I then multiply these two excesses together, and their product is 56. The two figures of this product, viz. 5 and 6, I add together, and their sum is 11, which exceeds 9 by 2; and this excess I place against the total product in the example. Lastly, I cast the 9's out of this product thus: I begin at the left hand, and say, 1 and 8 are 9; which I cast away: then, 7 and 3 are 10; I cast away 9, and 1 remains: then, 1 and 8 are 9, which I cast away: then, S and 5 are 13, which is 4 over 9-I cast away 9: then, 4 and 7 are 11, which is 2 over 9; and as this excess is the same as the excess standing against the product, I conclude the work is right. Third method of Proof. Multiplication may be proved by Division; viz. thus: Divide the product by either of the factors, and the quotient will be equal to the other factor, if the work be right. This is the safest method of proving multiplication, but it cannot be practised till the rule of Division is learned. More Examples in Simple Multiplication. 12. Multiply 40353607 by 16807. 13. Multiply 85173 Ans. 678223072849. by 78542. 14. Multiply 4897685 by 40003. Ans. 195922093055. 15. Multiply the number 8763 by itself. Ans. 76790169. 16. Required the continued product of the numbers Ans. 134217728. 4096×512×64. 17. Required the product of 1296×216×36×6. Ans. 60466176. CONTRACTIONS IN SIMPLE MULTIPLICATION. There are several methods of contraction which may be used in particular cases, in finding the products of numbers, by which the operations may be performed in a shorter manner than by the foregoing rules. The most useful of these contractions are the following. I. When there are ciphers on the right hand of one or both of the factors: Set down the factors in the same manner as if there were no ciphers on the right hand: then, in multiplying, neglect those ciphers, and multiply together the other figures as usual; and annex to the product as many ciphers as there are on the right hand of both the factors.* Prod. 282360000 3. Multiply 359260 by 3040. 4. Multiply 9826000 by 82530. : Prod. 112756000 Ans. 1092150400. Ans. 810939780000. Note. To multiply any number by 10, or 100, or 1000, c. Annex to the given multiplicand as many ciphers as there are in the multiplier, and it will then be the product required. So 54×10=540 And 342×100=34200 And 120x1000-120000 II. When the multiplier exceeds 12, and is a composite number which can be produced by multiplying together two or more small numbers not greater than 12: Then multiply the given multiplicand by one of those small numbers, or component parts of the multiplier, and that product by another of those numbers, or parts, and so on, till you *The reason of this method of contraction will easily appear from the demonstration of the rule for Case 2d, and from the nature of notation. A composite number is one that can be produced by multiplying together two or more smaller numbers; or, in other words, a composite number is one that can be divided by some smaller number besides unity, without a remainder. The factors which produce any composite number, are called the component parts of the number. Those numbers which are not composite, are called prime numbers. have multiplied by all of them; and the last product will be the total product required.* 7×6=42, and hence 1504×7×6=63168, the Ans. 3. Multiply 1504 by 42. 4. Multiply 8647 by 252. 4x7x9=252; hence 8647×4×7x9=2179044, the Ans. 5. Multiply 64321 by 81. Ans. 5210001. Ans. 68432256. Ans. 53884800. III. When all the figures of the multiplier are 9's: Annex as many ciphers to the multiplicand as there are figures of 9 in the multiplier; then, from this number subtract the given multiplicand, and the remainder will be the product required.† *The reason of this rule is obvious: for any number, multiplied by the component parts of another number, must give the same product as though it were multiplied at once by the whole number, or product of those parts: Thus, in example first, 5 times the product of 7 multiplied into 6423, makes 35 times that number, as plainly as 5 times 7 makes 35. It is evident that if any number be multiplied by 9, the product will be 9 tenths of the product of the same number multiplied by 10; and as the annexing of a cipher to the multiplicand increases it ten fold, it is evident that if the given multiplicand be subtracted from the tenfold multiplicand, the remainder will be ninefold the said given multiplicand, equal to the product of the same by 9; and the same will bold true of any number of 9's. Note. When the value, or weight, &c. of any article or thing is given, to find the value, &c. of any number of like articles Multiply together the value of one article and the number of them, and the product will be the answer. Either of the two factors may be made the multiplier, but it will generally be the most convenient to multiply by the less number. EXAMPLES. 1. A merchant sold 8 pieces of cloth, at 12 dollars a piece; how much money did he receive for the whole? Here 12×8-96 dollars, Ans. 2. If 27 barrels contain each 196 pounds of flour, how much flour do they all contain? Ans. 5292 pounds. 3. What do 210 hats come to, at 3 dollars each? Ans, 630 dollars. 4. What do 52 firkins of butter come to, at 7 dollars a firkin? Ans. 364 dollars. 5. My orchard contains 15 rows of trees, with 23 trees in each row; how many trees are there in it? Ans. 345. 6. What is the value of 526 acres of land, if each acre be worth 15 dollars? Ans. 7890 dollars. 7. What do 13 tons of hay come to, at 15 dollars a ton? Ans. 195 dollars. /8. If seventeen casks contain each thirty gallons of cider, how much cider do they all contain? Ans. Five hundred and ten gallons. 9. How many panes of glass will it take to make 22 win *The reason of this rule is very obvious; for it is plain that if one yard of cloth is worth 3 dollars, 2 yards are worth twice 3 dollars, or 6 dollars, and 3 yards are worth 3 times 3 dollars, or 9 dollars, and so on; and it is evident that the rule will hold true in all similar calculations. dows, with 24 panes in each window? Ans. 528. 10. A merchant bought 342 bales of linen; each bale contained 56 pieces of cloth, and each piece 25 yards: how many pieces, and how many yards of cloth were there? Ans. 19152 pieces, and 478800 yards. QUESTIONS ON THE FOREGOING. 1. What is Multiplication? 2. What is the number to be multiplied called? 3. What is the number by which we multiply called? 4. What is the general name for both numbers? 5. What is simple multiplication? 6. What is the first case in simple multiplication? 7. How do you set down the factors? 8. Where do you begin to multiply; and how do you proceed? 9. When do you carry, and to what do you add the number carried? 10. What is the second case? 11. How do you set down the factors in this case? 12. How many figures of the multiplier do you use at a time? 13. In multiplying by each figure of the multiplier, how do you set down the product? 14. When you have multiplied by all the figures of the multiplier, what do you do next? 15. What is the first method of proving multiplication? the second? the third? 16. How do you perform multiplication when there are ciphers at the right hand of the factors? 17. How do you multiply any number by 10, or 100, &c. ? 18. What is a composite number? 19. How may two numbers be multiplied together when the multiplier is a composite number? 20. What method of contraction may be used when all the figures of the multiplier are 9's? 21. How do you find the value of any number of similar articles when the value of one of them is known? DIVISION, Teaches how to separate any given number, or quantity, into any number of equal parts assigned; or to find how often one number is contained in another; and is a concise method of performing several subtractions. The number to be divided is called the Dividend; the number to divide by is called the Divisor; and the number |