49. How many yearlings, at 9 dollars a head, can be bought for 468 dollars? 50. How many acres of land, at 6 dollars an acre, can I buy for 973 dollars? 72. The preceding method of dividing is called Short Division. From the illustrations and principles now explained, we derive the following RULE FOR SHORT DIVISION. I. Write the divisor on the left hand of the dividend with a curve line between them. Then beginning at the left hand, divide successively each figure of the dividend by the divisor, and place each quotient figure directly under the figure divided. (Art. 68.) II. If there is a remainder after dividing any figure, prefix it to the next figure of the dividend and divide this number as before; and if the divisor is not contained in any figure of the dividend, place a cipher in the quotient and prefix this figure to the next one of the dividend, as if it were a remainder. (Arts. 69, 70.) III. When a remainder occurs after dividing the last figure, write it over the divisor and annex it to the quotient. (Art. 71.) 73. PROOF.-Multiply the divisor by the quotient, to the product add the remainder, and if the sum is equal to the dividend, the work is right. QUEST.-72. What is the rule for short division? 73. How is division proved? Obs. How does it appear that the product of the divisor and quotient should be equal to the dividend? What other way of proving division is mentioned? OBS. 1. Since the quotient shows how many times the divisor is contained in the dividend, (Art. 64,) it follows, that if the divisor is repeated as many times as there are units in the quotient, it must produce the dividend. 2. Division may also be proved by subtracting the remainder, if any, from the dividend, then dividing the result by the quotient. PROOF OF MULTIPLICATION BY DIVISION. 74. Divide the product by one of the factors, and if the quotient thus arising is equal to the other factor, the work is right. OBS. This method of proof depends on this obvious principle, viz: if the divisor and quotient, multiplied together, produce the dividend, the product of the two numbers, divided by one of those numbers, must give the other number. LONG DIVISION. Ex. 1. A father bought 741 acres of land, which he divided equally among his 3 sons: how many acres did each receive? Note. This example has been solved by short division. (Art. 70. Ex. 28.) We have introduced it here for the purpose of illustrating a different mode of dividing. Operation. Divisor. Divid. Quot. Having written the divisor and dividend as before, we find 3 is contained in 7, 2 times, and place the 2 on the right of the dividend, with a curve line between them. We next multiply the divisor by this quotient figure-2 times 3 are 6-and, placing the product under 7, the figure divided, subtract it therefrom. We now bring down the next figure of the dividend, and placing it on the right of the remainder 1, we have 14. 14, 4 times. Set the 4 on the right of the 3) 741 (247 6 14 12 21 21 Now 3 is in last quotient QUEST.-74. How is multiplication proved by division? Obs. Upon what principle does this proof depend? How are the numbers written for long division? Where begin to divide? Where is the quotient placed? figure, and multiply the divisor by it: 4 times 3 are 12. Write the product under 14, and subtract as before. Finally, bringing down the last figure of the dividend to the right of the last remainder, we have 21; and 3 is in 21, 7 times. Set the 7 in the quotient, then multiply and subtract as before. The quotient is 247, the same as in short division. 75. This method of dividing is called Long Division. It is the same in principle as Short Division. The only difference between them is, that in Long Division the result of each step in the operation is written down, while in Short Division we carry on the process in the mind, and simply write the quotient. Note. To prevent mistakes, it is advisable to put a dot below each figure of the dividend, when it is brought down. 2. How many times is 2 contained in 578 ? Ans. 289. Note. This and the following questions are designed to be performed by long division, and each operation should be proved. 3. How many times is 5 contained in 7560? Ans. 1512. 4. How many times is 4 contained in 126332? Ans. 31583. 5. How many times is 6 contained in 763251? 6. How many times is 3 contained in 4026942 ? 7. How many times is 8 contained in 2612488? 8. How many times is 5 contained in 1682840? 9. How many times is 7 contained in 45063284 ? 10. How many times is 9 contained in 650031507? 11. Divide 2234 by 21. Operation. 21)2234(106. Ans. 21 126 8 Rem. 21 is contained in 22 once.— Write the 1 in the quotient. Then multiplying and subtracting, the remainder is 1. Bringing down the next figure, we have 13 to be divided by 21. But 21 is not contained in 13, therefore we put a QUEST.-75. What is the difference between long and short division? Write cipher in the quotient, (Art. 70. Obs. 2,) and bring down the next figure. Then, 21 in 134, 6 times, &c. the remainder 8 over the divisor, and annex it to the quotient. (Art. 71.) 76. After the first quotient figure is obtained, for each figure of the dividend which is brought down, either a significant figure or a cipher must be put in the quotient. Ans. 23. Ans. 215 12. Divide 345 by 15. 15. Divide 59690 by 45. Ans. 142. OBS. When the divisor is not contained in the first two figures of the dividend, find how many times it is contained in the first three; and, generally, find how many times it is contained in the fewest figures which will contain it, and proceed as before. 19. How many times is 93 contained in 100469 ? 20. How many times is 156 contained in 140672 ? 77. From the preceding principles we derive the following RULE FOR LONG DIVISION. Begin on the left of the dividend, find how many times the divisor is contained in the fewest figures that will contain it, and place the quotient figure on the right of the dividend with a curve line between them. Then multiply the divisor by this figure and subtract the product from the figures divided; to the right of the remainder bring down the next figure of the dividend and divide this number as before. Proceed in this manner till all the figures of the dividend are divided. When there is a remainder after dividing the last figure, write it over the divisor and annex it to the quotient, as in short division. (Art. 71.) QUEST.-76. What is placed in the quotient, on bringing down each figure of the dividend? Obs. When the divisor is not contained in the first two figures of the dividend, what is to be done? 77. What is the rule for long division? OBS. When the divisor contains but one figure, the operation by Short Division is the most expeditious, and should always be practiced; but when the divisor contains two or more figures, it will generally be the most convenient to divide by Long Division. EXAMPLES FOR PRACTICE. 1. If a man travel 8 miles an hour, how long will it take him to travel 192 miles ? 2. How many yards of broadcloth, at 9 dollars a yard, can be bought for 324 dollars? 3. A farmer bought a lot of young cattle, at 11 dollars per head, and paid 473 dollars for them: how many did he buy? 4. How many tons of coal, at 7 dollars a ton, can be bought for 756 dollars? 5. At 12 dollars a month, how long will it take a man to earn 156 dollars? 6. In one day there are 24 hours: how many days are there in 480 hours? 7. A man traveled 215 miles in 21 hours: how many miles did he travel per hour? 8. At 16 dollars a ton, how many tons of hay can bé bought for 176 dollars? 9. How many casks of wine, at 25 dollars a cask, can be bought for 275 dollars? 10. The ship George Washington was 25 days in crossing the Atlantic Ocean, a distance of 3000 miles. How many miles did the ship sail per day? 11. The steamer Great Western crossed it in 15 days. How many miles did she sail per day? 12. The steamer Caledonia crossed it in 12 days. How many miles did she sail per day? 13. If a man can earn 32 dollars a month, how long will it take him to earn 420 dollars? 14. If 63 gallons make a hogshead, how many hogsheads will 1260 gallons make? 15. If a ship can sail 264 miles per day, how far can she sail in an hour? QUEST.-Obs. When should short division be used? When long di vision? |