EXAMPLES. 1. Multiply 146 by 10. Ans. 1460. DEM.-We join a cipher to the multiplicand, and our work is done, because it increases the figures of our multiplicand ten times, by giving units the place of tens; and tens the place of hundreds; and hundreds the place of thousands, &c. 2. Multiply 846 by 100. Ans. 84600. DEM.-Here, we join two ciphers to the multiplicand; and the figures of our multiplicand are increased 100 times, because 6, the first figure of our multiplicand, after the ciphers are joined, becomes hundreds; and the next left hand figure 4, becomes thousands, one hundred times the first value. 3. Multiply 8324 by 1000. CASE VI-To multiply by 9, 99, 999, &c. Ans. 8324000. Ans. 304600000. RULE:-Jon to the right of the multiplicand, as many ciphers, as your multiplier contains nines; and from the sum produced, subtract the multiplicand. Or you may simply multiply the multiplicand by the number of nines in the multiplier. The first method, frequently shortens the work. 1. Multiply 464 by 9. EXAMPLES. DEM.-Joining a cipher to the right 4640 of our multiplicand, is repeating our multiplicand 464 ten times, and by subtracting our multiplicand once, must evidently leave it nine times repeated; as ap Ans. 4176 pears, from the same example multiplied by 9. 4.6-4 9 Ans. 4 17 6 2. Multiply 4642 by 99. 464200 4642 Ans. 459558. 459558 DEM.-Here, it is plain, by joining two ciphers to the multiplicand, we increase it 100 times; and our multiplicand being once taken away, or subtracted, must leave it 99 times repeated. 3. Multiply 3472 by 999. 4. Multiply 34672 by 9999. Ans. 3468528. Ans. 346685328. CASE VII To multiply by a composite number; that is, when the multiplier can be produced by the multiplying of any two figures in the multiplication table. RULE. Multiply the multiplicand first by one of the figures, and that product by the other, and this last product will be the total product required. EXAMPLES. 1. Multiply 243 by 35. 243 1701 5 Ans. 8505 In this example, 7 and 5 are called the component parts of 35; because times 5 are 35; and it will make no difference which we multiply by first. DEM.-We first multiply by 7, which repeats our multiplicand 7 times; and then we multiply this product by 5, which is repeating, 5 times, a number 7 times as great as our first multiplicand, consequent ly, our multiplicand is 35 times repeated, because 5 times 7 are 35. Our work perhaps may appear more clear to the young student, by multiplying the same multiplicand directly by 35; thus, the same result, you perceive, is produced. NOTE. This method of work, will not be found of much service, in simple multiplication; but it is sometimes found very convenient in mul tiplying a compound quantity. 2. Multiply 480 by 36. 3. Multiply 1324 by 45. 4. Multiply 3684 by 72. 5. Multiply 3124 by 81 6. Multiply 4638 by 96 7. Multiply 3040 by 49. PROMISCUOUS EXAMPLES. Ans. 17280. Ans. 59580. Ans. 265248. Ans. 253044. Ans. 445248. Ans. 148960. 1. What will a year's board come to, at $2 a week; allow ing fifty-two weeks to the year? Ans. $104. 2. Suppose 80 seamen were concerned in taking a prize at sea, and on dividing their prize money, each seaman received $150; what was the amount of the prize? Ans. $12000. 3. A gentleman bought 307 horses for shipping, at the rate of $105 each; what did he pay for the whole? Ans. $32235. 4. A gentleman dividing his fortune among seven sons, found that the portion of each, was $1050; what was the gentleman's fortune? Ans. $7350. 5. There are 320 rods in a mile; now suppose the distance between the United States and Europe, to be 3000 miles 40 MULTIPLICATION OF DECIMAL OR FEDERAL MONEY. will you tell how many rods Europe is distant from the United States? Ans. 960000. 6. How much money must be distributed among 85 persons, so that each may have 110 dollars? Ans. $9350. 7. What is the product of 980, repeated 99 times? Ans. 97020. 8. If 1781 be multiplied by 1121 what will be the product? Ans. 1996501. 9. If I pay 9 dollars for one barrel of pork; what must I pay for 50 barrels, at that rate? Ans. $450. 10. If one tun of hay cost 8 dollars; what will be the cost of 50 tuns? Ans. $400, 11. What will 20 cows cost, at $18 each? Ans. $360. 12. A dollar in New-York contains 8 shillings; how many shillings in 96 dollars? Ans. 768 shillings. MULTIPLICATION OF DECIMAL OR FEDERAL MONEY, A knowledge of this money, will be found very beneficial in the transaction of business, when the price of one pound, one yard, &c. is given in dollars, cents and mills, to find the price of a quantity. RULE.-Multiply the price of one by the number expressing the quantity, the same as in whole numbers; and remember to place a separatrix, as many figures from the right hand in the product, as it is, in the given price. NOTE. All figures at the left of the separatrix, will be dollars; the next two at the right will be cents; and the next mills. 1st EXAMPLE. If 1 yard of calico, cost 35 cents; what will 5 yards cost? ,3 5 cts. $1, 7 5 cts. Ans. ,3 5 DEM.