EXAMPLES. Dividend. Operation. Dividend. 75 125 We first write down the dividend, and draw a curve line at the right and left hand sides of it, and write the divisor at the left side, (as you see.) We begin the operation, and see how many times 25'is contained in 87, (the same number of figures ;) finding it to be 3 times, we write it down at the right hand of the dividend, (for the highest figure in the quotient, the 87 or 7, being tens, it follows that the 3 must also be tens.) We now say 3 times 25 equals 75, which we write under the 87, draw a line underneath, and subtract the 75 from 87, and find 12 remainder; we now bring down the (next figure) 5, and write at the right hand of the remainder 12, and we have 125 : we say 25 is contained in 125 five times, (the 5 we are sure is units, because the 5 we brought down was units, we write it in the quotient, multiply as before, and the work is done. We find the share of each man to bc 35 dollars. Proof. 35 x 25=875 dividend. 2. How many times is 35 contained in 875 ? Ans. 25 times. 3. If a man's income be 1248 dollars a year, how much is that per week, there being 52 weeks in a year? Ans. 24 dollars per week. 4. How many times is 14 contained in 1624 ? Ans. 116 times. 5. What will be the quotient of 6880, divided by 32 ? Ans. 215 quotient. 6. If a man can travel 24 miles a day, how many days will it take him to travel 288 miles ? Ans. 12 days. 7. Divide 504 by 56. Ans. 9 quotient. 8. How many times are 16 in 2176 ? Ans. 136. 9. How often can 45 be subtracted from 270 ? Ans. 6 times. 10. The divisor 18, and the dividend 2556 ; what is the quotient? Ans. 142. 11. How many acres of land, at 22 dollars an acre, can be bought for 8514 dollars? Ans. 387 acres. 30 LONG DIVISION. 12. If a dividend be 46015, and the divisor 807, what is the quotient? And what is the remainder ? Ans. 57 quotient, and 16 remainder. 13. How many times is 18 contained in 12532 ? Ans. 696, and 4 remainder. 14. How many times is 15 contained in 3120? Ans. 208 times. N. B. In working the 14th example, the scholars will find, after they have brought down the figure 2 from the dividend, and annexed it to the remainder 1, making it 12, that 15 the divisor is not contained in it, therefore they must write a cipher in the quotient, and bring down the next figure. 15. A farmer planted 1526 trees in 14 equal rows--how many did he plant in a row? Ans. 109 trees. 16. How many times is 32 contained in 19520 ? Ans. 610 times.. 17. What number must I multiply by 125 that the product may be 176000? Ans. 1408 is the number. 18. How many times is 16 contained in 2368 ? 16) 2368 (149 16 76 128 144 In this example we have chosen 9 for the last figure of the quotient; but it appears that 9 tiines 16 are more than 128: therefore 16 is not contained 9 times in 128. The 9, and the product arising from it, must be rubbed out, and a smaller figure must be written in the quotient. 19. A man would put up 17484 pounds of tea into chests containing 94 pounds each-how many chests must he have ? Ans. 186 chests. 20. How many times 225 is equal to 16875 ? Ans. 75 times. 21. How many times is 56 contained in 18144 ? 56) 18144 (323 168 134 56 Tere we have chosen 3 for the last figure of the quotient; but, subtracting 3 times 56 from 224, there remains 56: this re mainder will contain the divisor once more; therefore, we must rub out the 3 and the work resulting from it, and must write 4 in the place of 3. 22. A man sold 36 cows for 792 dollars—how much was that for each cow ? Ans. 22 dollars. 23. A man wishes to travel 598 miles in 13 days—how far must he travel each day? Ans. 46 miles. 24. The inhabitants on the equator are carried by the rotation of the earth on its axis, at the rate of 25020 miles in 24 hours how far are they carried per hour ? Ans. 10424 miles. 25. How many times is 236 contained in 708944 ? Ans. 3004 times. 26. How many times is 321 contained in 231120 ? Ans. 720 times. 27. How many times is 556 contained in 309136? Ans. 556 times. 28. How many times is 620 contained in 1318120 ? Ans. 2126 times. 29. How many times is 56 contained in 24864 ? Ans. 444 times. NOTE. When two numbers can be found, which their product will produce the divisor, divide the dividend by one of the numbers, and the quotient thence arising by the other. 30. How many times is 72 contained in 11376? Ans. 158 times. 31. How many times is 144 contained in 979776? Ans. 6804 times. NorE.--When it occurs that both the dividend and divisor have ciphers on the right hand, omit an equal number of them in the operation. 