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Multiplication of Decimals. XXVII. How many yards of cloth are there in seven pieces, each piece containing 197 yards ? 197 = 19.875
Ans. 139.125 = 139 14 = 1394 yards. N. B. All the operations on decimals are performed in precisely the same manner as whole numbers. All the difficulty consists in finding where the separatrix, or decimal point, is to be placed. This is of the utmost importance, since if an error of a single place be made in this, their value is rendered ten times too large or ten times too small. The purpose of this article and the next is to show where the point must be placed in multiplying and dividing.
In the above example there are decimals in the multiplicand, but none in the multiplier. It is evident from what we have seen in adding and subtracting decimals, that in this case there must be as many decimal places in the product, as there are in the multiplicand. It may perhaps be more satisfactory if we analyze it.
7 times 5 thousandths are 35 thousandths, that is, 3 hundredths and 5 thousandths. Reserving the hundredths, I write the 5 thousandths. Then 7 times 7 hundredths are 49 hundredths, and 3 (which I reserved) are 52 hundredths, that is, 5 tenths and 2 hundredths. I write the two hun- . dredths, reserving the 5 tenths. Then 7 times 8 tenths are 56 tenths, and 5 (which I reserved) are 61 tenths, that is, 6 whole ones and 1 tenth. I write the 1 tenth, reserving the 6 units. Then 7 times 9 are 63, and 6 are 69, &c. It is evident then, that there must be thousandths in the product, as there are in the multiplicand. The point must be made between the third and fourth figure from the right, as in the multiplicand, and the answer will stand thus, 139.125 yards.
Rile. When there are decimal figures in the multipli cand only, cut off as many places from the right of the pro duct for decimals, as there are in the multiplicand.
If a ship is worth 24683 dollars, what is a man's share coorth, who owns šof her.
=.375 = 1876. The question then is, to find 16 of
24683 dollars. First find ooo of it, that is, divide it by 1000. This is done by cutting off three places from the right (Art. XI.) thus 24.683, that is, 24,98%, because 683 is a remainder and must be written over the divisor. In fact it is evident that tt'or of 24683 is 4483 = 24,08%. But since this fraction is thousandths, it may stand in the form of a decimal, thus 24.683.
It is a general rule then, that when we divide by 10, 100, 1000, fc. which is done by cutting off figures from the right, the figures so cut off may stand as decimals, because they will always be tenths, hundredths, fc.
Toro of 24683 then is 24.683 and 37 of it will be 375 times 24.683. Therefore 24.683 must be multiplied by 375. 24.683
$9256.125 Ans. $ 9256.125 This result must have three decimal places, because the multiplicand has three. The answer is 9256 dollars, 12 cents, and 5 mills. But the purpose was to multiply 24683 by .375, in which case the multiplier has three decimal places, and the multiplicand none. We pointed off as many places from the right of the multiplicand, as there were in the multiplier, and then used the multiplier as a whole number. This in fact makes the same number of decimal places in the product as there are in the multiplier.
We may arrive at this result by another mode of reasoning. Units multiplied by tenths will produce tenths ; units multiplied by hundredths will produce hundredths ; units multiplied by thousandths will produce thousandths, &c.
In the second operation of the above example, observe, that .375 is !, and ito, and [oʻsa, then too of 3 is toodi and Toro of 3 is voo, which is do and 1000, set down the 5 thousandths in the place of thousandths, reserving the do Then tooo of 80 is oft, or 18, and 5 times 16is and do (which was reserved) are too, equal to and the Set down the tho in the hundredth's place, &c. This shows also, that when there are no decimals in the multiplicand,
there must be as many decimal places in the product as in the multiplier
It was observed that when a whole number is to be multiplied by 10, 100, &c. it is done by annexing as many zeros to the right of the number as there are in the multiplier, and to divide by these numbers, it is done by cutting off as many places as there are zeros in the divisor. When a number containing decimals is to be multiplied or divided by 10, 100, &c. it is done by removing the decimal point as many places to the right for multiplication, and to the left for division, as there are zeros in the multiplier or divisor. If, for example, we wish to multiply 384.785 by 10, we remove the point one place to the right, thus, 3847.85, if by 100, we remove it two places, thus, 38478.5. If we wish to divide the same number by 10, we remove the point one place to the left, thus, 38.4785; if by 100, we remove it two places, thus, 3.84785. The reason is evident, for removing the point one place towards the right; units become tens, and the the tenths become units, and each figure in the number is increased tenfold, and when removed the other way each figure is diminished tenfold, &c.
How much cotton is there in 310 bales, each bale containing 4: cut.
3,10 = 3.7; 4 = 4.75. In this example there are decimals in both multiplicand and multiplier.
Ans. 17.575 cwt. 3.7 is the same as jy, we have to find jy of 4.75. Now to of 4.75, we have just seen, must be .475, and it is 37 times as much. We must therefore multiply.475 by 37, which gives 17.375 cwt.
We shall obtain the same result if we express the whole in the form of common fractions. 4.75 = 476 3.7 = it. Now according to Art. XVII. 1 of 475 is 4760 and j? will be 37 times as much, that is 2007
17.575 as before.
In looking over the above process we find, that the two numbers are multiplied together in the same manner as whole numbers, and as many places are pointed off for decimals in the product, as there are in the multiplicand and multiplier counted together.
It is plain that this must always be the case, for tenths multiplied by tenths must produce tenths of tenths, that is hundredths, which is two places ; tenths multiplied by hundredths must produce tenths of hundredths, or thousandths, which is three places; hundredths multiplied by hundredths
must produce hundredths of hundredths, that is ten-thou. gandths, which is four places, &c.
What cost 50 tons of hay, at $27.38 per ton ? 5 = 5.375.
$147.16750 Ans. In this example there are hundredths in the multiplicand, and thousandths in the multiplier. Now hundredths multiplied by thousandths must produce hundredths of thousandths, which is five decimal places, the number found by counting the places in the multiplicand and multiplier to. gether. The answer is 147 dollars, 16 cents, 7 mills, and is of a mill.
A man owned .03 of the stock in a bank, and sold .2 of his share. What part of the whole stock did he sell ?
It is evident that the answer to this question must be expressed in thousandths, for hundredths multiplied by tenths must produce thousandths. fo of Tåg are too. But if we multiply them in the form of decimals, we obtain only one figure, viz. 6. In order to make it express to% it will be necessary to write two zeros before it, thus, .006.
Ans. .006 of the whole stock.
This result is agreeable to the above rule.
The following is the general rule for multiplication, when there are decimals in either or both the numbers : Multiply as in whole numbers, and point off as many places from the right of the product for decimals, as there are decimal places in the multiplicand and multiplier counted together. If the product does not contain so many places, as many zeros must be written at the left, as are nécessary to make up the number.
Division of Decimals.
XXVIII. A man bought 8 yards of broadcloth for $75.376 ; how much was it per yard ?
In this example there are decimals in the dividend only.
I consider $75.376 as 75376 mills. Then dividing by 8, either by long or short division, I obtain 9422 mills per yard, which is $9.422. The answer has the same number of decimal places as the dividend.
ivide 117.54 bushels of corn equally among 18 meri. How much will each have ?
117.54 = 117=1178* ; this divided by 18 gives 1=6,10% = 6.53.