EXAMPLE.-Multiply 27.14986 by 92.41035, and retain only five places of deci Ans. 2508.92806.

mals.

DIVISION OF DECIMALS.

RULE.-Divide as in whole numbers, and point off in the quotient as many places for decimals as the decimal places in the dividend exceed those in the divisor; but if there are not so many places, supply the deficiency by prefixing ciphers. EXAMPLE.-Divide 53.00 by 6.75.

6.75) 53.00 (7.851+

Here 3 ciphers were annexed to carry out the division.

BY CONTRACTION.

RULE.-Take only as many figures of the divisor as will be equal to the number of figures, both integers and decimals, to be in the quotient, and find how many times they may be contained in the first figures of the dividend, as usual.

Let each remainder be a new dividend; and for every such dividend leave out one figure more on the right-hand side of the divisor, carrying for the figures cut off as in Contraction of Multiplication.

NOTE. When there are not so many figures in the divisor as are required to be in the quotient, continue the first operation till the number of figures in the divisor be equal to those remaining to be found in the quotient, after which begin the contraction. EXAMPLE.-Divide 2508.92806 by 92.41035, so as to have only four places of deci. mals in the quotient.

92.410315) 2508.928/06 (27.1498

1848 207 +1

660 721

646 872+2

13 849

9 241

4 608

3 696

912

832+-4

80

74+2

6

EXAMPLE.-Divide 4109.2351 by 230.409, retaining only four decimals in the quo

tient.

REDUCTION OF DECIMALS.

Ans. 17.8345.

To reduce a Vulgar Fraction to its equivalent Decimal. RULE.-Divide the numerator by the denominator, annexing ciphers to the numerator as far as necessary.

To find the Value of a Decimal in Terms of an Inferior Denomi

nation.

RULE.-Multiply the decimal by the number of parts in the next lower denomination, and cut off as many places for a remainder, to the right hand, as there are places in the given decimal.

Multiply that remainder by the parts in the next lower denomination, again cutting off for a remainder, and so on through all the parts of the integer.

EXAMPLE.-What is the value of .875 dollars?

EXAMPLE.-What is the content of .140 cubic feet in inches?

.140

1728 cubic inches in a cubic foot.

241.920

EXAMPLE.-What is the value of .00129 of a foot?

Ans. 241.920 cubic inches. 1000

EXAMPLE.-What is the value of 1.075 tons in pounds?

Ans. .01548 inches.

Ans. 2408.

To reduce Decimals to equivalent Decimals of higher Denomina

tions.

RULE.-Divide by the number of parts in the next higher denomination, continuing the operation as far as required.

EXAMPLE.-Reduce 1 inch to the decimal of a foot.

12 1.00000

.08333, &c., Ans.

EXAMPLE.-Reduce 14 minutes to the decimal of a day.

EXAMPLE.-Reduce 14" 12"" to the decimal of a minute.

14" 12"

60

601852."
m

60 14.2"

.23666', &c., Ans.

NOTE. When there are several numbers, to be reduced all to the decimal of the highest.

Reduce them all to the lowest denomination, and proceed as for one denomi nation.

EXAMPLE.-Reduce 5 feet 10 inches and 3 barleycorns to the decimal of a yard

RULE OF THREE IN DECIMALS.

RULE.-Prepare the terms by reducing the vulgar fractions to decimals, compound numbers to decimals of the highest denomination, the first and third terms to the same name; then proceed as in whole numbers. See Rule, page 31. EXAMPLE.-If a ton of iron cost of a dollar, what will .625 of a ton cost?

DUODECIMALS.

In Duodecimals, or Cross Multiplication, the dimensions are taken in feet, inch es, and twelfths of an inch.

RULE.-Set down the dimensions to be multiplied together, one under the other, so that feet may stand under feet, inches under inches, &c.

Multiply each term of the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each immediately under its corresponding term, carrying 1 for every 12, from one term to the other. In like manner, multiply all the multiplicand by the inches of the multiplier, and then by the twelfth parts, setting the result of each term one place farther to the right hand for every multiplier. The sum of the products is the answer.

EXAMPLE.-Multiply 1 foot 3 inches by 1 foot one inch.

PROOF.-1 foot 3 inches is 15 inches, and 1 foot 1 inch is 13 inches; and 15X13 195 square inches. Now the above product reads 1 foot 4 inches and 3 twelfths of an inch, and

EXAMPLE.-How many square inches are there in a board 35 feet 4 inches long and 12 feet 3 inches wide?

EXAMPLE.-Multiply 20 feet 6 inches by 40 feet 6 inches.

By duodecimals, Ans. 831 feet 11 inches 3 twelfths equal 831 square feet
and 135 square inches.
By decimals

Feet

40 feet 6 inches = 40.5
20 "64

=20.541666, &c.

831.937499
144

Table showing the value of Duodecimals in Square Feet, and

What number of square inches are there in a floor 100 feet broad and 25 feet 6 inches and 6 twelfths long?

