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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 is contained in 22 once. 21)2234(10674. Ans. Write the l in the quotient. Then 21
multiplying and subtracting, the
remainder is 1. 134
Bringing down 126
the next figure, we have 13 to be
divided by 21. But 21 is not con8 Rem.
tained in 13, therefore we put a
QUEST.-75. What is the difference between long and short division?
cipher in the quotient, (Art. 70. Obs. 2,) and bring down the next figure. Then, 21 in 134, 6 times, &c. Write 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. 12. Divide 345 by 15.
Ans. 23. 13. Divide 5378 by 25.
Ans. 21525 14. Divide 7840 by 32. 15. Divide 59690 by 45. 16. Divide 81229 by 67. 17. Divide 99435 by 81. 18. How many times is 131 contained in 18602 ?
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 coniuined 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 mulliply 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
hour? 8. At 16 dollars a ton, how many tons of hay can be 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 division ?
16. How many times 12 in 172, and how many over! 17. How many times 15 in 630, and how many over ? 18. How many times 22 in 865, and how many over ? 19. 1236 is how many times 17, and how many over ? 20. 7652 is how many times 13, and how many over ? 21. 3061 is how many times 125, and how many over ? 22. 1861 is how many times 231, and how many over ? 23. 8 times 256 is how many times 9 ? 24. 12 times 157 is how many times 7 ? 25. 15 times 2251 is how many times 12 ? 26. 19 times 136 is how many times 75 ? 27. 63 times 102 is how many times 37 ? 28. 78 times 276 is how many times 136 ? 29. 115 times 321 is how many times 95 ? 30. 144 times 137 is how many times 312 ?
CONTRACTIONS IN DIVISION. 77, a. The operations in division, as well as in multiplication, may in many instances be shortened by a careful attention to the application of the preceding principles.
Case I.—When the divisor is a composite number.
Ex. 1. A gentleman divided 168 oranges equally among 14 grandchildren who belonged to 2 families, each family containing 7 children: how many oranges did he give to each child ?
Suggestion. First find how many each family received, then how
each child received. If 2 families receive 168 oranges, one will receive as many as 2 is contained times in Operation. 168, viz: 84. Now there are 7 children in
2)168 each family. If 7 children receive 84 oranges, one child will receive as many as 7
7)84 is contained times in 84, viz: 12. He there 12 Ans. fore gave 12 oranges to each child.
Note.—This operation is exactly the reverse of that in Ex. 1. Art. 55. The divisor 14 being a composite number, we divide first by one of its factors, and the quotient thus found by the other. The final result would have been the same, if we had divided by 7 first, then by 2. Hence,
78. To divide by a composite number.
Divide the dividend by one of the factors of the divisor, and the quotient thus obtained by the other factor. The last quotient will be the answer required.
To find the true remainder, should there be any.
Multiply the last remainder by the first divisor, and to the product add the first remainder.
Obs. If the divisor can be resolved into more than two factors, we may divide by them successively, as above.
To find the true remainder when more than two factors are employed, multiply each remainder by all the preceding divisors, and to the sum of the products add the first remainder. 2. Divide 465 by 35.
1 last remainder. Operation.
7 first divisor. 7)465 5)66–3 rem.
3 first Rem. added. 13-1 rem.
10 true Rem. Ans. 133$. 3. A teacher having 36 scholars arranged in 4 equal classes, wishes to distribute 216 pears among them equally : how many can he give to each scholar?
4. How many cows, at 27 dollars a head, can be bought for 945 dollars ?
5. How many times is 64 contained in 453 ? 6. How many times is 72 contained in 237 ?
Case II.— When the divisor is 1 with ciphers annexed to it.
79. It has been shown that annexing a cipher to a number increases its value ten times, or multiplies it by 10. (Art. 58.) Now reversing this process ; that is, removing a cipher from the right hand of a number, will evidently diminish its value ten times, or divide it by 10; for,
Quest.–78. How proceed when the divisor is a composite number? How find the true remainder? Obs. How proceed when the divisor can be resolved into more than two factors? How find the remainder in this case? 79. What is the effect of annexing a cipher to a number? What is the effect of removing a cipher from the right of a number? How does this appear?