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will be hundredths, and so on. Therefore, if when one zero is annexed, the dividend is not so large as the divisor, a zero must be put in the quotient with a point before it, and in the same manner after two or more zeros are annexed, if it is not yet divisible, as many zeros must be placed in the quotient.
Two men talking of their ages, one said he was 37294315 years old, and the other said he was 64413 years old. What was the difference of their ages?
If it is required to find an answer within 1 minute, it will be necessary to continue the decimals to seven places, for 1 minute is of a year. If the answer is required only within hours, five places are sufficient; if only within days, four places are sufficient.
64418 = 64.8520000 17783
= 37.2602313 +
37 38 47
Ans. 27.5917687 years. It is evident that units must be subtracted from units, tenths from tenths, &c. If the decimal places in the two numbers are not alike, they may be made alike by annexing zeros. After the numbers are prepared, subtraction is performed precisely as in whole numbers.
Multiplication of Decimals. XXVII. How many yards of cloth are there in seveni pieces, each piece containing 193 yards ?
19 = 19.875
13946 = 1391 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 separatris, 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 multiplieand. 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 hundredihs, I write the 5 thousandths. Then 7 times 7 hundredths are 49 hundredths, and 3 (which I reserved)
52 hundredths, that is, 5 tenths and 2 hundredths. I write the 2 hundredths, 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.
Rule. When there are decimal figures in the multiplicand only, cut off as many places from the right of the product for decimals as there are in the multiplicand.
If a ship is worth 24683 dollars, what is a man's share worth, who owns of her ?
= .375 = 37. The question then is, to find 342 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, 24.08.20 because 683 is a remainder and must be written 'over the divisor. In fact it is evident that Toto of 24683 is 2 4 6 8 3
But since this fraction is thou
24 6 8 3
sandths, 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, &c., which is done by cutting of figures from the right, the figures so cut off may stand as decimals, because they will always be tenths, hundredths, &c.
Toto of 24683 then is 24.683 and 10% 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, aod 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 ihe 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; uniis multiplied by thousandths will produce thousandths, &c.
In the second operation of the above example, observe, that .375 is jó, and to, and to try then yooo of 3 is too, and toto of 3 is too, which is too and Toot, set down the 5 thousandths in the place of thousandths, reserving the ido. Then goto of 30 is To Oy or it, and 5 times 187 is , and do (which was reserved) are op, equal to 1o and Tor Set down the tio in the hundredths' 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 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 3-10 bales, each bale containing 4 cwt.?
=4.75. In this example there are decimals in both multiplicand and multiplier.
17.575 cwt. 3.7 is the same as f], we have to find 17 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 '17575 cwt.
1 7 5 7 5
= 17 575
We shall obtain the same result if we express the whole in the form of common fractions. 4.75 = 416 = 174, and 3.7 = 1. Now according to Art. XVII. It of 117 is 177o, and it will be 37 times as much,
iono = 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 iś 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-thousandths, which is four places, &c.
What cost 5 tons of hay, at $27.38 per ton? 5 = 5.375.
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 together. 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