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 proJuct? 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 MONEÝ. 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. 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 decimals in the given number, is plain; because the decimals 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. ,3 5 ,3 5 $ 1, 7 5 MULTIPLICATION OF DECIMAL OR FEDERAL MONEY, 41 ten tenths, are equal to one unit; then the left hand figure of tenths in our example, (17 tenths,) must be 1 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. $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. acres, at $24,50 Ans. 3675 dollars. to, at 9 cts. a dozAns. 81 cts. 11. What cost a farm containing 150 cents per acre? 12. What will 9 dozen of eggs come en ? lars 13. Bought 20 horses for shipping, per head; what did they all cost? 14. What are 35 bushels of apples per bushel ? 15. What will 56 pounds of butter per pound? 16. If a man spend ,75 cts. per day; in 365 days, at that rate? at seventy two dolAns. $1440. worth at ,30 cents Ans. 10,50 cts. come to, at ,22 cts. Ans. 12,32 cts. what will he spend 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 cts. per bushel? Ans. $20,25 cts. 19. If a man receive $1,25 cts. a day; what must he receive for 48 day's 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 multiplier 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 ex. presses 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 figure, 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, set down the right hand figure of the product and add the left, to the product of the next figuré, 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 multiplicand, a multiplier. Why should that prove it? A. Because it is evident, 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 addition, be equal to the product in multiplication, the work has been correctly performed; because it is plain, if we take 8 for a multiplicand, 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 repeated; it must diminish it to nothing. How is multiplication proved by division? A. By dividing the product by either of the factors; and if the quotient be equal to the other, the work is right; because division is only the reverse of multiplication, or a short way of performing subtraction; and if we divide the product by the multiplicand the quotient must equal the multiplier, which shows how often the multiplicand has been subtracted from the product; and as it can only be subtracted, as often as the multiplicand has been repeated, the quotient must always equal the multiplier, leaving no remainder. When the multiplier consists of more than one figure; how do you proceed? A. First, multiply the multiplicand by the right hand figure of the multiplier; then by the next left hand figure of the multiplier, and so on, till the multiplicand has been repeated by each figure of the multiplier, placing the product of each directly under the multiplying figure; then add the several products, in the same order in which they stand, for the total product. What must be done, when ciphers occur between the significant figures of the multiplier? A. Care must be taken to commence the product of the first figure of the multiplier, in each place, directly under the figure by which we multiply. If our multiplier is 36, and instead of removing the product of the 3, one figure further to the left, than the product of the 6; we place the product of the 3 directly under the product of the 6, and then add the products: what would it be the same as multiplying by? A. By 9; because we first repeat our multiplicand 6 times, and then 3 times; and not giving the product of the 3, its proper local place, the multiplicand stands 6 times and 3 times repeated; that is, when added, 9 times repeated; but had we placed the product of the 3, one figure further to the left, we would have given it its local value; the products then would have expressed the multiplicand 6 times repeated, and 30 times repeated, and when added, the multiplicand 36 times repeated. When ciphers occur at the right hand of one or both the factors, what must be done? A. The significant figures must be multiplied the same as if no ciphers were at the right; then to the right hand of the total product of the significant figures, we annex as many ciphers, as are equal to the ciphers, at the right of both the factors; in order, to give the significant figures their local place in the product. How do you multiply by 10, 100, 1000, &c.? A. To multiply by 10, we join a cipher, to the right of the multiplicand, which increases it ten times; because figures increase in a ten fold proportion, and the cipher removes the place of each, one figure further to the left; the figure which first expressed units, now expresses tens, and the rest at the left, increase in the same proportion; and when we join two ciphers, the unit figure becomes hundreds; and the figure in the place of hundreds, expresses 100 times its simple value, and when we join three ciphers, the unit figure becomes thousands, that is, one thousand times its simple value, and the figures, at the left, being in a like manner removed to the left, increase in the same proportion. Can you multiply by 9, 99, 999, &c. in any other way than by directly multiplying the multiplicand by the number of nines? A. Yes, we may join as many ciphers to the right of our multiplicand, as our multiplier has nines, and from the sum thus produced, subtract the multiplicand; because by joining one cipher to the multiplicand, we repeat it 10 times; and then subtracting the multiplicand once, leaves it 9 times repeated; and if our multiplier is 99 we join two ciphers which increases our multiplicand 100 times; then subtracting the multiplicand once, leaves it 99 times repeated; and we observe the same rule, in multiplying by 3 nines, or any number of nines. How do you proceed to multiply by a composite number? A. First multiply the multiplicand by one of the coinponent parts, and then that product by the other; and the last product will be the total product. When is a number said to be a composite number? A. When it is produced by multiplying together any two figures in the multiplication table. Is thirty five a composite number? A. Yes. What are its component parts? A. 7 and 5. Why? A. Because 7 times 5 are 35. In multiplying by the component parts of 35, why should it produce the same product, as multiplying directly by 35? A. Because when we multiply by 7, we repeat our multiplicand 7 times; and when we multiply this product by 5, we repeat a number 5 times, which is 7 times greater than our first multiplicand; and 5 times 7 times must make 35 times repeated.-What is multiplication of decimal, or federal money? A. It is repeating the parts of an integer, or integers and parts of integers, a given number of times.What are integers in federal money? A. Dollars. What are the decimal parts of an integer? A. Dimes, cents and mills, or cents and mills, as they are commonly expressed. What do you understand by a decimal? A. I understand, that it is something less than a unit; in calling tenths, I understand that it takes 10 for a unit; and hundredths, that it takes 100 for a unit; and thousandths, that it takes 1000 to equal a unit; so that the name of each figure, shows the exact number required to equal a unit. How do you multiply decimal money? A. The same as whole numbers. How do you distinguish decimals from whole numbers? A. By a separatrix placed between them. Why are decimals multiplied the same as whole numbers? A. Because they increase in a tenfold proportion, the same as whole numbers. How do you point off decimals in your product? A. As far from the right, as the separatrix is in the multiplicand. When the price of one yard, one pound, or the price of a unit of any kind, is given; how do you obtain the price of a quantity? A. by multiplying the price of one, by the number expressing the quantity. Why should that give the price of the quantity? A. Because it is repeating the price of one as many times, as the quantity expresses, or contains units. If one yard cost $4,50 cts.; how would you find the price of 4 yards? A. I would multiply $4,50 cts., the price of one yard, by the units expressed in the number of yards, that is, by 4. "Why should that give the price of 4 yards? A. Because, it is plain, that 4 yards must cost 4 times as much, as one yard; and multiplying the price of one yard by 4, is repeating the price of one yard, 4 times. SIMPLE DIVISION, Is the shortest method of performing subtraction, where the same number is to be taken away a given number of times. It shows in a |