Ans. 2566 feet 11 inches 3 twelfths equal 2566 feet 135 inches.

RULE OF THREE.

The RULE OF THREE teaches how to find a fourth proportional to three given numbers.

It is either DIRECT OF INVERSE.

It is Direct when more requires more, or less requires less. Thus, if 3 barrels of flour cost $18, what will 10 barrels cost? Or, if 300 lbs. of lead cost $25.50, what will 10 lbs. cost?

In both of these cases the Proportion is Direct, and the stating must be,

As 3:18 ::10:
300: 25.50::10:

Ans. 60.
Ans. .85.

It is Inverse when more requires less, or less requires more. Thus, if 6 men build a certain quantity of wall in 10 days, in how many days will 8 men build the like quantity? Or, if 3 men dig 100 feet of trench in 7 days, in how many days will 2 men perform the same work?

Here the Proportion is Inverse, and the stating must be,

As 8:10::6:
2: 7::3:

Ans. 7.
Ans. 10.

The fourth term is always found by multiplying the 2d and 3d terms together, and dividing the product by the 1st term.

Of the three given numbers necessary for the stating, two of them contain the supposition, and the third a demand.

RULE. STATE the question by setting down in a straight line the three necessary numbers in the following manner:

Let the 2d term be that number of supposition which is of the same denomination as that the answer, or 4th term, is to be, making the demanding number the 3d term, and the other number the 1st term when the question is in Direct Proportion, but contrariwise if in Inverse Proportion, that is, let the demanding number be the 1st term.

Then multiply the 2d and 3d terms together, and divide by the 1st, and the product will be the answer, or 4th term sought, of the same denomination as the 2d term.

NOTE.-If the first and third terms are of different denominations, reduce them to the same. If, after division, there be any remainder, reduce it to the next lower denomination, and divide by the same divisor as before, and the quotient will be of this last denomination.

Sometimes two or more statings are necessary, which may always be known by the nature of the question.

EXAMPLE 1.-If 20 tons of iron cost $225, what will 500 tons cost?

Tons. Dolls. Tons.

20: 225: 500
500

210) 11250,0

5625 dollars, Ans.

EXAMPLE 2.-If 15 men raise 100 tons of iron ore in 12 days, how many men will raise a like quantity in 5 days?

Days. Men. Days.
As 5:15 12

12 5) 180

36 men, Ans.

EXAMPLE 3.-A wall that is to be built to the height of 36 feet was raised 9 feet high by 16 men in 6 days: how many men could finish it in 4 days at the same rate working?

Days. Men. Days. Men.
4 166: 24 Ans.

Then, if 9 feet require 24 men, what will 27 feet require ?

9: 24: 27: 72 Ans.

EXAMPLE 4.-If the third of six be three, what will the fourth of twenty be?

Ans. 7.

COMPOUND PROPORTION.

COMPOUND PROPORTION is the rule by means of which such questions as would require two or more statings in simple proportion (Rule of Three) can be resolved in one.

As the rule, however, is but little used, and not easily acquired, it is deemed preferable to omit it here, and to show the operation by two or more statings. EXAMPLE.-How many men can dig a trench 135 feet long in 8 days, when 16 men can dig 54 feet in 6 days?

First

Second

Feet. Men. Feet. Men.

As 54 16:: 135: 40

Days Men. Days. Men.

As 8:40: 6: 30 Ans.

EXAMPLE.-If a man travel 130 miles in 3 days of twelve hours each, in how many days of 10 hours each would he require to travel 360 miles ?

EXAMPLE.-If 12 men in 15 days of 12 hours build a wall 30 feet long, 6 wide, and 3 deep, in how many days of 8 hours will 60 men build a wall 300 feet long, 8 wide, and 6 deep? Ans. 120 days.

INVOLUTION.

INVOLUTION is the multiplying any number into itself a certain number of times. The products obtained are called POWERS. The number is called the Root, or first power.

When a number is multiplied by itself once, the product is the square of that number; twice, the cube; three times, the biquadrate, &c. Thus, of the number 5. 5 is the Root, or 1st power.

66 5X5 25

The little figure denoting the power is called the INDEX or EXPONENT.

Square, or 2d power, and is expressed 52.
Cube, or 3d power, and is expressed 53.
Biquadrate, or 4th power, and is expressed 54.

EVOLUTION is finding the Roor of any number.

The sign✓placed before any number, indicates the square root of that number is required or shown.

The same character expresses any other root by placing the index above it.

Thus, 255, and 4+2=√36.
And, 3/27=3, and 3/64=

Roots which only approximate are called Surd Roots.

4.

TO EXTRACT THE SQUARE ROOT.

RULE.-Point off the given number from units' place into periods of two figures

each.

Find the greatest square in the left-hand period, and place its root in the quotient; subtract the square number from the left-hand period, and to the remainder bring down the next period for a dividend.

Double the root already found for a divisor; find how many times the divisor is contained in the dividend, exclusive of the right-hand figure, place the result in the quotient, and at the right hand of the divisor.