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Thus if A vary inversely as B, and A1, B1 be other corresponding values of A and B, A : A1 = : 1. This is expressed thus:- A ∞ Def. 12. One quantity is said to vary as two others jointly when, if the first be changed, the product of the others is changed in the same ratio. Def. 13. One quantity is said to vary directly as one quantity and inversely as another, when, if the first be changed, the product of the second and the reciprocal of the third is changed in the same ratio.

I. TO FIND THE RATIO OF ONE NUMBER TO ANOTHER.

Rule. Divide the 1st by the 2nd, the quotient, whether integral or fractional, will be the ratio required.

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II. TO SIMPLIFY THE TERMS OF A RATIO, When fractional.

Rule. Multiply both terms by the L. C. M. of the denominators of the fractions.

EXAMPLE.

Simplify the terms of the ratio 9: 10.

Here L. C. M. = 42.

3 5 129 215

..9. -:10-=—-:-- = 387: 430.

14 21 14 21

Ill. TO COMPOUND TWO OR MORE RATIOS.

Rule. Multiply the antecedents together for a new antecedent, and the consequents for a new consequent; and divide the terms by their G.C.M. EXAMPLE.

Find the ratio compounded of the ratios 8: 15; 12: 32.

Compound ratio = 8 X 12: 15 X 32

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Rule. Multiply the antecedent of the one by the consequent of the other, that ratio will be the greater, the product of whose antecedent with the other consequent, is the greater.

EXAMPLE.

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V.

TO DIVIDE A NUMBER INTO PARTS WHICH SHALL BEAR TO
EACH OTHER GIVEN RATIOS.

Rule. Divide the number by the sum of the numbers composing the given ratios, and multiply the quotient by each of the numbers. The products will be the parts required.

EXAMPLE.

Divide 286 into 3 parts in the ratio of 1, 3, 9.

1+3+9=13:

28613 22

22 X 366: 22 X 9198.

.. The parts required are 22, 66, 198.

VI. TO DETERMINE WHETHER FOUR NUMBERS BE PROPORTIONALS IN A GIVEN ORDER.

Rule. Convert the first two and last two numbers into ratios, and compare them. If they be equal, the numbers are proportional; if unequal, they are not.

Or-Multiply the two extremes together and the two means, if these products are equal, the numbers are proportional; if unequal, they are not.

EXAMPLE.

Determine whether the numbers 3, 9, 6, 12, be proportionals.

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VII. HAVING GIVEN THREE NUMBERS, TO FIND A FOURTH SUCH, THAT ALL SHALL BE PROPORTIONAL.

Rule. Multiply the 2nd and 3rd together, and divide the product by the 1st, the quotient is the number required.

EXAMPLES.

1. Find a fourth proportional to the numbers 8, 6, 16.

6 X 16 96

No. required=

=== 12.

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VIII. TO FIND A THIRD PROPORTIONAL TO TWO GIVEN NUMBERS.

Rule. Multiply the 2nd by itself, and divide by the 1st; the quotient is the number required.

EXAMPLES.

Find a third proportional to 25, 30: also to 33, 41.

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Def. 1. When a number is multiplied by itself, it is said to be involved. Def. 2. The product of a number multiplied by itself is called a "power" of the number.

Def. 3. If the number be numbered by itself once, the power is called the "square," or 2nd power; if twice, the power is called the "cube," or 3rd power; and the powers are called the 4th, 5th, &c. according as there are 4, 5, &c. factors, which enter into the composition of the product.

TO INVOLVE A NUMBER TO A GIVEN POWER.

Rule-1. To square a number, multiply it by itself.

2. To cube a number, multiply the square by the number.

3. If the given power be an odd one, raise the number to the next greater power than the half, and multiply this by the next less. For example, to raise a number to the 5th power, cube it, and multiply the cube by the square.

4. If the given power be an even one, raise the number to a power equal to half the given power, and square it. For example, to raise a number to the Sth power, raise it to the 4th power and square.

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Def. 1. The process of finding what number, raised to a given power, will produce a given number, is called Evolution.

Def. 2. The number which, raised to a given power, will produce a given number, is called the 'Root' of the given number, and is styled the square root, cube root, 4th, 5th, &c. root, according as it has to be raised to the 2nd, 3rd, 4th, 5th, &c. power to produce the number.

Def. 3. A root is represented by the sign V. Thus 19: of 9; 3/27 cube root of 27, &c.

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1. TO FIND THE SQUARE ROOT OF A GIVEN NUMBER.

Rule-1. Divide the number into periods of two figures each, by placing a dot over every alternate figure, beginning with the units' place.

2. Find the largest number, whose square does not exceed the first period. This is the 1st figure of the required root.

3. Subtract the square of this from the 1st period.

4. To the remainder annex the 2nd period; call this number the 1st dividend.

5. Double the 1st figure of the root, and write it to the left of the 1st dividend, placing a line between them: call this the 1st trial-divisor.

6. Divide the 1st dividend, with the last figure cut off, by the 1st trialdivisor.

7. Write the quotient as the 2nd figure of the root, and also to the right of the first trial-divisor: call this number the 1st divisor.

8. Multiply the 1st divisor by the 2nd figure of the root, and subtract the product from the 1st dividend.

9. To the remainder annex the 3rd period, forming the 2nd dividend.

10. To the last divisor add the last figure found in the root, and place it to the left of the last dividend, thus forming the 2nd trial-divisor.

11. Proceed in the same manner as directed in 6, 7, &c. and repeat the process, till all the periods have been used, when, if there is no remainder, the square root will have been found; but if there be one, there is no square root.

12. Whenever any quotient, obtained as directed, gives a subtrahend greater than the dividend, a less number must be taken.

13. This Rule applies to decimals as well as to whole numbers; only in decimals, the number of decimal places must be even, and the root must be made to have half as many as the number.

14. The square root of a fraction is found by extracting the square root of numerator and denominator; or by reducing the fraction to a decimal, and then extracting the root.

15. In finding the square root of a number to a certain number of decimal places, when more than half the figures have been found, the remaining figures may be found by dividing the dividend with the last figure cut off by the last trial-divisor.

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