strong.

To reduce Fractions to Decimals.

Rule. - 1. Annex ciphers to the numerator, and divide by the denominator.

2. Point off in the quotient as many, decimal places as there-have been ciphers annexed.

Example. - Reduce 1/8 to a decimal.

Ans. .125 (125 thousandths).. 8)1.000

.125

Simple Proportion, or Rule of Three.

Simple Proportion is an equality between two simple ratios.

Ratio is the relation, in respect to magnitude or value, which one quantity or number has to another of the same kind; or the quotient arising from the division of one number by another: thus, the ratio of 8 to 4 is 2, since 8 is 2 times 4; the ratio of 4 to 8 is 1/2, since 4 is 1/2 of 8.

Rule. - Make that number the third term which is of the same kind as the answer; and if, from the nature of the question, the third term must be greater than the fourth term, or answer, make the greater of the two remaining terms the first term, and the smaller, the second; but, if the third term must be less than the fourth, make the less of the two remaining terms the first, and the greater, the second; then multiply the second and third terms together, and divide their product by the first term: the quotient will be the fourth term, or answer.

Examples. - If a man receives $15 for a week's work, how much shall he have for 7 days' work?

da. da. $ $ 6 : 7 :: 15 : ( ) 6)105 $17 3/6 = $ 17.50. Ans.

If 5 men can build a house in 45 days, how long will it take 8 men?

no. m. da. da.

8 : 5 :: 45 : ( ) 5

-----------

8)225

28 1/8 days. Ans.

Compound Proportion.

Compound Proportion is an expression of equality between a compound and a simple ratio.

Rule. - Make that number the third term which is of the same kind as the answer; of the remaining numbers, take any two that are of the same kind, and consider whether an answer depending upon these alone would be greater or less than the third term, and place them as directed in simple proportion.

Then take any other two of the same kind, and consider whether an answer depending only upon them would be greater or less than the third term, and arrange them accordingly; and so on until all are used. Multiply the product of the second terms by the third term, and divide the result by the product of the first terms: the quotient will be the fourth term, or answer.

Example. - If 6 men can build an 8-inch brick wall, 95 feet long and 15 feet high, in 3 days, how long will it take 5 men to build a 12-inch wall, 40 feet long and 9 feet high, the days being 10 hours long in both cases?

5 x 8 x 95 x 15 = 57000

6x12 x40x9x3= 77760 ÷ 57000 = 1.36 + days = 1 day 3 hours 36 + minutes.

Example. - I paid $35 for the labor of 2 men for 6 days, they working 12 hours daily. How much ought I to pay 4 men for 7 days' work, 10 hours being reckoned a day's work, and paying at the same rate per hour as I paid the first men?

2 x 6 x 12 = 144

4 x 7 x 10 x 35 == 9800 ÷ 144 = 68.05. Ans. - $68.05.

Square Root.

The Square Root of any number is that number which, multiplied by itself, will produce the given number.

Rule for extracting the Square Root. - 1. Point off the given number into periods of two figures each; counting from units' place toward the left in whole numbers, and toward the right in decimals.

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

3. At the left of the dividend write twice the first figure of the root, and annex one cipher for a trial divisor; divide the dividend by the trial divisor, and write the quotient for a trial figure in the root.

4. Add the trial figure of the root to the trial divisor for a complete divisor; multiply the complete divisor by the trial figure in the root, and subtract the product from the dividend; and to the remainder bring down the next period for a new dividend.

5. To the last complete divisor add the last figure of the root, and to the .sum annex one cipher for a new trial divisor, with which proceed as before.

Note 1. - If at any time the product be greater than the dividend, diminish the trial figure of the root, and correct the erroneous work.

Note 2. - The left-hand period may contain but one figure.

Note 3. - If the dividend does not contain the divisor, a cipher must be placed in the root, ami also at the right of the divisor; then, after bringing down the next period, this last divisor must be used as the divisor of the new dividend.

Note 4- - When there is a remainder after extracting the root of a number, periods of ciphers may be annexed; and the figures of the root thus obtained will be decimals.

Note 5. - The square root of a fraction may be obtained by extracting the square roots of the numerator and denominator separately, providing the terms are perfect squares; otherwise the fractions must first be reduced to decimals.

Examples. - What is the square root of 406457.2516?

Operation

What is the square root of 2?

Application of Square Root.

A Triangle is a figure having three sides and three angles oi corners.

A Right-angled Triangle is a figure having three sides and three angles, one of which is a right angle.

In every right-angled, triangle, the square of the hypothenuse is equal to the sum of the squares of the base and perpendicular.