ADDITION OF DECIMALS.

Rule. Arrange the numbers under each other, according to their several values, find the sum as in addition of whole numbers, and cut off for decimal as many figures to the right hand as there are decimals in any one of the given numbers.

23.450 7.849

8.6234

253.004

Examples.

What is the sum of 23.45, 7.849, 543.2, 8.6234, 253.004. If any of the decimals be repetends, continue them 543.200 beyond the others, and make them end together; then in adding, increase the sum of the first column 836.1264 by as many units as there are nines therein, as— .75 Here the first sum contains 18, two nines; there.6666 fore 2 added to 18 = 20; the rest of the work is the same as usual in others.

.8888

.875
.4444

3.6250

If some of the decimals be repetends, and others circulates, continue them both beyond those that are finite, and until their periods end together; then to the sum of the first column add as many as would arise to carry on the sum, thus:

2.5

3.666666

The repetend of .6, the circulate of .69, and .372, continued until their periods end together. It may 7.696969 14.372372 easily be observed that there would be 1 to carry to 28.236008 the first column if it were carried any farther.

Note. It is always necessary to attend to the rules for repetends and circulates; three or four decimal figures, according to the rule, being sufficiently near the truth for common calculations.

SUBTRACTION OF DECIMALS.

Rule. Place the numbers according to their value, subtract as in whole numbers, and cut off for decimals as in addition.

132.005

35.87043

96.13457

Example.

Subtract 35.87043 from 132.005.

If both be single repetends, make them end together and if there be occasion to borrow at the first figure, borrow 9 only, instead of 10, thus:

.8333 .6666

If both be circulates, or one a repetend and the other a circulate, continue both until their periods .1666 end together and if there should be occasion to borrow at the following figure, where they continued that figure farther, carry 1 to the first figure, and if the numbers be in different denominations, reduce them until they are alike. Subtract from 13.

៖ ៖

200000 ÷ 3 = 1.666666

834000 999 = .834834

.831831

MULTIPLICATION OF DECIMALS.

Rule. Place the factors under each other, and multiply them together as in whole numbers: then point off as many figures from the product (counting from right to left) as there are decimal places in both factors: observing if there be not enough, to annex as many cyphers to the left hand of the product as will supply the deficiency.

.2715
.253

8145

Example.

Multiply .2715 by .253.

In this example the product is .686895: the number of decimals in the multiplicand are 4, and the 13575 multiplier 3, equal to 7, therefore a cypher must be placed on the left hand of the product to reduce it to .0686895 its proper terms.

5430

To multiply a repetend by a single figure, add 1 to the first product for every 9 therein, so will you have a repetend in the product; if there be several figures in the multiple, do so with each product, and continue them until they end together; then add them as so many repetends.

If the multiplicand be a circulate, consider the increase that would arise to the first product if the multiplicand was continued farther: thus do with each product, make them end together, and add them by the rule for adding circulates.

When any number of decimals are to be multiplied by 10, 100, 1000, &c., remove the separating point in the multiplicand so many places towards the right hand as there are cyphers in the multiplier; thus: .578 x 10 5.78, .578 x 100 = 57.8, .578 × 1000 = 578.

=

DIVISION OF DECIMALS.

This rule is also worked as in whole numbers; the only difficulty is in valuing the quotient, which is done by any of the following rules:

Rule 1. The first figure of the quotient is always of the same value with that figure of the dividend which answers or stands over the place of units in the divisor.

2. The quotient must always have as many decimal places as the dividend has more than the divisor.

Note 1.-If the divisor and dividend have both the same number of decimal parts, the quotient will be a whole number.

2.-If the dividend has not so many places of decimals as are in the divisor, then so many cyphers must be annexed to the dividend as will make them equal; the quotient will then be a whole number.

3. But if when the division is done, and the quotient has not so many figures as it should have places of decimals, then so many cyphers must be prefixed as there are places wanting.

Divide 173.5425 by 3.75. 173.54253.75 = 46.278

Divide .14856 by 2.476. .148562.476 = .6000

Examples.

Divide 5.714 by 8275.

5.7148275 = .00069

In this example the quotient was deficient in the number of decimals required, and are placed on the left hand.

When numbers are to be divided by 10, 100, 1000, &c., it is performed by placing the separating point in the dividend so many places towards the left hand as there are cyphers in the divisor, thus:

By reduction we change vulgar fractions, and the lesser parts

of coin, weights and measures, &c., into decimals, and find the value of any decimal given.

Because decimals increase their value towards the left hand, and decrease their value towards the right hand, in the same tenfold proportion with integers, or whole numbers, they may be annexed to the whole numbers and worked in all respects as whole numbers; and if simple arithmetic be well understood, there is little more to be learned than the placing of the separating points, the rule for which ought to be well attended to.

To reduce a vulgar fraction to a decimal of an equal value. Rule. Add a cypher or cyphers to the numerator, and divide by the denominator, the quotient will be the decimal required.

Reduce to a decimal:

Example.

7.0000 16.4375

Thus you may take any number of cyphers at pleasure, but their number will be best ascertained when the work is finished, then you must have as many decimal figures as you have taken annexed cyphers in dividing, and if there are not so many in the quotient, you may prefix cyphers to the left hand of the product, thus: = .03125.

3

1.00000

32

Sometimes the quotient figures will repeat continually, as thus 2000 .666, then it is called the repetend, and the last figure may be dashed or marked, to distinguish it from a terminate decimal.

12.000

33

Sometimes two, three, or more figures will repeat, as thus

=

= .3636, such are called compound repetends, or circulates, and the first and last figure may be dashed or marked.

To reduce the lesser parts of coin, weights and measures, &c., to a decimal.

Rule. Divide the least name by such number as will reduce it to the next greater; to the decimal so obtained prefix the given number of the same name, then divide by such number as will reduce it to the next greater; always annex cyphers to

C

the dividend as occasion may require, then proceed till it be reduced to the decimal of the required integer; or reduce the given parts to a single quantity, by reducing them to the lowest name mentioned; annex cyphers thereto, and divide by such numbers as will reduce them to the name required; or reduce the given parts to a vulgar fraction, and that fraction to a decimal.

Example 1.

Reduce 17s. 104d. to the decimal of a pound sterling:

Reduce 2 ft. 9 in. to the decimal of a yard :

To find the value of any given decimal.

Rule. Multiply the decimal given by the number of parts of the next inferior denomination, cutting off the decimals from the product; then multiply the remainder by the next inferior denomination; thus proceeding till you have brought the least known parts of the integer.

Example 1.

Required the value of .89375 of a pound sterling:

.89375
20

17.87500

12

10.50000

2.0000017s. 10 d.

Example 2.

Reduce .625 of a hundred-weight to its proper term: .625 × 4 = 2.500 × 28 = 14.000; or 2 quarters and 14 lbs.