Rule. Find the gain or loss on one of each constituent, then take as much of each as will make the gain and loss equal. Or,

Write the values of the constituents in a column on the right of a vertical line, and that of the compound on the left. Draw a line connecting each value less than that of the compound with one or more of those greater.

Write the difference between the value of the compound and that of each constituent opposite every value with which it is connected.

The sum of the differences opposite any value may be taken as the quantity of that constituent. The ratio of this sum to the sum of all the differences is the ratio of that constituent to the whole.

PROOF.-Multiply each quantity by its value, and divide the sum of the products by the sum of the quantities. The quotient should be the value of the mixture.

NOTE 1.-After finding all the ratios developed by all possible methods of connecting and comparing the values, new ratios may be developed by finding the sums or differences of corresponding values in the different sets.

NOTE 2.-Such questions have an unlimited number of particular answers, because each of all the possible ratios may be filled by an unlimited number of supposed numbers.

NOTE 3.-When the value of a constituent is that of the compound, it need not be used in the written process, because any quantity of it may be added to the compound without affecting its value per unit. In such a case, this rule finds the proportions of the others among themselves.

NOTE 4.-Any common factor may be omitted from the proportionals.

NOTE 5.-This case is often called Alligation Alternate.

EXAMPLES FOR PRACTICE.

Find seven answers to each of the following:

3. How much sugar worth 10, 12, and 15 cents a pound, must be mixed together, so that the mixture shall be worth Ans. 2, 2, and 4, or 1, 1, and 2.

13 cents a pound?

4. How many pounds of tea at $.50, $.60, $.80, and $1.20 a pound, must be mixed together, so that the mixture shall be worth $.75 a pound? Ans. 10, 9, 5, and 8.

5. What relative quantities of alcohol, 86, 92, 95, and 98% strong, will make a mixture 93% strong?

6. What relative quantities of gold, 16, 18, 19, 22, and 23 carats fine, will make a metal 20 carats fine?

7. Bought cows at $10 each, hogs at $3, sheep at $1, so that the average cost per head was $1; how many of each did I buy? Ans. 22 sheep, 1 hog, 1 cow.

CASE III.

Art. 422. To find the other quantities, when that of any constituent, or of the compound, is known.

Ex. 1. In Ex. 2, Case II., suppose 10 lb. of the 10 ct. constituent; find the other quantities.

ANALYSIS BY THE FIRST PROCESS.--If 10 lb. is of the whole, then of 10 lb., or 2 lb., is ' of the whole, and 22 times 2 lb., or 44 lb., is the whole. Hence the 12 ct. constituent is 10 lb., and the 15 and 18 ct. constituents are each of 44, or 12 lb.

NOTE. Analyze this supposition by the other processes.

Ex. 2. In Ex. 2, Case II., suppose 88 bushels to be the compound; find the quantities of the constituents.

ANALYSIS BY THE SECOND PROCESS.-If 88 bu. is the whole, then the 10 ct. and 18 ct. constituents are each of 88 bu., or 32 bu., the 12 ct. constituent is of 88 bu., or 8 bu., and the 15 ct. constituent is of 88 bu., or 16 bu.

NOTE.—Analyze this supposition by the other processes.

Rule.—Find the proportions by Case II., and with these find the required quantities from the given quantity.

EXAMPLES FOR PRACTICE.

3. How many pounds of teas at 60, 80, and 90 cents a pound, must be mixed with 29 lb. at $1.25 a pound, so that the mixture may be worth $.96 a pound? Ans. 14 lb.

4. What relative quantities of sugar at 6, 8, 10, and 15 cts. will make a mixture of 66 lb. at 9 cts. a pound?

5. How much alcohol, 75%, 80%, 90%, and 100% strong

make 72 gal. 85% strong?

6. A merchant sold salt for $2, flour for $9, sugar for $30, and 40 barrels of molasses for $40 a barrel; what was the number of barrels of each, if the average price was $20 a barrel?

7. A farmer sold sheep for $5, hogs for $8, cows for $20, and 26 mules for $40 each; the average price of all was $18 each; how many sheep, hogs and cows did he sell?

Ans. 64, 1, 5, or 64, 2, 10, or 64, 3, 15, &c.

8. A merchant bought some hats for $4 each, some vests for $6 each, and 24 coats at $16 each; the average cost of all was $10; how many hats and vests did he buy?

One answer, 8 hats, 24 vests.

9. A grocer wishes to mix 20 lb. of sugar at 8 cts. and 30 lb. at 10 cts. with some at 15 cts. and 20 cts., so that the mixture may be worth 12 cts.; how much of the latter kind must he take? Ans. 20 lb. at 15 cts.; 10 lb. at 20 cts.

NOTE. We find by Case II. that the number at 8 and 20 cts. are as 2 to 1, and at 10 and 15 cts. as 3 to 2; hence, as often as we take 2 lb. at 8 cts. we take 1 lb. at 20 cts., and as often as we take 3 lb. at 10 cts. we take 2 lb. at 15 cts.

10. A jeweller has 2 pwt. 15 gr. of gold, 16 carats fine; how much gold, 21g carats fine, must be mixed with it, to make the mixture 18 carats fine? Ans. 1 pwt. 11 gr.

11. How many pounds of tea at $.50 and $.60 per pound must be mixed with 24 lb. at $.80 and 30 lb. at $.90, so that it may average $.75 a pound?

Ans. 18 lb. at $.50; 8 lb. at $.60. 12. A's farm cost him, on the average, $60 an acre. He gave for 100 acres of it $50 an acre, and for the rest of it $85 an acre. How many acres in his farm? Ans. 140.

13. A stock-dealer bought 270 head of sheep for $1365, paying $4, $4, $51, and $71⁄2 a head; how many did he buy at each rate? One answer, 40, 110, 95, 25.

14. A boy has some small coins whose denominations are respectively 1 ct., 2 cts., 5 cts., and 10 cts., which he wishes to exchange for 60 three-cent pieces; how many must he exchange of each kind?

15. A farmer bought 100 head of stock for $100, paying $each for sheep, $3 for hogs, and $10 for cows; how many of each did he buy?

Ans. 94 sheep, 1 hog, 5 cows.

SYNOPSIS OF APPLICATIONS OF AVERAGE.

Average.Rate Bills-Marine Average,
Compounds-Alligation.

1

CHAPTER XVIII.

INVOLUTION. EVOLUTION.

INVOLUTION.

Art. 423. Involution is the process of finding a power of a number.

A power of a number is either the number itself, or a product whose component factors are that number. Thus, 32 is a power of 2, being equal to 2 × 2 × 2 × 2 × 2; also, 2 is the first power of itself. (See Art. 44.)

Involving a number is using it to produce its power.

A root of a given number is either the number itself, or that number whose involution produces the given number. Thus, 2 is a root of 32, because involving 2 produces 32; also, 32 is the first root of itself.

Art. 424. Powers are named from the number of times the root is used in producing them.

The first power of a number is the number itself.

The second power, or square, of a number is the product obtained by using that number twice as a factor. Thus, 49 is the second power, or square, of 7, because 7 × 7 = €49.

The third power, or cube, of a number is the product obtained by using that number three times as a factor. Thus, 27 is the third power, or cube, of 3, because 3 x 3 x 3 = 27.

NOTE. The second power is called a square, because the area of a square is the product of two equal factors. (See Art. 213.) The third power is called a cube, because the volume of a cube is the product of three equal factors. (See Art. 217.)