EXAMPLE. Reduce 11 hours, 59 minutes, 60 seconds, to the frac

- Reduce 15 h., 25 m., 429 sec., to the fraction of a

7X 60 X 60 X 24 604800

To work fractions, or whole numbers and fractions, by the Rule of Three, or Proportion.

RULE. Reduce the mixed terms to simple fractions, state the question as in whole numbers, invert the divisor, and multiply and divide as in whole numbers.

If 2 yards of cassimere cost $44, what will of a ;47; then,

EXAMPLE. vard cost? 2 :::

=

4722102 $1.27,5.

$1.27,5. Ans.

DECIMAL FRACTIONS.

A decimal fraction is written with its numerator only. Its denominator is understood. It occupies one or more places of figures, and has a point or dot (.) prefixed or placed before it. The dot (.) alone distinguishes it from an integer or whole number. It supposes a denominator whose value is a UNIT broken into parts, having a tenfold relation to the number of places the numerator occupies. The denominator, therefore, of any decimal is always a unit (1) with as many ciphers annexed as the numerator has places of figures. Thus, the denominator of .1, .2, .3, &c., is 10, and the fractions are read, one tenth, two tenths, three tenths, &c. The denominator of .01, .11, .12, &c., is 100, and these are read, one hundredth, eleven hundredths,

twelve hundredths, &c. The denominator of .001, .101, .125, &c., is 1000, and these are read one thousandth, one hundred and one thousandths, one hundred and twenty-five thousandths, &c. The denominator of a decimal occupying four places of figures as .7525 is 10000, and so on continually.

The first figure on the right of the decimal point is in the place of tenths, the second in the place of tenths of tenths, or hundredths, the third in the place of tenths of tenths of tenths, or thousandths, &c. Thus the value of a decimal occupying four places of figures, as

+

1

1 100

A decimal is converted into a vulgar fraction of

equal value, by affixing its denominator.

Ciphers placed on the right of decimals do not change their value. Thus, .1850.185, plainly for the reason that the denominator of the latter bears the same relation to that of the former that 185 bears to 1850; from both terms of the fraction a ten fold has been dropped. Ciphers placed on the left of decimals decrease their value ten fold for every cipher so placed. Thus, .1 = To, •01 = Toʊ, .001 = Tooo, &c.

A mixed number is a whole number and a decimal. Thus, 4.25 is a mixed number. Its value is 4 units, or ones, and 25 of 1, = 425 = 44. The number on the left of the separatrix is always a whole number - that on its right, always a decimal.

ADDITION OF DECIMALS.

RULE. Set the numbers directly under each other according to their values, whole numbers under whole numbers, and decimals under decimals; add as in whole numbers, and point off as many places for decimals in the sum as there are figures in that decimal occupying the greatest number of places.

Add together .125, .34,.1, .8672. Also, 125, 34.11,

.1

125.
34.11
.235

.8672

1.4322

1.4322 Ans.

160.7772 Ans.

SUBTRACTION OF DECIMALS.

RULE.

Set the numbers, the less under the greater, and in other respects as directed for addition; subtract as in whole numbers, and

point off as many places for decimals in the remainder as the decimal having the greatest number of figures occupies places.

EXAMPLES. Subtract .2653 from .8.

.8
.2653

Also, 11.5 from 238.134.

238.134
11.5

.5347 Ans.

226.634 Ans.

MULTIPLICATION OF DECIMALS.

RULE. - Multiply as in whole numbers, and point off as many places for decimals in the product as there are decimal places in the multiplicand and multiplier both. If the product has not so many places, prefix ciphers to supply the deficiency.

EXAMPLES. Multiply 14.125 by 3.4. Also, 5.14 by .007.

NOTE. - Multiplying by a decimal is equivalent to dividing by a whole number that bears the same relation to a UNIT that a unit bears to a decimal. Multiplying by a decimal, therefore, is equivalent to dividing by the denominator of a fraction of equal value whose numerator is 1, or of dividing by the denominator of a fraction of equal value whose numerator is more than 1, and multiplying the quotient by the numerator. Thus, the decimal .25= 25 , and the decimal .875= And 14.23.25 = 3.5575, and 14.234-3.5575. So, also, 14.23.875 12.45125, and 14.23÷8= 1.778757=12.45125. It is sometimes a saving of labor and matter of convenience to achieve multiplication by this process.

100

875 1000

DIVISION OF DECIMALS.

RULE. Write the numbers as for division of whole numbers, then remove the separatrix in the dividend as many places of figures to the right, (supplying the places with ciphers if they are not occupied,) as there are decimal figures in the divisor; consider the divisor a whole number and divide as in division of whole numbers.

EXAMPLES.

Divide 16.5 by 1.232. Also, 1.2145 by 12.231.

1.232,)16.500, (13.3928+. Ans. 12.231,) 1.214.50 (-09999 +}

1232

79

Ans.

NOTE. - Dividing by a decimal is equivalent to multiplying by a whole number that bears the same proportion to a UNIT that a unit bears to the decimal. Dividing by a deci mal, therefore, is equivalent to multiplying by the denominator of a fraction of equal value whose numerator is 1, or multiplying by the denominator of a fraction of equal value whose numerator is more than 1, and dividing the product by the numerator. Dividing by a fraction is equivalent to multiplying by its denominator and dividing the product by its numerator, or dividing by its numerator and multiplying the quotient by its denominator. Thus, .5= TO 1 , and .7500: And 12.24.5=24.48, and 12.24224.48. So, also, 12.24 — .75=16. 32, and 12.24448.963= 16.32. This method of accomplishing division may often be resorted to with convenience.

REDUCTION OF DECIMALS.

To reduce a decimal in a higher to whole numbers in successive lower denominations.

RULE.-Multiply the decimal by that number in the next lower denomination that equals ONE of the denomination of the decimal, and point off as many places for a remainder as the decimal so multiplied has places. Multiply the remainder by the number in the next lower denomination that equals 1 of the denomination of the remainder, and point off as before; so continue, until the reduction is carried to the Ĵowest denomination required.

EXAMPLE. - What is the value of .62525 of a dollar?

EXAMPLE. -What is the value of .46325 of a barrel?

To reduce decimals, or whole numbers and decimals, in lower denominations, to their value in a higher denomination.

RULE. Reduce all the given denominations to their value in the lowest denomination, then divide their sum by the number required of the lowest denomination to make ONE of the denomination to which the whole is to be reduced.

EXAMPLE.

mal of a barrel.

14 X 4

Reduce 14 gallons, 3 quarts, 2.368 gills, to the deci

56359 X 8 472 + 2.368474.368.

8 X 4 X 32:

=

1024) 474.368 (.46325.

Ans.

To work decimals, or whole numbers and decimals, by the Rule of Three, or Proportion.

RULE. State the question and work it as in whole numbers, taking care to point off as many places for decimals in the product to be used as the dividend, as there are decimals in the two terms which form it, and to remove the decimal point therein as many places to the right as there are decimals in the term to be used as a divisor, before the division is had.

EXAMPLE. If .75 of a pound of copper is worth .31 of a dollar how much is 3.75 lbs. worth?