29799. Remainder 425. Remainder 62. CONTRACTIONS IN DIVISION. CASE I. When there are ciphers at the right hand of the divisor. RULE. Cut off the ciphers in the divisor, and the same number of figures from the right hand of the dividend; then divide the remaining ones as usual, and to the right hand of the remainder, (if any,) place the figures which were cut off from the dividend, and you will have the true remainder. EXAMPLES. 1. Divide 7380964 by 23100. 2. Divide 8978485 by 8000. Divis. Divid. Quot. Divis. Divid. 231'00)73809/64(31923064 8'000)8978'485 4. Divide 75694 by 2500. Ans. 30,9 2500 5. Divide 1255000 dollars equally among 5000 men, and how many dollars will each one have? Ans. 251. 6. Divide 49184145 by 3600. 945 Ans. 136623600 7. How many times is 49000 contained in 421400000? Ans. 8600. Ans. 5432410008 8. Divide 3259450000 by 60000. 10. Divide 436250000 by 125000. MORE EXAMPLES. 1. Divide 149596478 by 120000. 2. Divide 654347230 by 901000. 3. Divide 457878695 by 9736000. Ans. 131 112 89000 Ans. 3490. Rem. 76478. CASE II. When the divisor is a composite number; that is, when it is the product of any two numbers in the Table. RULE. Divide the dividend by one of the component parts, or factors, and the quotient thence arising by the other, the last quotient will be the answer. Note. The true remainder is found by multiplying the last remainder by the first divisor, and adding in the first remainder. |