If before multiplying, the multiplicand is made a certain number of times larger, the product is made as much lar. ger. If the multiplicand is made too large, the product is as much too large. For example; If we wish to find how much twice 2,3 is, we can change it to whole numbers, and multiply it by 2, and we know the answer is 10 times too large. For 23 is 10 times lar. ger than 2,3, and therefore when it is multiplied by 2, its product is 10 times too large. If then we make it 10 times smaller, we shall have the right answer. When. ever, therefore, we wish to multiply a decimal, we can change it to whole numbers, and multiply it by the rule for common multiplication. We then can make the product as much smaller, as we made the multiplicand larger, by changing it to whole numbers. For instance, if we wish to multiply 3,6 by 3, we can expunge the separatrix, and the multiplicand becomes 10 times too large. We then multiply it as in whole num. bers thus, 36 108 This product is also 10 times too large, and we find the right answer, by placing a separatrix so as to divide it by 10, thus making it ten times smaller. In like manner, if the multiplier is increased a certain number of times, the product is increased in the same pro. portion. If we are to multiply 32 by 2,3, and should by expung. ing the separatrix, change the multiplier to whole num. bers, it would make the product 10 times too large, and to obtain the right answer we must divide the product by 10 with a separatrix, thus making it 10 times smaller. Multiply 2,5 by 4. What is the effect on the product, if the multiplicand is made a certain number of times larger? How is the right product to be obtained ? What is the effect on the product, if the multiplier is increased a certain number of times ? By what number do you multiply, when you expunge the separatrix of the decimal ? What is the product of the multiplication after the se. paratrix is expunged? How much too large is this product? How do you divide this product by the same number as you multiplied the decimal ? Explain each process as above ? Multiply 12,46 by 5 | Multiply 3,2 by 6 18,23 8 52,23 7 ,346 9 286,45 8 36,2 ny 123,678 9 25,36 5 32,92 12 44,429 4 64,64 11 92,1234 7 988,931 9 Multiply 329 by 2,4 | Multiply 764 by 8,925 426 3,5 2875 72,63 362 39,5 30021 984,4 4689 2,36 8643 6,529 4693 5,462 2875 ,462 32678 " 6,8246 7628 " ,3596 Let the multiplier be 2, 4, and the multiplicand is 3,6. Changing the multiplier to whole numbers, would make the product ten times too large. Should the multiplicand be changed to whole numbers, the product would again be made ten times larger, so that it would be made 100 times too large. Therefore to bring the answer right, we must divide it by 100, thus making it 100 times smaller. This is done by the use of a separatrix. 3,6 and 2,4, when changed to whole numbers and multiplied together, are 864. This is 100 times too large, and is brought right, by dividing it by 100, thus, 8,64. RULE FOR EXPLAINING DECIMAL MULTIPLICATION. Change the Decimals to whole numbers by expunging the How is the right product obtained ? What is the rule for explaining the process of decimal multiplication ? separatrix. Multiply as in whole numbers. Divide the an. swer by the product of the two numbers by which the factors were multiplied, -in expunging the separatrix. EXAMPLE. Multiply 8,61 by 4,7. Change these to whole numbers, and they become 861 and 47. (Here the multiplicand, in expunging the separatrix, is multiplied by 100, and the multiplier by 10.) Multiplying them together, they produce 40467. The product of the two numbers by which the factors were multiplied, (10 and 100), is 1000. Dividing 40467 by it, gives the answer 40,467. EXAMPLES. Multiply 2,37 by 4,6. Soubora 362,68 6895,40 2195,334 937,8 L 86,4 765,3 1,23 89123,002 ,591 The following common rule for decimal multiplication, includes all the others, and may be used after understand. ing the preceding. COMMON RULE FOR DECIMAL MULTIPLICATION. Multiply as in whole numbers, and then point of in the product, as many orders of decimals, as are found in both the factors. If 4 grains, 3 penny-weights, are repeated 3 times, what is the product? If 3 yards, 1 quarter, be repeated 3 times, what is the product ? If 5 feet, 2 inches, be repeated 4 times, what is the product ? If 2 hogsheads, 5 gallons, be repeated 5 times, what is the product ? If 4 drams, 2 ounces, be repeated 3 times, what is the product ? What is 4 times 2 days, 7 hours ? RULE FOR COMPOUND MULTIPLICATION. Place the multiplier below the multiplicand. Multiply each order separately, beginning with the lowest. In the product of each order, find how many units there are of the next higher order. Carry these units to the next product, and set the remainder under the order tiplied. What is the common rule? What is the rule for compound multiplica. per cwt. ? Proceed thus :--Four times six pence are 24 pence, which is 2 units of the next higher order, (or shillings;) to be carried to that order; and as no pence remain, a ci. pher is to be placed in the order of pence. Four times 9 shillings are 36 shillings, and the 2 carried make 38 shillings, which is 1 pound, to be carried to the next product, and 18 shillings to be written in the shilling order. Four times 1 pound is 4 pounds, and the 1 carried, makes 5, which is to be written in the order of pounds. Let the pupil do the following sums, stating the process while doing it, as above. What cost 9 yards of cloth, and 5s. 6d. per yard? What is the weight of 6 chests of tea, each weighing 3 cwt. 2 qrs. 9 lbs. ? What is the weight of 7 hogsheads of sugar, each weighing 9 cwt. 3 qrs. 12 lbs.! How much brandy in 9 casks, each containing 41 gals. 3 qts. 1 pt. ? ANSWERS. yds. qr. na. yds. qr. na. 1. Multiply 14 3 2 by 11 163 2 2 at. pt. hhd. g. qt. pt. 2. Multiply 21 15 2 1 by 12 254 61 2 0 le. m. fur. po. le. m. fur. po. 3. Multiply 81 2 6 21 by 8 655 1 4 8 p. a. p. 4. Multiply 41 2 11 by 18 748 38 yr. m. w. d. yr. 5. Multiply 20 5 3 6 by 14 286 11 2 0 S. s. 6. Multiply 1 15 48 24 by 5 ny 19 20 hhd. g. 1. m. w. d. |