32. How many times is 2500 contained in 52500 ? Ans. 21 times. 33. An army of 1500 men, having plundered a city, took 2625000 dollars-what was each man's share ? Ans. 1750 dollars. 34. What number must I multiply by 135 that the product may be 505845 ? Ans. 3747. 35. If the product of two numbers be 14700, and the multiplier 84, what is the multiplicand ? Ans. 175 multiplicand. 36. If the product be 14700, and the multiplicand 175, what is the multiplier ? Ans. 84 the multiplier. 37. The divisor is 144, and the dividend 2592; what is the quotient? Ans. 18 is the quotient. 38. The quotient of two numbers is 18, and the dividend 2592, what is the divisor? Ans. 144 the divisor. 39. Divide 218875 by 425. Ans. 515 is the quotient. 40. Divide 5257050 by 8675. Ans. 606 quotient. 32 PROBLEMS AND MISCELLANEOUS QUESTIONS. Problems and Miscellaneous Questions involving the prin ciples of the preceding Rules. Prob. 4.-Having the sum and difference of two nunibers given, to find those numbers. RULE. To half the sum add half the difference, and the sum is the larger; and from half the sum take half the difference, and the remainder is the less. Or, from the sum take the difference, and half the remainder is the least. To the least add the given difference, and the sum is the larger. EXAMPLE. 1. What are those two numbers, whose sum are 48 and difference 14? Prob. 5.--Having the sum of two numbers and the difference of their squares given, to find those numbers. NOTE.—The square of a number is the product of it multiplied into itself. RULE. Divide the difference of their squares by the sum of the numbers, and the quotient will be their difference: you will then have their sum and difference, to find the numbers, by Prob. 4. EXAMPLE. 1. What two numbers are those, whose sum is 32, and the difference of their squares is 256 ? Ans. the larger number 20, and the less 12. Prob. 6.-Having the difference of two numbers, and the difference of their squares given to find those numbers. RULE. Divide the difference of their squares by the difference of the numbers, and the quotient will be their sum; then proceed by Prob. 4. EXAMPLE 1. What are those two numbers, whose difference is 20, and the difference of their squares is 2000 ? Ans. 60 the larger, and 40 the less. Prob. 7.--Having the product of two numbers, and one of them given, to find the other. RULE. Divide the product by the given number, and the quotient will be the number required. EXAMPLES 1. The sum of two numbers is 135, and one of the numbers is 65 What is the other? Ans. 70. 2. How many barrels of flour can you buy for 1880 dollars, at 5 dollars per barrel ? Ans. 376 barrels. 3. What will be the value of 6 pounds of butter at 14 cents per pound, and 2 pair of fowls at 35 cents per pair ? Ans. 154 cents. 4. What number multiplied by 25 will produce 18825 ? Ans. 753. 5. If a trader buy 548 barrels of flour, at 5 dollars a barrel, and pay 6 dollars for storage, for how much must he sell the flour, to gain 100 dollars ? Ans. 2846 dollars. 6. Four men and three boys found a sum of money, and divided it so that each man had 45 dollars, and eachr boy 15 dollars—what sum did they find ? Ans. 225 dollars. 7. If a man earns 36 dollars a month, how many months will it take him to earn 540 dollars? Ans. 15 months. 8. A farmer bought a field, valued at 125 dollars, for which he gave 6 cows, valued at 15 dollars apiece, and the rest in moneyhow much money did he pay? Ans. 35 dollars. 9. A drover having 599 dollars, wishes to buy all the oxen he can pay for, at 34 dollars a head, and then lay out the remainder of his money for sheep, at 3 dollars a head-how many of each must he buy? Ans. 17 oxen and 7 sheep. 10. A trader bought 240 sheep at 4 dollars a head, and paid for them in cows at 20 dollars a head-how many cows did he give? Ans. 48 cows. 11. What number must be divided by 15, in order that the quotient shall be 640 ? Ans. 9600. 12. What number must be subtracted from 7342, in order that the remainder shall be 456? Ans. 6886. 13. What number must be multiplied by 30, in order that the product shall be 2130 ? Ans. 71. 14. From 1000 take 346, and multiply the remainder by 12. Ans. 7848. 15. What is the difference between twice five and twenty, and twice twenty-five? Ans. 20 difference. 16. To the one eighth of 1000 add 325. Ans. 450. 17. If 35 yards of cloth cost 420 dollars, what does one yard cost? Ans. 12 dollars. 18. What is the value of 6 bushels of apples, at 35 cents per bughel, and 3 pounds of butter at 22 cents per pound? Ans. 276 cents. |