Product, Brought up. 32 also understand, that subtraction is made to prove 16 8 0 1st. EXAMPLE-Proved according to our 4th method. DEMONSTRATION.-Division being exactly the reverse of multiplica 4 8324 32 tion, is consequently made to prove it; because, when we multiply 8 by 4, the 8 is 4 times repeated; and when this 8 is made a divisor, we find that 32 contains it 4 times, which gives us our other factor; and repeating the 8 by this factor, it again produces 32, the same as our first product; which must always be the case; because, it is only repeating the same number a second time, and must produce the ame product; and when we come to subtract this product, we can have no remainder; because, taking the same number from itself can leave no remainder. 2. What will five yards of broadcloth cost, at three dollars a yard? Proof by Addition. 9 Multiplicand. 5 Multiplier. $15 Ans. 3 3 Proof by 5 5 3 3 15 Here, by multiplying the 3 by 5, we repeat the 3 five times; and it is evident, that five yards are worth five times as much as one. If we were not acquainted with multiplication, we would be under the necessity of setting 3 down five times, and adding as our proof shows. From this sum, we learn, that when the price of one is given; as 1 yard, 1 pound, 1 ounce, &c. we may obtain the price of the quantity, by multiplying the price of a unit, by the quantity; for the quantity when made a multiplier is considered to be a number containing as many units as the quantity contains yards, pounds, ounces, &c. 3 What will 9 calves come to, at 3 dollars each? Ans. $27. 4. What is the worth of 9 bushels of clover seed, at 9 dol. lars a bushel? Ans. $81. CASE II-When the multiplier consists of more than one figure. RULE.-Place your multiplier under the right hand figures of the multiplicand, so that units shall stand under units, tens under tens, &c. Then, multiply the multiplicand by each figure of the multiplier, commencing at the right hand; and remember to commence the product of each, as you pass to the left, directly under the multiplying figure. If a cipher, or ciphers occur between the significant figures of the multiplicand, you will bring them down in the product, in their proper place, in order to give the product of the figures at the left their proper local value; but if you have any thing to carry, instead of bringing down a cipher, you set down the figure which you have to carry; then add these several products for the whole, or total product. 1. Multiply 30684 by 43. 43 92052 122736 Product. 1319412 Here, when we multiply the multiplicand by 3, we repeat it 3 times. When we multiply by 4, we remove the product one fig ure further to the left, which gives it ten times the value it would have, placed directly under; and it should have ten time, the value; because it is not, in fact, multiplying the multiplicand by 4, but by 40, for the 4 expresses 4 tens. Had the product of the se cond figure of the multiplier been placed directly under the product of the first figure, it would have been repeating it 4 times; but placing it one figure to the left increases it ten times, which makes 40; because 4 times 10' are 40. If we take the multiplier_apart, and multi ply the multiplicand separately by the parts of the multiplier, and add the products, we will have the same result; thus, 30684 3 92052 30684 40 1227360 Here you will perceive we multiply by the parts of our first multiplier; first our multiplicand by 3, and then our multiplicand by 40, which is the local value of the left hand figure of our multiplier; and multiplying the multiplicand by 4, increases it 4 times; then bringing, a cipher at the right, increases it 10 times; which evidently makes 40 times repeated. Then, when we add the products, we obtain the same result; but in our first example, giving the product of our left hand figure its local place, was the same in effect. 1319412 Had we placed the product of the left hand figure of our multiplier, directly under the product of the first, and then have added, it would have only been repeating the multiplicand 7 times; for when we multiply by 3, (the first figure of the multiplier,) we repeat the multiplicand 3 times; then if we multiply by 4, placing the product directly under the product of the first figure of the multiplier, we repeat it 4 times; and 3 times repeated, and 4 times repeated, make 7 times repeated; as the following work shows. Thus, you see the same result is produced. 8. A merchant bought 481 bales of linen; in each bale there were 36 pieces; and in each piece, 24 yards; how many pieces, and how many yards, were there? Ans. 17316 pieces, 415584 yards. 9. Three hundred and forty seven men shared equally a prize, and each received forty-nine dollars; what was the whole prize? Ans. $17003. 10. If four bushels of wheat make a barrel of flour, and the price of wheat, one dollar a bushel; what will 375 barrels come to ?: Ans. $1500 CASE III-When ciphers occur between the significant figures of the multiplier, omit them in the operation RULE.-Place the product of each significant figure, directly under that by which you multiply; then add the products together, › & their sum will be the total product. EXAMPLES. 1st. Multiply 384 by 203 Here, when we multiply by 2, the 3-84 1.1 5.2 7768 7.7,95 2: Ans. · 2. 3604' left hand figure of the multiplier, we commence the product directly under itself, which brings it in the place of hundreds; and it should stand in the place of hundreds; 'for it is repeating the multiplicand by 200 the multiplying figure, (2) standing in the place of hundreds. 3-56805978864 Ans. 6 27 68087412 Ans.` 7. What is the product of 365432 multiplied by 7608? Ans. 2780206656. 8. Multiply eight thousand five hundred and sixteen, by seventy-six thousand and two. 9. Multiply 120345 by 9004. Ans: 647233032. Ans. 1083586380+ 10. What is the product of 24393 multiplied by 402? Ans 9805986 CASE IV. When one or both of the factors have ciphers at the right hand. ́ RULE.—Multiply the significant ngures the same as if there were no ciphers at the right, and add their products; then, join to the right nand of the total product of the significant figures, as many ciphers as .here are at the right hand of both the factors. EXAMPLES. 1. Multiply 4 6 3 by 4 0 40 1 8 5 20. DEM.-Here, we first multipl by 4, which repeats the multipli cand 4 times; but we wish to have our multiplicand repeated forty times, and to effect this, we bring the cipher of our multiplier at the right of the product, which gives the product of the 4 its local value, which is ten times more than its simple value; and ten times the simple value of 4, is forty; consequently we have our multiplicand forty times repeated. 2 Multiply 3460 by 4200. 3460 692 1384 14532000 DEM.-Here, we first multiply our significant figures, as our rule directs. And since we multiply only by 42, the hundredth part of our multiplier, our pro duct is only a hundredth part of what it should be; we then increase it a hundred times, by joining two ciphers at the right hand of our product. We now have repeated the significant figures of our multiplicand 4200 times; but still, our product is only one tenth of what it should be; because we have only multiplied a tenth part of our multiplicand; then to give our product its proper value, we increase it ten times, by joining another cipher at the right hand; which is for the cipher at the right of our multiplicand. 3. Multiply 21200 by 70. 4. Multiply 340300 by 13400. Ans. 1484000. Ans. 4560020000. 5. Twenty shillings make one pound; how many shillings in four hundred and fifty pounds? Ans. 9000 s. 6. In eighty pounds, how many shillings? Ans. 1600. 7. One hundred and sixty square rods make one acre; how many square rods in 320 acres? Ans. 51200 rods. 8. Six hundred and 40 acres make one square mile; how acres in three hundred square miles? many 9. Multiply 480000 by 12000. Ans. 192000 acres. Ans: 5760000000 CASE V. To multiply by 10, 100, 1000, &e. RULE. Add as many ciphers to your multiplicand, as there are ciphers in the multiplier; and the multiplying is performed. D |