CHAPTER XI.

ALLIGATION.

289. Alligation treats of the mixing or combining of two or more articles of different values.

290. Alligation Medial is the process of finding the average value or quality of the several articles.

291. Alligation Alternate is the process of determining the proportion of the several articles used in forming a mixture.

The word alligation is derived from the Latin alligo, "I bind," and is so called because in solving some of the problems the numbers may be joined or linked together.

CASE I.

To find the Average.

ORAL EXERCISE.

1. If a grocer mix 3 lb. of sugar at 10 cts., 5 lb. at 8 cts. and 2 lb. at 15 cts., what is the average price per pound?

SOLUTION.-3 lb. at 10 cts. are worth 30 cts.; 5 lb. at 8 cts. are worth 40 cts., and 2 lb. at 15 cts. are worth 30 cts.: 30 cts. plus 40 cts. plus 30 cts. are 100 cts.; 3 lb. +5 lb. +2 lb. equal 10 lb. If 10 lb. are worth 100 cts., 1 lb. is worth of 100 cts., or 10 cts.

2. If a wine-dealer mix 6 gal. of wine worth $2 a gallon with 4 gal. worth $3 a gallon, what must he sell it at per gallon so as not to lose?

3. A farmer bought 20 sheep at 5 dollars a head, 10 at 41 dollars a head, and 20 at 3 dollars a head: what was the average price?

WRITTEN PROBLEMS.

4. A miller has 4 loads of wheat, as follows: 60 bu. at $1.25, 47 bu. at $1.30, 55 bu. at $1.32, and 38 bu. at $1.20 : how must he sell it per bushel so as not to lose? Ans. $1.27.

5. A merchant bought 150 yd. muslin at 10 cts., 120 yd. at 74 cts., 144 yd. at 144 cts., and 160 yd. at 15 cts. : what must he sell it at so as to gain 25% ? Ans. 15 cts. per yard.

6. A grocer bought 600 lb. coffee at 23 cts., 800 lb. at 26 cts., and 960 lb. at 30 cts.: he sold it at an average price of 31 cts. a pound: what was the entire gain? Ans. $97.60.

CASE II.

To find the Proportion of Different Articles of Given Value or Quality when the Average Value or Quality is given.

WRITTEN PROBLEMS.

1. How much muslin at 10, 13 and 15 cts. a yard must a merchant sell that the average price may be 12 cts. a yard?

SOLUTION.

10

12 13 15

1

121

1

2

= 2

Analysis.-On the 10-ct. muslin the gain is 2 cts. a yard; hence, to gain 1 ct. yd. must be sold. the loss is 1 ct. on muslin the loss is

On the 13-ct. muslin yd.; and on the 15-ct. 3 cts. a yard, or a cent on of a yard. Since the gain on yd. of 10-ct. muslin is 1 ct., and the loss on yd. of 15-ct. muslin is 1 ct., the loss and the gain are equal, and we take yd. at 10 cts. as often as yd. at 15 cts., or 3 yd. at 10 cts. as often as 2 yd. at 15 cts. In a similar manner, since the gain on yd. at 10 cts. balances the loss on 1 yd. at 13 cts., we take yd. at 10 cts. as often as 1 yd. at 13 cts., or 1 yd. at 10 cts. as often as 2 yd. at 13 cts. The requisite numbers of yards are, therefore, 3+1, or 4 at 10 cts., 2 at 13 cts., and 2 at 15 cts.

2. In what proportion must sugar worth 9 cts. a pound be mixed with that worth 14 cts., so that the mixture may be worth 12 cts. a pound? Ans. 2 lb. at 9, and 3 lb. at 14.

16 cts.?

3. How many dozens of eggs at 12, 15, 18 and 20 cts. a dozen must be sold that the average price per dozen may be Ans. 1 at 12, 2 at 15, 1 at 18, and 1 at 20. 4. If I have wine worth $1.50, $1.75, $2.20 and $2.40 a gallon, in what proportions must it be mixed that I may sell it at $2 a gallon?

Ans. 4 at $1.50, 4 at $1.75, 5 at $2.20, and 5 at $2.40.

5. A wine-merchant has wine worth respectively $3 and $3.50 a gallon. He wishes to mix these two and sell a grade at $2.50 a gallon: what proportion of each and what proportion of water must he take?

Ans. 5 gal. at $3, 5 gal. at $3.50, and 3 gal. of water. 6. A farmer bought sheep at $5, $6, $7 and $10 a head: how many of each did he buy if the average price was $7.50? Ans. 1 at $5, 5 at $6, 5 at $7, and 5 at $10.

