3 36 16 9 29 12 4 X 1 = 4 4 86 126 12 16 7 X 1 2017 24 2 3 4 219 12 108 12. 9 28 10 X 110 12, 10 2012, 11 549 co ob co co co co co co co co co coj 10 10 10 10 10 10 10 2 1 1 2 10 44 11 64 12 487 8 5 X 15 10 5 2 10 17 11 125 3 14 5 208 X 1810, 10 1610, 11 110 24/10, 12 4017 7 49 10 4 40 7 10 12 84 10 9 90 100 2 3 120 What will 3 apples come to, at 3 cents each? What will 8 bushels of wheat cost, at 8 shillings a bushel times 8 are how many? What will 10 bushels of oats come to, at 2 shillings a bushel? What are 9 yards of calico worth, at 3 shillings a yard? 3 times 9 are how many ? What must you pay for 12 cows, at 12 dollars each? What will 4 cows come to, at 10 dollars each? What will 5 cows? what will 6 cows? what will 7 cows? what will 8 cows? what will 10 cows? What will 11 ploughs cost, at 11 dollars each ? What will 9 hogs cost, at 9 dollars each? A pound contains 20 shillings; how many shillings in 2 pounds? in 3 pounds? in 4 pounds? in 5 pounds? How many are twice 20? how many are 3 times 20? 4 times 20? 5 times 20? Twelve pence make 1 shilling; how many pence in 2 shillings? in 3 shillings? in 4 shillings? in 5 shillings? in 6 shillings in 7 shillings? in 8 shillings? in 9 shillings? in 10 shillings in 11 shillings? in 12 shillings? How many are twice 12? 3 times 12? 4 times 12? 5 times 12? 6 times 12? 7 times 12? 8 times 12? 9 times 12? 10 times 12? 11 times 12? 12 times 12? CASE I-When the multiplier is a single, significant figure. RULE.-Place the multiplier under the right hand figure of the multiplicand, and draw a line underneath. First, multiply the right hand figure of the multiplicand by the multiplier; and when the product does not exceed 9, place it directly under; but if the product exceed 9, place down the right hand figure of the product; and add the left to the product of the next figure of the multiplicand, and so proceed, till you have multiplied all the figures of the multiplicand by the multiplier; remembering to set down the whole product of the Jeft hand figure. METHODS OF PROOF.-1st. Make the multiplicand a multiplier, and if it produce the same result, the work is right. 2nd. Multiplication may be proved by addition. Write the multiplicand down as many times, as the multiplier eKpresses a unit; then add, and if the same result be produced, the work is right. 3d. Multiplication may be proved by subtraction. From the product, subtract the multiplicand as many times, as the multiplier expresses a unit; and if it diminish it to nothing, the work is right. 4th. Lastly, multiplication may be proved by division. Divide the product by either of the factors, and if the operation produce the other, the work is right. Although this is the best meth od of proof, yet it should be omitted, till the student has become acquainted with division. Other methods of proof might be given; but these are introduced on account of their being the most simple, and best suited to our purpose in illustrating the principles of this rule; and showing the relation which it bears to the other simple rules. EXAMPLES. 1. Multiply 8 by 4. Multiplicand, 8 Multiplier, 4 Product, 32 We place the multiplier under the multiplicand, as our rule directs. Then we say 4 times S are 32, placing the product under; we then have 32 for a product, which is 8, 4 times repeated. DEMONSTRATION.-That this is a short way of performing addition, is very evident: for we arrive at the result at once, which, in addition, requires the multiplicand 8, to be set down four times and added; thus, 8+8+8+8=32. 8 1st. EXAMPLE.-Proved according to the first method. Multiplicand, 4 What was before the multiplicand, Multiplier, now becomes the multiplier. We now say 8 times 4 are 32, the same pro32 duct as was produced by the first op Product, ⚫eration. DEMONSTRATION.-It is plain, that it can make no difference, whether 8 be repeated 4 times, as, 4 times 8 are 32; or 4 repeated 8 times, as, 8 times 4 are 32, the same result is produced. 1st. EXAMPLE.-Proved according to the 2nd method. DEMONSTRATION-Here, we set down the multiplicand 4 times: 8 8 because in the example, our multiplier expresses 4 units; we then add, and the result clearly demonstrates, that when the same number is repeated by multiplying, it may be repeated by adding; and consequently proves the work. But proof by addition is a more tedious method than proof by multiplication, and is therefore only introduced to show the relation between the two rules. It is also plain, that it can make no difference, whether we add the multiplicand as many times as our multiplier expresses a unit; thus, 8+8+8+8=32; or, add the multiplier as many times as our multiplicand expresses a unit, thus, 4+4+4+4+4+4+4+4=32; the same result, you perceive, must be produced." 32 Amount. 1st. EXAMPLE.-Proved according to the 3d method. Multiplicand, 8 DEM. You have just learned, that multiplicaMultiplier, 4 tion is a short way of performing addition. You C'arried up. Product, Brought up. 32 also understand, that subtraction is made to prove 8 addition; consequently it may be made to prove -multiplication; for subtraction is the reverse of ad 24 8 dition, and $2, (the product, in our example,) is 8, four times expressed; then if 8 be taken away from 16 32, (the product,) 4 times, it must evidently dimin 8 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. 1st. EXAMPLE.-Proved according to our 4th method. DEMONSTRATION.--Division being exactly the reverse of 8 8324 32 multiplication, 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 same 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? $ 3 Multiplicand, $15 Ans. 3 addition, Silco co co co coCA by Proof Proof by 5 3 15. Here by multiplying the 3 by 5, we repeat the 3, five times; and it is evident, that five yards is 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, Qunces, &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 dollars a bushel? Ans. $81. 5. Multiply 48 by 3. Multiplicand, 48 Multiplier, 3 14 4 Here, we say 3 times 8 are 24, placing down 4, the right hand figure of the product, and keeping the left hand figure in mind; then we say 3 times 4, are 12, and 2 are 14; placing down the 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. Multiplicand, 3 4 32 46783 8373463 7834 Multiplier, 3 10296 Ans. 5 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 figure further to the left, which gives it ten times the value it would have, placed directly under; and it should have ten times 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 second 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 |