13. What is the product of 78 multiplied by 25? It will be seen that 5 and 5 are the component parts of 25. 14. There are 365 days in a year. years, how many days has he lived? 15. Multiply 7684 by 112. 78 5 390 5 1950 Ans. If a man live 48 8X7X2=112. 8410736 by 56. 17548671 by 81. 16. 17. 18. 19. 20. 998673214 by 1864. 99998887777 by 445566. 88900236789456 by 77889123. 21. If it take 142 stones to build a rod of wall, how many will it take to build 10 rods? It will be seen that the answer in this ex ample is the same as the multiplicand, with the addition of one cipher. If, then, we an 142 10 1420 nex a cipher to the multiplicand, it is the same as multiplying by 10; since annexing a cipher to the right hand of any number increases the value tenfold. We have, then, this rule for multiplying by 10, 100, 1000, or 1 with any number of ciphers. Annex as many ciphers to the multiplicand as there are ciphers at the right hand of the multiplier, and it will give the answer required. In this example, we multiply by the significant figures only, placing as many ciphers at the right hand of the product as there are ciphers in the multiplier and multiplicand. 48 5280000 26. What is the product of 68400 multiplied by 18000? Ans. 1231200000. QUESTIONS. 18. When the multiplier is 10, 100, 1000, &c. how may you proceed? 19. When there are ciphers at the right hand of the multiplier and multiplicand? When there are ciphers between the significant figures, reject them, observing to place the first product of every figure directly under that figure-thus: 27. Multiply 2008 by 604. Operation. 2008 604 8032 12048 1212832 31. What will 412 hogsheads of molasses cost at 31 dollars per hhd. ? Ans. 12772 dolls. 32. What number is that of which 8, 9, 11, are factors? 33. If 80 men dig a canal in 92 days, could dig the same in one day? 34. If they receive 5 shillings per day, lings do 7520 men receive? Ans. 792. how many men Ans. 7520. how many shilAns. 37600. 35. A merchant bought 28 boxes of sugar, each weighing 235 lbs., at 8 cents per lb. How many cents did they cost? Ans. 52640. 36. How many shillings will 89 cords of wood cost, at 15 shillings per cord? Ans. 1335. 37. A merchant bought 15 pieces of cloth, each piece containing 27 yards at 7 dollars per yard. How much did he pay for the whole? 38. If a ship sail 12 miles per sail in 12 days? Ans. 2835 dollars. hour, how far will she Ans. 3456 miles. 39. If a man hoe 3 rows, 28 hills each, in one hour, how many hills will he hoe in 12 days, working 8 hours Ans. 8064. in a day? 40. What will 50 firkins of butter cost, weighing 54 lbs. each, at 14 cents per lb. Ans. 37800. 42. A man has 9 piles of wood, 16 cords in a pile. What is it worth at 7 dollars per cord? Ans. 1008 dolls. 20. How do you proceed when there are ciphers between the QUESTIONS. significant figures? DIVISION. 1. John has 5 oranges given him. He keeps one himself, and divides the others equally between his two sisters. How many did each receive? 2. Samuel divided 9 walnuts equally between three boys. How many did each receive? 3. James' father gave him 8 butternuts to divide equally between himself and his 3 brothers. How many did each receive? 4. If Harriet gave 24 cents for 6 pictures, what did she pay for 1? 5. Mary divided 36 cents equally between 6 poor children. What did each receive? 6. If 8 yards of cloth cost 56 cents, what did one cost? 7. How many yards of broadcloth can be bought for 72 dollars, at the rate of 8 dollars per yard? 8. How many bushels of apples can be bought for 100 cents, at 25 cents per bushel? 9. How many barrels of flour may be bought for 77 dollars, at 7 dollars per barrel? 10. In how many hours will a man travel 48 miles, at the rate of 4 miles per hour? 11. Eighty cords of wood piled in 8 different piles. How many cords in each pile? 12. A farmer sold wool to the amount of 81 dollars, for 9 shillings a fleece. How many fleeces did he sell? 13. How many thousand of boards may be bought for 144 dollars, at 12 dollars per thousand? 14. How many books may be bought for 84 cents, at 7 cents apiece? 15. If 16 apples be divided equally between 4 boys, how many does each receive? It is evident, that, as many times as 4 is contained in 16, or, as many times as it can be subtracted from it, so many apples each boy will receive, thus: 16 12 We find by trial, that 4 is contained in 16 four times, which is the number of apples each boy is to receive. 10 If 4 boys receive 4 apples each, how many do they all receive? It is plain that 4 boys will receive 4 times as many as one; therefore, if one boy receive 4 apples, 4 boys will receive 4X4=16. From this example, it is evident that Division is a concise method of performing many subtractions, or, the reverse of Multiplication. As in multiplication 2 numbers are required to perform the operation, so in division. The number to be divided is called the Dividend; the number by which you divide is called the Divisor. The Dividend is to be regarded as the product of two factors, of which the Divisor is one, and the other is sought, which is the Quotient after division. The divisor and quotient multiplied together produce the dividend. Thus, it appears that Division and Multiplication mutually prove each other. SHORT DIVISION. 16. If 336 dollars be divided equally among 3 men, how many dollars will each receive? To subtract 3 from 336 as many times as would be necessary to give each man his share, would be long and tedious; but, by Short Division the operation becomes simple: 336 is 3 hundreds, 3 tens, and 6 units. It will be perceived, that, if 300 be divided into 3 equal parts, one of these parts will be 100, and three tens divided in like manner becomes 1 ten, and 6 units divided by 3 become 2 units. The answer, then, is 1 hundred 1 ten and 2 units, or 112. QUESTIONS. 1. What is Division? 2. How many numbers are required to perform the operation? 3. and what are they called? 4. How is the dividend to be regarded? 5. What are the two factors, which, multiplied together will produce the dividend? 6. How is Division proved? By carrying on the process partly in the mind, the operation may be made still shorter, thus: 3)336 112 Answer. 17. If 4 shares of bank stock cost 456 dollars, what will one share cost? Operation. 114 Answer. We first set down the number to be divided, or the dividend; at the left of this number, place the number by which we divide, or the divisor. Taking the first left hand figure, or hundreds, we find how many times the divisor is contained in it. The number of times, or 1, we place directly under the divided figure. We next divide the tens of the dividend: 4 is contained in 5 once, and 1 ten over, which also must be divided. If this ten be added to the unit, it will make 1 ten and 6 units-equal to 16 units; 4, then, is contained in 16, 4 times, which we place in the column of units; and in 456, 114 times. 18. Six brothers received a legacy of 1512 dollars. What was the share of each? Operation. 252 Answer. In this question, the divisor is not contained in the first figure of the dividend. We, therefore, take the next figure, 5, which with the 1 makes 15 hundred; 6 is contained in 1500, 200 times, and 30 tens, or 300 over. This, added to the 1 ten in the next column, is 31 tens; 6 is contained in 31, five times, or in 31 tens, 50 times, and 1 ten over, which, with the two units, makes 12 units: this, divided by 6, is 2 units. The answer, then, is 200+50+2 =252. From the foregoing, we derive the following RULE. Write the divisor at the left hand of the dividend, with a line drawn between them. Find how many times the divisor is contained in the first figure or figures of the dividend, setting the result directly QUESTIONS. 7. How is the process partly carried on in Short Division? 8. What is the method of procedure in the 17th question? 9. What is the rule for Short Division ? D |