7. If I have cloth worth $21, $2, $2 and $4 a yard, how many yards of each must I sell that the average price may be $34 a yard? Ans. 3 at $21, 1 at $21, 3 at $24, and 7 at $4.

CASE III.

To find the Proportions of the Different Parts of a Given Value or Quality when the Average Value or Quality is given, and the Proportion of One or more of the Parts.

1. A merchant sold 25 yd. of muslin at 12 cts. a yard: how much must he sell at 8 and 15 cts. respectively that the average price may be 13 cts.?

13 12 1

SOLUTION.

2 × 121=

{

25

25

75

1515+1)

Analysis.-By Case II. we determine that 2 yd. of the 8-ct. muslin are taken as often as 2 yd. at 12 cts. and 6 yd. at 15 cts. The required number of yards

at 12 cts. is 25, or 12 times 2; hence we take 123 times 2 yd., or 25 yd., at 8 cts., and 12 times 6 yd., or 75 yd., at 15 cts.

2. A dry-goods merchant has 30 yd. of damaged merino which he wishes to sell at 40 cts. a yard: how many yards must he sell worth $1.20 that he may get cost for his goods, which was 90 cts. a yard? Ans. 50 yd.

3. A miller buys corn at 60, 62 and 65 cts. a bushel: how much did he buy of each if he bought 70 bu. at 70 cts., and the average price is 67 cts.? Ans. 15 bu. of each.

4. A merchant bought cloth at $4, $3.50, $2.20 and $2 a yard, and 30 yd. at $2.50: how many yards of each did he buy if the average price is $3.20 a yard?

Ans. 50 at $4, 110 at $3.50, 40 at $2.20, 10 at $2

5. If a druggist have 6 gal. of alcohol 95% strong, and some 90% strong, how much water and how much of the 90% alcohol must be mixed with the 6 gal., that the strength may be reduced to 80% ?

Ans. 3 gal. at 90%, and 14 gal. water.

CASE IV.

To find the Proportion of the Different Parts of a Given Value or Quality when the Entire Quantity is given.

1. How much sugar at 8, 9, 12 and 15 cts. a pound must be mixed so as to fill a barrel of 300 lb. at 10 cts.?

Analysis. By the analysis, as in the previous cases, we determine that 5 lb. of the 8-ct. sugar shall be taken as often as 2 lb. of the

15-ct.; and 2 lb. of the 9-ct. as often as 1 lb. of the 12-ct. The sum of these quantities is 10 lb.

To make a mixture of 300 lb. will therefore require 30 times as many pounds of each, or 150 lb. of the 8-ct., 60 lb. of the 9-ct., 30 lb. of the 12ct., and 60 lb. of the 15-ct.

Another form of solution may be given as follows:

bers is 8, which is not an exact

divisor of 300. Either column a or column b, or both, may be multiplied by any number so that the sum of the two columns may form an exact divisor of 300. Multiplying column a by 2, we have 2 lb. at 8 cts., 5 at 9, 2 at 12, and 1 at 15, or a total of 10 lb., which is of 300 lb. Multiplying the resulting columns by 30, we have 60 lb. at 8 cts., 150 at 9 cts., 60 at 12 cts., and 30 at 15 cts. a pound.

2. A man hired men at $15 a week, and boys at $5 a week,

to do some work; they worked 5 weeks, and he paid $450: how many were there of each if the number of laborers was 10? Ans. 4 men, 6 boys.

3. Sold 100 lb. of butter for $33; some was sold at 30 cts., some at 28 cts., and some at 35 cts. a pound: how many pounds were there of each?

Ans. 12 lb. at 30 cts., 20 lb. at 28 cts., and 68 lb. at 35 cts. 4. Sold 50 doz. of eggs for $8: how many dozens of each were there at 13, 14, 18 and 21 cts. respectively?

5. Bought turkeys at 85 cts., ducks at 30 cts., and chickens at 40 cts. each; in all, 80 fowls for 48 dollars: how many of each were there? Ans. 40 turkeys, 20 ducks, 20 chickens.

CHAPTER XII.

POWERS AND ROOTS.

SECTION I.

POWERS.

292. A Power of a number is the number itself, or that number taken two or more times as a factor. Thus, 3 is the first power of 3; 3×39, or 3 taken twice as a factor is the second power, or square, of 3.

3×3×3=27, or 3 taken three times as a factor, is the third power, or cube, of 3.

293. The power of a number is usually indicated by a small figure, called the exponent, written to the right and above it; thus, the fourth power of 5 is written 51.

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