163. Second Method. When numbers are large, or when they are not perfect cubes, the method to be used in extracting the cube root is developed from the following equation:

(t + u)3 = t3 + 3 t2u + 3 tu2 + u3 = t3 + (3t2 + 3 tu + u2) u.

RULE.—To find the cube root of a number:

STEPS:

1. Beginning with the units, point off the number into periods of three figures each (principle 2, 148) to find the number of figures in the root.

2. Find the greatest cube in the left-hand period and write its root for the first figure in the

root.

3. Subtract the cube from the period, and to the remainder annex the next period for a dividend. 4. For a trial divisor, write three times the square of the root found, considered as tens.

5. Divide, and write the quotient for the trial figure in the root.

6. For the complete divisor, add to the trial divisor 3 times the product of the last trial figure by the preceding part of the root considered as tens, and to this sum add the square of the trial term.

7. Multiply the complete divisor by the trial term in the root, subtract the product from the dividend, and to the remainder annex the next period for a new dividend.

8. Repeat, beginning with the 4th step.

NOTE. The learner should study the examples above, together with the "steps." It would be well, also, to show that the following formula is an algebraic expression of the "steps":

(t + u)3 = t3 +3t3u + 3tu2 + u3 = t3 + (3ť2 + 3tu + u3)u.

164. RULE.-To find the cube root of a fraction: FIRST METHOD: Find the cube root of each term of the fraction.

SECOND METHOD: Reduce the fraction to a decimal, and proceed as in whole numbers.

10. A room is twice as long as high, and the width is equal to the height: what is the height if the capacity of the room is 3456 cubic feet?

11. A cubical box contains 474552 cubic inches. What is the area of one of its sides?

12. What is the entire surface of a cube whose volume is 91125 cu. ft.?

13. What is one dimension of a cubical bin that will hold

the same number of bushels of wheat as a bin that is 924 inches long, 56 inches wide, and 6 feet deep?

14. A rectangular reservoir with square base is 7 times as high as wide, and has a capacity of 1089 gallons. Find its dimensions.

15. A cubic inch of iron weighs 4.5 oz. A bar of iron, twice as wide as thick and three times as long as wide, weighs 27 pounds. Find its dimensions.

16. What will it cost to plaster the bottom and sides of a cubical reservoir which will hold 53361 gallons of water, at 44 a square foot?

CHAPTER VI

RATIO AND PROPORTION

I. RATIO

165. Ratio is the relation of one number to another of the same kind. It is expressed by the quotient of the first divided by the second.

Thus, the ratio of 3 to 4 is 4.

166. The sign of a ratio is the colon (:), which is the sign of division with the horizontal line omitted.

Thus, 87 signifies the ratio of 8 to 7.

167. The terms of a ratio are the two numbers compared, and are called antecedent and consequent. The two terms form a couplet.

Thus, 35 is a couplet.

168. The antecedent (Latin antecedere, to go before) is the first term, or dividend.

169. The consequent (Latin consequi, to follow) is the second term, or divisor.

170. The value of a ratio is found by dividing the antecedent by the consequent, and it is always an abstract number.

171. A simple ratio is a ratio consisting of two terms; as, 6:58.

172. A compound ratio is a ratio whose terms are the