Brought up. 16 Product, 32 also understand, that subtraction is made to prove 8_addition; consequently it may be made to prove 24 multiplication; for subtraction is the reverse of ad8 dition, and 32, (the product in our example,) is 8, -four tirnes expressed; then if 8 be taken away from 32, (the product,) 4 times, it must evidently dimin8 ish it to nothing; because it is taking away 8, the 8 multiplicand, as many times as it has been repeated . 8 by 4, the multiplier. 0 1st. EXAMPLE.-Proved according to our 4th method. DEMONSTRATION.—Division being exactly the reverse of muttiplica & tion, is consequently, made to prove it; because, 4 when we multiply 8 by 4, the 8 is 4 times repeated ; and when this 8 is made a divisor, we find that 32 8|32|4 contains it 4 times, which gives us our other factor; 32 and repeating the 8 by this factor, it again produ ces 32, the same as our first product; which must always be the case; because, it is only repeating the same nunber a second time, and must produce the ame product; and when we con to subtract this product, we can dave 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? Here, by multiplying the 3 by 5, we 3 Multiplicand. repeat the 3 five times, and it is evi. 5 Multiplier. dent, that five yards are worth five times as much as one. If we were not 'ac$15 Ans. quainted with multiplication, we would Proof by be under the necessity of setting 3 down five times, and adding as our proof shows. Multiplication. From this sum, we learn, that when 5 the price of one is given; as I yard, 1 3 pound, 1 ounce, &c. we may obtain the price of the quantity, by multiplying the price of a unit, by the quantity; for the 15 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. Proof by Addition. Cil.coco.co co co €; 5. Multiplý 48 by 3. Here, we say 3 times 8 aré 24; pie Multiplicand, 48 cing down 4, the right hand figure of Multiplier, the product, and keeping the left hand 3 figure in mind; then we say 3 times 4 are 12, and 2 are 14; placing down the 4 4 4 whole product, (it being the product of the left hand figure;) this is carrying by 10, or adding the left hand figure to the product of the next figure, which is the same. 6. 8. 9. Multiplicand, 3 4 3 2 4 6 7 83 8 3 7 3 4 6 3 7834 Multiplier, 3 5 6 1 Caše II.-When the multiplier consists of more an one figure. RULE.-Place your multiplier under the right hånd 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, înstead of bring ing down a cipher, you set down the Agure which you håve to carry; then add these several products for the whole, oi total product. 1. Multiply 30684 by 43. Here, when we multiply the 43 multiplicand by 3, we repeat it 3 times. When we multiply by 4; 92052 we remove the product one fig122736 ure further to the left, which Product. 1319412 gives it ten times the value it would have, placed directly un. der; 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 mereases 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 30684 Here you will perceive we multiply 3 40 by the parts of our first multiplier ; first our multiplicand by 3, and then our nul92052 1227360 tiplicand by 40, which is the local value 92052, of the left hand figure of our multiplier; and multiplying the multiplicand by 4, 1319412 increases it 4 times; then bringing. a ci pher 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. 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 re.. peated; as the following work shows. 3 0 6 8 4 3 0 6 8 4 7 2 1 4 7 8 8 4. 6 4 3 3.2.6 9. 1.6.1 9. 3 4 O 4.6 4 6 7 0 2 3 5 9 3 1 1 9 2 1 Ans.. 5 6. 7. 38 6 3 0.4 3 7:6 0.0.4.5. .: 679 4.0 4 3 2 3 2 1 1 6 8 4 1.3 6 4 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 ihe 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 IÍI-When. ciphers occur between the significanb figurës: of the multiplier, omit them in the operation RULE.-Pface 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 left hand figure of the multiplier, 250:3 we commence the product directly under itself, which brings it in the 1.1 5 2 place of hundreds; and it should 71 6 8 stand in the place of hundreds; 'for it is repeating the multiplicand by 7.7.9. 52: Ans. · 300 : the multiplying figure, (2) standing in the place of hundreds. 2... 3. 4. 10304 205 1700420 3604 1360336 367608 Ans. 1362036420 AÁs. 2112320 Ans. : 5. 6: 3 4 6 34 6 3 2* 30 4 6 4 0 3 1 0 3 0 2 20604 3:5 6 8 0 5 9 7 8 8 6 4 Ans. 6 2 7 6.8087 4 1 2 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. Ans: 647233032. 9. Multiply 120345 by 9004. Ans. 10835863801 10. What is the produet of 24393 multiplied by 402? Ans.98059862 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 DEM.-Here, we first multip) 4 0 by 4, which repeats the multipli cand 4 times; but we wish to 1 8 5 2 0 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. DEM.-Here, we first multiply. 3460 our significant figures, as our rule directs. And since we mul4200 tiply only by 42, the hundredth 692 part of our multiplier, our pro 1384 duct is only a hundredth part of what it should be; we then in14532000 crease it å 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 vah:e, 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. Ans. 1484000, 4. Multiply 340300 by 13400. 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. 5+200 rods. 8. Six hundred and 40 acres make one square mite; how many acres in three hundred square miles ? Ans, 192000 acres. 9. Multiply 480000 by 12000. Ans. 5760000000 CASE V.-To multiply by 10, 100, 1000, &e. RULE,Add as many diphers to your multiplicand, as there are olphers in the multiplier; and the multiplying is performed. D |