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same ten-fold proportion ; as in the following Scale or Table of Notation.
Set the numbers under each other according to the value of their places, like as in whole numbers; in which state the decimal separating points will stand all exactly under each other. Then. beginning at the right-hand, add up all the columns of numbers as in integers; and point off as many places, for deciinals, as are in the greatest number of decimal places in any of the lines that are added ; or place the point directly below all the other points
EXAMPLES. 1. To add together 29.0146, and 3146 5, and 2109, and •62417, and 14.16
29.0146 3146 5 2109
5299 29877 the Sum.
Ex. 2. What is the sum of 276, 39-213, 72014:9, 417, and 5032 ?
3 What is the sum of 7530, 16•201, 3.0142, 957:13, 6.72119 and .03014.
4. What is the sum of 312.09, 3.5711, 7195-6 71.498, 9739:215, 179, and .0027?
SUBTRACTION OF DECIMALS.
PLACE the numbers under each other according to the value of their places, as in the last Rule. Then, beginning at the right-hand, subtract as in whole numbers, and point off the decimals as in Addition.
1. To find the difference between 91.73 and 2.138.
Ans 89.592 the Difference.
2. Find the diff. between 1.9185 and 2.73. Ans 0:8115. 3. To subtract 4.90142 from 214.81. Ans. 209.90858. 4. Find the diff. between 27 14 and .916. Ans. 27 13 084.
MULTIPLICATION OF DECIMALS.
Place the factors, and multiply them together the same as if they were whole numbers. –Then point off in the product just as many places of decimals as there are decimals in both the factors. But if there be not so many figures in the product, then supply the defect by prefixing ciphers.
* The Rule will be evident from this example :-Let it be required to multiply ·12 by :361 ; these numbers are equivalent to 12 and 361; the product of which is
= '04332, by the nature of Nolation, which consists of as many placi s as there are ciphers, that is, of as many places as there are in both numbers. And in like manner for any other numbers,
To multiply Decimals by 1 with any number of Ciphers, as by
10, or 100, or 1000, &c.
This is done by only removing the decimal point so many places farther to the right-hand, as there are ciphers in the multiplier: and subjoining ciphers if need be.
1. The product of 51.3 and 1000 is 51300.
To Contract the Operation, 80 as to retain only as many Decimals
in the Product as may be thought Necessary, when the Product would naturally contain several more Places.
Set the units' place of the multiplier under that figure of the multiplicand whose place is the same as is to be retained for the last in the product; and dispose of the rest of the figures in the inverted or contrary order to what they are usually placed in. - Then, in multiplying, reject all the figures that are more to the right-hand than each multiplying figure, and set down the products, so that their right-hand figures
may fall in a column straight below each other ; but observing to increase the first figure of every line with what would arise from the figures omiited, in this manner, namely I frum 5 to 14. 2 from 15 to 24, 3 from 25 10 34. &c; and the sum of all the lines will be ine product as required, cummoniy to . the nearest unit in the last figure.
1. To multiply 27.14986 by 92 41035, so as to retain only four places of decimals in the product. Contracted Way.
Common Way. 27:14986
27; 4986 530.4.29
2 Multiply 480:14936 by 2 72416, retaining only tour decimals in the product.
3. Multiply 2490-3048 by •573286, retaining only five decimals in the product.
4. Multiply 325 70 428 by 7218393, retaining only three decimals in the product.
DIVISION OF DECIMALS.
Divide as in whole numbers ; and point off in the quotient as many places for decimals, as i he decimal places in the divi. dend exceed those in the divisor*.
• The reason of this Rule is evident ; for, since the divisor multiplied by the quotient gives the dividi-nd, therefore the number of decimal places in the dividend, is equal to those in ihe divisor and quotient, taken together, by the nature of Multiplication ; and consequently the quotient itself must contain as many as the dividend exceeds the diVisor
Another way to know the place for the decimal point, is this : The first figure of the quotient must be made to occupy the same place, of integers or decimals, as doth that figure of the dividend which stands over the unit's figure of the first product.
When the places of the quotient are not so many as the Rule requires, the defect is to be supplied by prefixing ciphers.
When there happens to be a remainder after the division, or when the decimal places in the divisor are more than those in the dividend ; then ciphers may be annexed to the diyidend, and the quouent carried on as far as required.
When the divisor is an integer, with any number of ciphers annexed: cut off those ciphers, and remove the deci. mal poinı in the dividend as many places farther to the left as there are ciphers cut off, prefixing ciphers if need be; then proceed as before.
This is no more than dividing both divisor and dividend by the same number, either 10, or 100, or 1000, &c. according to the number of ciphers cnt off, which, leaving them in the same proportion, does not affect the quotient. And, in the same way, the decimal point may be moved the same number of places in both the divisor and dividende either to the right or left, whether they have ciphers or not