30 by 1 RULE FOR MULTIPLYING WHEN THE FACTORS ARE TERMINATED BY CIPHERS. Multiply the significant figures together, and to their product annex as many ciphers as terminate both the factors. Note.—All figures are called significant, except ciphers. Multiply 20 400 by 60 3000 9 30 200 6 90 2000 40 100 100 (“ 2400 2000) (" 160 4200) When any number is made by multiplying two numbers together, it is called a composite number. Thus 12 is a composite number, because it is made by multiplying 3 and 4 together. Is 18 a composite number? What two numbers multi . plied together make 18 ? Is 14 a composite number? Is 13 a composite number? Is 9 a composite number? If 12 is multiplied by 8, what is the product? What are the factors which compose 8? If you multiply 12 by one of these numbers, and the product by the other, will the answer be the same as if you multiply 12 by 8? Let the pupil try and see. Multiply 123 by 18. Multiply it by one of the num. bers that compose 18, and the product by the other num. ber, and what is the result ? EX RULE FOR MULTIPLYING, WHEN THE MULTIPLIER CEEDS 12, AND IS A COMPOSITE NUMBER. Resolve the multiplier into the factors which compose it, and multiply the multiplicand by one, and the product by the other. What is the rule for multiplying when both factors are terminated by ciphers? What is a composite number? What is the rule for multiplying when the multiplier exceeds 12, and is a composite number? Let the following sums be done by the above rule. by 20 Multiply 587 by 16 268 49 6543 24 329 54 521 27 426 32 72 30 2345 96 793 36 7654 64 6543 40 In multiplication it makes no difference in the product, which of the factors is used for multiplier or multiplicand; for 3 times 4, and 4 times 3, give the same product, and thus with all other factors. It is in most cases most con. venient to place the largest number as multiplicand. DECIMAL MULTIPLICATION. In explaining decimal multiplication, it is needful to un. derstand the mode of multiplyiog and dividing by the sep. aratrix. If we have 2,34 we can make it ten times greater, by moving the separatrix one order to the right, thus, 23,4. For 23 units, 4 tenths, is ten times as much as 2 units, 34 hundredths. It is therefore multiplied by 10. We can multiply it by 100 by removing the separatrix entirely, thus, 234, for the 2 units and 34 hundredths, be. come 234 units, and are thus multiplied by 100. Whenever therefore we wish to multiply a mixed or pure decimal, by any number composed of i and ciphers, we can do it by moving the separatrix as many orders to the right, as there are ciphers in the multiplier. 66 EXAMPLES. 100 1000 4,63956 10000 How can decimals be multiplied by any number composed of 10 ciphers. But if the decimal has not as many figures at the right, as are needful in moving the separatrix, ciphers can be ad. ded thus. Multiply 2,5 by 1000. Then in order to mul. tiply by a cipher, it is necessary to move the separatrix as many orders to the right, as there are ciphers in the multiplier, 1000; in order to do this, two ciphers must be added thus, 2500, Here 2 units, and 5 tenths, are changed to 2 thousands and 5 hundreds, and of course are made 1000 times larger, or multiplied by 1000. In the following examples, in order to multiply by moving the separatrix, it is necessary to add ciphers to the right of the multiplicand. EXAMPLES Multiply 3,7 by 100 Multiply 5,2 by 100 1000 36,3 « 1000 10000 3,869 10000 34,200“ 100000 5,6469 ~ 100000 66 2,35 « Division also, can be performed on decimals, by the use of the separatrix. Whenever we divide a number, we make it as much smaller, as the divisor is greater than one. If we divide by 10, as 10 is ten times greater than one, we make the number 10 times smaller. If we divide by 100, we make the numbers 100 times smaller. If therefore we make a number 10 or 100 times smaller, we divide by 10 or 100. If we make it 1000 times smaller, we divide by 1000, &c. What is done if the decimal has not as many figures at the right as are required ? When we divide a number, how much smaller do we make it? How can we divide a decimal by any number composed of 1 and ciphers ? If then we are to divide 323,4 by 10, we must make it 10 times smaller. This we can do by moving the separatrix one order to the left, thus, 32,34. If we are to di. vide by 100, we can do it by moving the separatrix twa orders to the left, thus, 3,234. If we are to divide by 10,000, we can do it by moving the separatrix 4 orders to the left, thus, ,3234. Whenever therefore, we wish to divide a pure or mixed decimal, by a number composed of 1 and ciphers, we can do it by moving the separatrix as many orders to the left, as there are ciphers in the divisor. EXAMPLES. Divide 32,5 by 10 | Divide 32,69 by 10 342,6 « 100 3269,1 " 100 469,3 “ 1000 2396,4 1000 46936,7 « 10000 12346,95 “ 10000 23469,8 5 100000 15463,96“ 100000 But if the decimal has not enough figures to enable the separatrix to be moved, according to the rule, ciphers must be prefixed. Thus if we wish to divide 3,2 by 100, we do it thus, ,032. Here the 3 is changed from 3 units, to 3 hundredths, and of course made 100 times less. 32,4 66 21,6 « 600,7 66 32,366 EXAMPLES. Divide 2,4 by 100 Divide 23,4 by 10000 1000 246,4 " 100000 932,5 66 10000 293,6 « 100000 100000 546,9 100000 1000000 1000000 286,96 10000000 100,4 « 10000000 542,8 6 100000000 3694,9 “ 1000000 A decimal can also be multiplied, by expunging the se. paratrix. Thus 2,4 is multiplied by 10, by expunging the separatrix, thus, 24. What is done if the decimal has not figures enough? What effect is produced by expunging the separatrix of a decimal ? 66 2,56 is multiplied by 100, by expunging the separatrix, thus, 256. In all these cases, the decimal is multiplied by a num. ber composed of 1, and as many ciphers as there are deci. mals at the right of the separatrix which is expunged. If you expunge the separatrix of the following decimals, by what number are they multiplied ? 2,46. 3,295. 54,6823. 54,63. 89,46321. 5,6432. How can you multiply 3,1 by 10? What is it after this multiplication ? How do you multiply 3,12 by 100? What is it after this multiplication ? How do you multiply 9,567 by 1000? What is it after this multiplication ? If the separatrix is expunged from 2,52, by what is it multiplied ? If the separatrix is expunged from 2,56934, by what is it multiplied ? If the separatrix is removed from 5,943216, by what is it multiplied ? If the separatrix is removed from 3,4621, by what is it multiplied ? If the separatrix is removed from 3,5, by what is it multiplied ? If a man supposes he owes $54,23, and finds he owes 10 times as much, what is the sum he owes? How do you perform the multiplication with the separatrix ? What does the number become after being thus multiplied ? Multiply in the above mode $244,635 by 10, by 100, and by 1000. What does the sum become, by each of these operations ? Divide $244,635 by 10, by 100, and by 1000, with the separatrix. What does the sum become by each of these operations ? Divide and multiply, with a separatris, $2556,436, by 10, by 100, and 1000. In this case by what number is the decimal multiplied ? |