15. A, B, C, and D put $5700 in trade. A's money was in 8 months, and his gain was $160; B's was in 5 months, and his gain was $200; C's was in 2 months, and his gain was $18); D's was in 6 months, and his gain was $240. What stock did each have in? Ans. A, $600; B, $1200; C, $2700; D, $1200 For Dictation Exercises, see Key.

378. QUESTIONS FOR REVIEW.

RATIO. What is ratio? what is arithmetical ratio? geometrical ratio? What is the first term of a ratio called? the second term? both terms when taken together? What is a ratio of equality? of greater inequality? of less inequality? Give an example. In what respect do ratios resemble fractions? How, then, may ratios, at any time, be written? How do you multiply a ratio? how divide a ratio? Suppose you multiply or divide both terms by the same number? What is a simple ratio? a complex? a compound ratio? How do you rcduce a complex ratio to a simple one? a compound ratio? Write a simple ratio; a complex ratio; a compound ratio.

PROPORTION. - What is proportion? Explain the proportion 2:4 7:14. What are the 1st and 4th terms called? the 2d and 3d ? the 1st and 3d? the 2d and 4th? the 1st and 2d? What is inverse proportion? compound proportion? What is a mean proportiona) between two numbers? Upon what important principle does the solving of examples by proportion depend? Write a proportion, and illustrate that principle. How can you find an extreme, when the other three terms are given? how a mean? how a mean proportional between two given numbers? Give the rule for solving an example by simple proportion, and illustrate it by an example of your own. Perform the same example by analysis. What is meant by analysis? Give your rule for solving an example by compound proportion, and illustrate it. In solving any example by proportion, the two terms of a ratio must be of the same kind; why?

Who are the partners?

PARTNERSHIP. What is partnership? How are profits and losses usually shared? What is simple partnership? (Ans. It is partnership where persons enter into business for the same time.) How do you find each person's share of gain or loss in simple partnership? What is compound partnership? How do you find the shares of gain or loss in compound partnership? Why is partnership sometimes called partitive proportion?

INVOLUTION.

379. involution consists in raising a number to a required power. (Art. 89.)

380. The required power is indicated by a small figure called the index or exponent, placed at the right, and a little above the number. (Art. 90.)

381. The first power of a number is the number itself. The second power or square of a number is obtained by using the number as a factor twice. The third power or the cube results from using the number as a factor three times, and so on.

-

NOTE. The most important applications of Involution are in the use of the second and third powers.

382. Any power may be obtained by the following RULE. Employ the given number as a factor as many as there are units in the exponent of the required power.

EXAMPLES.

timcs

1. Find the squares of the integers from 1 to 25 inclusive, and commit them to memory.

*

Numbers,

1,

2,

3,

4,

5,

Squares, 1, 4,

9,

Ans.

8,

16,

6, 7, 16, 25, 36, 49, 64, Numbers, 9, 10, 11, 12, 13, 14, 15, Squares, 81, 100, 121, 144, 169, 196, 225, 256, 19, 20, 21, 22, 23, 24, Squares, 289, 324, 361, 400, 441, 484, 529, 576, 625.

Numbers, 17, 18,

25.

2. Find the cubes of the integers from 1 to 10 inclusive, and ommit them to memory.*

Ans. {

Numbers, 1, 2, 3,

Cubes, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

At the option of the teacher.

23. What is the difference between the square and the cube of 24.

24. What is the compound interest of $1.10 for 4 years, at 10 per cent.?

25. How many paving stones 13 inches square will be required to pave 100 rods of a street 3 rods in width?

26. How many dice measuring an inch each way may be made from a cubical foot of ivory, allowing for waste in the manufacture?

EVOLUTION.

383. Evolution consists in finding the roots of numbers. 384. The root of a number is one of the equal factors which produce that number.

The square root is one of the two equal factors; the cube root, one of the three equal factors; the fourth root, one of the four equal factors, and so on.

385. √ is the radical sign, and by itself signifies the square root, and with a figure above it, signifies the degree of the root indicated by the figure; thus, 27 signifies the third root of 27. The root may also be indicated by a fractional exponent; thus, 16 (read, 16 to the power) √16 = 2.

=

SQUARE ROOT.

386. Table, showing the places occupied by the square of any number of units, tens or hundreds.

387. By the above we perceive that the square root of any whole number expressed by one or two figures, must be units; expressed by three or four figures, must be units and tens; expressed by five or six figures, must be units, tens, and hundreds. Hence, generally, if a number be separated into periods of two figures each, beginning with the units, the number of figures in the square root will be indicated by the number of periods.

NOTE I.. The left hand period may contain but one figure. NOTE II. The principle above elucidated applies also to decimal fractions; but every period in decimal fractions must contain two figures.

388. That the pupil may comprehend the method of extracting the square root of a number, we will multiply 64 by itself, i. e., square it, and keep the separate products, instead of reducing them and adding as in ordinary multiplication.

64 X 64 (604) X (60+ 4).

×

6023600.

= 2 × (60 × 4) = 480.

=4096.

42 = 16.

By the above it will appear that a square whose root is com

posed of tens and units, contains

(1.) The square of the tens ;

(2.) Twice the tens multiplied by the units; and (3.) The square of the units.

(

389. We will now extract the square root of 4098.

will consist of two figures,

and the square of the tens must be contained in the 40 (hundreds); the largest square contained in 40 (hundreds) is 36 (hundreds), the root of which is 6 (tens); this we write as the tens' figure of the root, and subtract ita

square 36 (hundreds) from the 40 (hundreds), and to the remainder 4 (hundreds), bring down the next period, 96.

This remainder (496) must contain two times the tens of the root multiplied by the units, plus the square of the units, or the product of two times the tens, plus the units, multiplied by the units. If it contained only two times the tens multiplied by the units, we should obtain the units' figure by dividing the remainder (496) by two times the tens. We make 2 X 6 (tens) = =12 (tens) the trial divisor, which is contained in 49 (tens) 4 times. We write 4 as the units' figure of the root, and also at the right of the 12 (tens), and have 124 for the true divisor. This we multiply by 4, and thus complete the square, obtaining at once, twice the product of the tens by the units, and the square of the units. If the root consists of more than two figures, having obtained the first two, we can consider them as tens in reference to the next figure, and proceed with them in all respects as above. Thus, suppose it be

OPERATION.

4138.94 (64.33+

36

124) 538

496

1283) 4294
3849

12863) 44500
38589

5911

required to extract the square root of 4138.94: find the first two places as before; bring down the next period, 94, and form a new trial divisor by doubling 64 (the root already found); find how many times this, considered as tens, is contained in 429 tens, for the third