-It is plain, that 5 yards should cost 5 times as much as 1 yard; and by multiplying the price of 1 yard by 5, our product is five times the price of one yard, as will plainly appear by adding the price of one yard, 5 times. The reason of our pointing off as many figures for the decimal parts of a dollar, as there are deicmals in the given number, is plain; because the de cimals increase in a tenfold proportion, the same as whole numbers; and when we multiply the decimal that expresses tenths, it is plain, that the left hand figure of the product is a whole number; because, By Addition. ,3 5 ,35 ,3 5 81, 75 ten tenths, are equal to one unit; then the left hand figure of tenths in our example, (17 tenths,) must be I unit; because ten tenths make a unit; and we then have 7 tenths over, which stands in the place of tenths, from our pointing. 2. What will 5 yards of broadcloth come to, at $4,75 cts. a yard? Ans. $23,75 cts. 3. What will 35 bushels of corn come to, at ,60 cts. a bushel? Ans. $21,00 cts. 4. What is the value of 57 yards of muslin, at ,37 cts. a yard? Ans. $21,09 cts. 5. What is the value of 375 acres of land, at $15,50 cts. per acre? Ans. $5812,50 cts. 6. What cost 100 bushels of barley at ,37 cents per bushel ? Ans. $37,00. 7. At 2 dollars and 38 cents per ream of paper, what is the worth of 150 reams? Ans. 81 cts. Ans. $357. 8. What are three hundred barrels of beef worth, at $9,85 cts. per barrel? Ans. $2955. 9. What will 8 bushels of potatoes come to, at 37 cts. per bushel? Ans. $2,96 cts. 10. What will 10 cords of wood come to, at $1,25 cts. per cord? Ans. $12,50 cts. 11. What cost a farm containing 150 acres, at 824,50 cents per acre? Ans. 3675 dollars. 12. What will 9 dozen of eggs come to, at 9 cents a dozen? 13. Bought 20 horses for shipping, at seventy-two dollars per head; what did they all cost? Ans. $1440. 14. What are 35 bushels of apples worth, at ,30 cents per bushel? Ans. $10,50 cts. 15. What will 56 pounds of butter come to, at ,22 cts. per pound? Ans. $12,32 cts. 16. If a man spend 75 cts, per day; what will he spend in 365 days, at that rate? Ans. $273,75 cts. 17. What will 30 sheep cost at 1 dollar and 25 cents per head? Ans. $37,50 cts, 18. What will 45 bushels of corn come to, at ,45 cents per bushel? Ans. $20,25 cts. 19. If a man receive $1,25 cts. a day, what must he receive for 48 days' labour? Ans. $60. QUESTIONS ON SIMPLE MULTIPLICATION What is multiplication? A. The shortest method of performing addition, when the same number is to be repeated a given number of times. What are the two given numbers called? A. Multiplicand and multiplier. What is the number called arising from the operation of the work? A. The answer or product. What is the multiplicand? A. A number to be repeated by another. What is the multiplier? A. The repeater, or number by which the multiplicand is to be repeated. What do you understand by the product? A. I understand, that it is the same as setting down the multiplicand as many times as the mudtiplier expresses a unit, and-adding the amount in addition, and the product in multiplication, must be the same. If you have four for a multiplier, how many times greater is the product than the multiplicand? A. Four times; because the multiplicand has been repeated four times. Could the same be performed by addition? A. It might by setting down the multiplicand as many times as the multiplier expresses a unit; and then adding, the amount would be the same as the product in multiplication; but it would be a tedious method to work by addition, especially when the multiplier is large. If you multiply a number by five, what part of the product is the multiplicand? A. A fifth part; because the multiplicand has been repeated five times to produce the product. How do you place the given numbers for work? A. The multiplier under the multiplicand, so that units stand under units, tens under tens, &c. If your multiplier consists of but one fig. ure, how do you proceed in the work? A. Multiply the right hand figure of the multiplicand by the multiplier, and if the product should not exceed 9, place it directly under; but if the product exceed 9, setdown the right hand figure of the product, and add the left to the product of the next figure, because it is of the same kind of the next at the left; and thus proceed with all the figures of the multiplicand; remembering to set down the whole product of the left hand figure. How may multiplication be proved A. The most simple methods of proof are, by multiplication, addition, subtraction, and division. How do you prove multiplication by itself? A. By making the multipli cand, a multiplier. Why should that prove it? A. Because it is evi dent, that it can make no difference which of the given numbers we take for the multiplier; thus, we may say, 6 times 8 are 48; or we may make 6 the multiplicand, and say, 8 times 6 are 48; the same result is produced in either case. How do you prove multiplication by addition? A. By setting down the multiplicand as many times as the multiplier expresses a unit, and then adding; and if the amount in ad dition, be equal to the product in multiplication, the work has been correctly performed; because it is plain, if we take 8 for a multipli cand, and 4 for a multiplier, that the product will be 32; and if we set 8 down 4 times, and add, the amount will be 32; but by multiplication we arrive at the result with less labour. How is multiplication proved by subtraction? A. By subtracting the multiplicand from the product, as many times as the multiplier expresses a unit; and should the operation diminish the product to nothing, the work is right; for it is clear, if the multiplicand be taken away as many times as it has been |