5. How many acres of land worth 35 dollars an acre must be added to a farm of 75 acres, worth $50 an acre, that the average value may be $40 an acre? Ans. 150 acres.

6. A merchant mixed 80 pounds of sugar worth 61 cents per pound with some worth 84 cents and 10 cents per pound, so that the mixture was worth 7 cents per pound; how much of each kind did he use?

CASE III.

418. When the quantity of the whole compound is limited.

1. A grocer has sugars worth 6 cents, 7 cents, 12 cents, and 13 cents per pound. He wishes to make a mixture of 120 pounds worth 10 cents a pound; how many pounds of each kind must he use?

the mixture would be but 12 lbs. while the required mixture is 120, or 10 times 12. If the whole mixture is to be 10 times as much as the sum of the proportional quantities, then the quantity of each simple used must be 10 times as much as its respective proportional, which would require 30 lbs. at 6 cts., 20 lbs. at 7 cts., 30 lbs. at 12 cts., and 40 lbs. at 13 cts.

RULE. Find the proportional numbers as in Case I. Divide the given quantity by the sum of the proportional quantities, and multiply each of the proportional quantities by the quotient thus obtained.

EXAMPLES FOR PRACTICE.

2. A farmer sold 170 sheep at an average price of 14 shillings a head; for some he received 9s., for some 12s., for some 18s., and for others 20s.; how many of each did he ? Ans. 60 at 9s., 40 at 12s., 20 at 18s., and 50 at 20s.

3. A jeweler melted together gold 16, 18, 21, and 24 carats fine, so as to make a compound of 51 ounces 22 carats fine; how much of each sort did he take? Ans. 6 ounces each of the first three, and 33 ounces of the last.

4. A man bought 210 bushels of oats, corn, and wheat, and paid for the whole $178.50; for the oats he paid $1, for the corn $, and for the wheat $14 per bushel; how many bushels of each kind did he buy? Ans. 78 bushels each of oats and corn, and 54 bushels of wheat.

5. A, B, and C are under á joint contract to furnish 6000 bushels of corn, at 48 cts. a bushel; A's corn is worth 45 cts., B's 51 cts., and C's 54 cts.; how many bushels must each put into the mixture that the contract may be fulfilled?

6. One man and 3 boys received $84 for 56 days' labor; the man received $3 per day, and the boys $1, $4, and $14 respectively; how many days did each labor? Ans. The man 16 days, and the boys 24, 4, and 12 days respectively.

419. A Power is the product arising from multiplying a number by itself, or repeating it several times as a factor; thus, in 2 x 2 x 28, the product, 8, is a power of 2.

420. The Exponent of a power is the number denoting how many times the factor is repeated to produce the power, and is written above and a little to the right of the factor; thus, 2 x 2 x 2 is written 23, in which 3 is the exponent. Exponents likewise give names to the powers, as will be seen in the following illustrations:

3

3 x 3

3 x 3 x 3

=31= 3, the first power of 3; =329, the second power of 3; = 3327, the third power of 3. 421. The Square of a number is its second power. 422. The Cube of a number is its third power.

423. Involution is the process of raising a number t a given power.

R.P.

14

424. A Perfect Power is a number that can be exactly produced by the involution of some number as a root; thus, 25 and 32 are perfect powers, since 25 = 5 x 5, and 32 = 2 x 2 x 2 x 2 x 2.

1. What is the cube of 15 ?

OPERATION.'

15 x 15 x 15 =3375, Ans.

ANALYSIS. We multiply 15 by

15, and the product by 15, and obtain 3375, which is the 3d power,

or cube of 15, since 15 has been taken 3 times as a factor.

RULE. Multiply the number by itself as many times, less 1, as there are units in the exponent of the required power

EXAMPLES FOR PRACTICE.

2. What is the square of 25? 3. What is the square of 135? 4. What is the cube of 72? 5. What is the 4th power of 24? 6. Raise 7.2 to the third power. 7. Involve 1.06 to the 4th power. 8. Involve .12 to the 5th power. 9. Involve 1.0002 to the 2d power. 10. What is the cube of

?

OPERATION.

Ans. 625.

Ans. 18225.

Ans. 373248.

Ans. 331776.

Ans. 373.248. Ans. 1.26247696. Ans. .0000248832. Ans. 1.00040004.

It is evident from the above operation, that

A common fraction may be raised to any power, by raising

each of its terms, separately, to the required power.

425. A Root is a factor repeated to produce a power; thus, in the expression 5 x 5 x 5 = 125, 5 is the root from

which the power, 125, is produced.

426. Evolution is the process of extracting the root of a number considered as a power, and is the reverse of Involution.

427. The Radical Sign is the character, /, which, placed before a number, denotes that its root is to be extracted.

428. The Index of the root is the figure placed above the radical sign, to denote what root is to be taken. When no index is written, the index 2 is always understood.

429. A Surd is the indicated root of an imperfect power. 430. Roots are named from the corresponding powers, as will be seen in the following illustrations:

The square root of 9 is 3, written √93.
The cube root of 27 is 3, written

27 = 3. The fourth root of 81 is 3, written 813.

431. Any number whatever may be considered a power whose root is to be extracted; but only the perfect powers can have exact roots.

SQUARE ROOT.

432. The Square Root of a number is one of the two equal factors that produce the number; thus the square root of 49 is 7, for 7 x7 = 49.

433. In extracting the square root, the first thing to be determined is the relative number of places in a given number and its square root. The law governing this relation is exhibited in the following examples:

From these examples we perceive

1st. That a root consisting of 1 place may have 1 or 2

places in the square.

2d. That in all cases the addition of 1 place to the root adds 2 places to the square. Hence,

If we point off a number into two-figure periods, commencing at the right hand, the number of full periods and the left hand full or partial period will indicate the number of places in the square root; the highest period answering to the highest figure of the root.

434. 1. What is the length of one side of a square plat containing an area of 5417 sq. ft.

Fig. I.
70

4

ANALYSIS. Since the given figure is a square, its side will be the square root of its area, which we will proceed to compute. Pointing off the given number, the 2 periods show that there will be two integral figures, tens and units, in the root. The tens of the root must be extracted from the first or left hand period, 54 hundreds. The greatest square in 54 hundreds is 49 hundreds, the square of 7 tens; we therefore write 7 tens in the root, at the right of the given number.

=

Since the entire root is to be the side of a square, let us form a square (Fig. I), the side of which is 70 feet long. The area of this square is 70 × 70 4900 sq. ft., which we subtract from the given number. This is done in the operation by subtracting the square number, 49, from the first period, 54, and to the remainder bringing down the second period, making the entire remainder 517.

If we now enlarge our square (Fig. I) by the addition of 517 square feet, in such a manner as to preserve the square form, its size will be that of the required square. To preserve the square form, the addition must be so made as to extend the square equally in two directions; it will therefore be composed of 2 oblong figures at the sides, and a little square at the corner (Fig. II). Now, the width of this addition will be the additional length to the side of the square, and consequently the next figure in the root. To find width we divide square contents, or area, by length. But the length of one side of the little square cannot be found till the width of the addition be determined, because it is equal to this width. We will therefore add the lengths of the 2 oblong figures, and the sum will be sufficiently near the whole length to be used as a trial divisor.