number of units of each ingredient that must be in the mixture that the gain and loss may balance each other.

IV. Proceed in the same manner with other ingredients; the results will be the proportional parts.

722. 1. How much sugar at 10, 9, 7, and 5 ct. will produce a mixture worth 8 cents a pound?

2. A man wishes to mix sufficient water with molasses worth 40 cents a gallon to make the mixture worth 24 cents a gallon; what amount must he take of each ?

3. A jeweller has gold 16, 18, 22, and 24 carats fine; how much of each must he use to form gold 20 carats fine?

4. A merchant desires to mix flour worth $6, $71, and $10 a barrel so as to sell the mixture at $9; what proportion of each kind can he use ?

5. A farmer has wheat worth 40, 55, 80, and 90 cents a bushel; how many bushels of each must be mixed with 270 @40 ct. to form a mixture worth 70 cents a bushel?

Examples like this where the quantity of one or more ingredients is limited may be solved thus:

First, we find the gain or loss on one unit as in (720).

Second, we balance the whole gain or loss on an ingredient where the quantity is limited, by using any ingredient giving an opposite result thus:

PRODUCING GAIN.

GAINED AND LOST.

PRODUCING Loss.

(1.) 270 bu. at 30 ct. per bu. gain.=$81.00=405 bu. at 20 ct. per bu. loss. (2.) 2 bu. at 15 ct. per bu. gain.

.30 3 bu. at 10 ct. per bu. loss. 408 bu. 680 bu. in mixture.

Observe, the gain on the 270 bu. may be balanced with the other ingredient that produces a loss, or with both ingredients that produce a loss, and these may be put in the mixture in different proportions; hence a series of different mixtures may thus be formed.

6. A merchant having good flour worth $7, $9, and $12 a barrel, and 240 barrels of a poorer quality worth $5 a barrel, wishes to sell enough of each kind to realize an average price of $10 a barrel on the entire quantity sold. How many barrels of each kind can he sell?

7. I wish to mix vinegar worth 18, 21, and 27 cents a gallon with 8 gallons of water, making a mixture worth 25 cents a gallon; how much of each kind of vinegar can I use?

8. A man bought a lot of sheep at an average price of $2 apiece. He paid for 50 of them $2.50 per head, and for the rest $1.50, $1.75, and $3.25 per head; how many sheep could there be in the lot at each price?

9. A milkman mixes milk worth 8 cents a quart with water, making 24 quarts worth 6 cents a quart; how much water did he use?

Examples like this, where the quantity of the mixture is limited, may be solved thus:

SOLUTION.-1. We find, according to (720), the smallest proportional parts that can be used, namely, 3 quarts of milk and 1 quart of water, making a mixture of 4 quarts.

2. Now, since in 4 qt. of the mixture there are 3 qt. of milk and 1 qt. of water, in 24 qt. there must be as many times 3 qt. of milk and 1 qt. of water as 4 qt. are contained times in 24 qt. Consequently we have as the first step 24 qt. ÷ 4 qt. 6, second step 3 qt. × 6 18 qt. and 1 qt. × 6 6 qt. Hence in 24 qt. of the mixture there are 18 qt. of milk and 6 qt. of water.

=

10. A grocer has four kinds of coffee worth 20, 25, 35, and 40 cents a pound, from which he fills an order for 135 pounds worth 32 cents a pound; how may he form the mixture?

11. A jeweler melts together gold 14, 18, and 24 carats fine, so as to make 240 oz. 22 carats fine; how much of each kind did it require?

12. I wish to fill an order for 224 lb. of sugar at 12 cents, by forming a mixture from 8, 10, and 16 cent sugar; how much of each must I take?

INVOLUTION

DEFINITIONS.

23. A Power of a number is either the number itself or the product obtained by taking the number two or more times as a factor.

Thus 25 is the product of 5×5 or of 5 taken twice as a factor; hence 25 is a power of 5.

24. An Exponent is a number written at the right and a little above a number to indicate:

(1.) The number of times the given number is taken as a factor. Thus in 73 the 3 indicates that the 7 is taken 3 times as a factor; hence 737x7x7 = 343.

(2.) The degree of the power or the order of the power with reference to the other powers of the given number. Thus, in 54 the 4 indicates that the given power is the fourth power of 5, and hence there are three powers of 5 below 54; namely, 5, 52, and 53.

725. The Square of a number is its second power, so called because in finding the superficial contents of a given square we take the second power of the number of linear units in one of its sides (404).

726. The Cube of a number is its third power, so called because in finding the cubic contents of a given cube we take the third power of the number of linear units in one of its edges (412).

727. Involution is the process of finding any required power of a given number.

PROBLEMS IN INVOLUTION.

728. PROB. I.

number.

-To find any power of any given

1. Find the fourth power of 17.

SOLUTION.-Since according to (721) the fourth power of 17 is the product of 17 taken as a factor 4 times, we have 17 × 17 × 17 × 17=83521, the required power.

2. Find the second power of 48. Of 65. Of 432.

3. Find the square of 294.

4. Find the cube of 63.

Of 386.

Of 25. Of 76. Of 392.

Of 497. Of 253.

Of. Of.8.

Observe, any power of a fraction is found by involving each of its terms separately to the required power (267).

Find the required power of the following:

6. 2372. 8. (13)3. 10. (.25)4. 12. (.7)2. 14. (.005). 7. 454. 9. (1) 11. (.3%).

13. (.14. 15. .03022. 729. PROB. II.-To find the exponent of the product of two or more powers of a given number.

1. Find the exponent of product of 73 and 72.

SOLUTION. Since 73 = 7 × 7 × 7 and 72 = 7 × 7, the product of 73 and 72 must be (7 × 7 × 7) × (7 × 7), or 7 taken as a factor as many times as the sum of the exponents 3 and 2. Hence to find the exponent of the product of two or more powers of a given number, we take the sum of the given exponents.

Find the exponent of the product

2. Of 354 x 353.

4. Of 182 x 187.

6. Of 237 x 235.

5. Of (4) × ($)8.

7. Of (†1)1× (†1)o.

8. Of (74)2.

3. Of ()x()2.

Observe, (74)2=74 × 74 = 74×2 = 78.

Hence the required exponent is the product of the given exponents.

9. Of (123)4.

10. Of (96)5. 11. Of (168)8. 12. Of [(4)3]4

EVOLUTION

DEFINITIONS.

730. A Root of a number is either the number itself or one of the equal factors into which it can be resolved.

Thus, since 7 x7 = 49, the factor 7 is a root of 49.

731. The Second or Square Root is one of the two equal factors of a number. Thus, 5 is the square root of 25.

732. The Third or Cube Root is one of the three equal factors of a number. Thus, 2 is the cube root of 8.

733. The Radical or Root Sign is √, or a fractional exponent.

When the sign, /, is used, the degree or name of the root is indicated by a small figure written over the sign; when the fractional exponent is used, the denominator indicates the name of the root; thus,

/9 or 9 indicates that the second or square root is to be found. 3/27 or 273 indicates that the third or cube root is to be found. Any required root is expressed in the same manner. usually omitted when the square root is required.

The index is

734. A Perfect Power is a number whose exact root can be found.

735. An Imperfect Power is a number whose exact root cannot be found.

The indicated root of an imperfect power is called a surd; thus √5.

736. Evolution is the process of finding the roots of numbers.