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EXAMPLES FOR PRACTICE. 1. What will an annuity of $ 378 amount to in 5 6 per cent. compound interest ?
Ans. $ 2130.821+.
·5.6 37093 X 378=$213 0.821+. 2. What will an annuity of $ 1728 amount to in 4 years, at 5
per cent. compound interest ? Ans. $7447.89,6+.
3. What will an annuity of $ 87 amount to in 7 years, at 6 per cent. compound interest?
Ans. $ 730.26,3+. 4. What will an annuity of $ 500 amount to in 6 years, at 6 per cent. compound interest? Ans. $ 3487.65,9+.
5. What will an annuity of $ 96 amount to in 10 years at 6 per cent. compound interest ?
Ans. $ 1265.35,6+. 6. What will an annuity of $ 1000 amount to in 3 6 per cent. compound interest ?
Ans. $ 3183.60. 7. July 4, 1842, H. Piper deposited in an annuity office, for his daughter, the sum of $ 56, and continued his deposites each year, until July 4, 1848. Required the sum in the office July 4, 1848, allowing 6 per cent. compound interest.
Ans. $470.05,4+. 8. C. Greenleaf has two sons, Samuel and William. 'On Samuel's birth-day, when he was 15 years old, he deposited for him, in an annuity office, which paid 5 per cent. compound interest, the sum of $ 25, and this he continued yearly, until he was 21 years of age. On William's birth-day, when he was 12 years old, he deposited for him, in an office which paid 6 per cent. compound interest, the sum of $20, and continued this until he was 21 years of age. Which will receive the larger sum, when 21
? Ans. $60.06,5+ William receives more than Samuel. 9. I gave my daughter Lydia, $ 10, when she was 8
years old, and the same sum on her birth-day each year, until she
years old. This sum was deposited in the savings bank, which pays 5 per cent. annually. Now, supposing each deposite to remain on interest until she is 21 years of age, required the amount in the bank.
§ XLII. ALLIGATION. Art. 302. Alligation signifies the act of tying together, and is a rule employed in the solution of questions relating to the mixture of several ingredients of different prices or qualities. It is of two kinds, Aligation Medial and Alligation Alternate.
Art. 303. Alligation Medial is the method of finding the mean price of a mixture composed of several articles or ingredients, the quantity and price of each being given.
Art. 304. To find the mean price of several articles or ingredients, at different prices, or of different qualities.
RULE. - - Find the value of each of the ingredients, and divide the amount of their values by the sum of the ingredients; the quotient will be the price of the mixture.
EXAMPLES FOR PRACTICE. 1. A grocer mixed 201b. of tea worth $0.50 a pound, with 30lb. worth $ 0.75 a pound, and 50lb. worth $0.45 a pound; what is 1 pound of the mixture worth ? Ans. $ 0.55.
$0.50 X 20 = $ 10.00
$ 0.45 x 50 $ 2 2.5 0
Then, $ 55.0 0 ; 100 $ 0.55 per pound.
$ 0.55 X 30 lb. * 1 6.50 : $ 55.0 0.
$ 0.55 X 50 lb. $ 2 7.50 2. I have four kinds of molasses, and a different quantity of each, as follows: 30 gal. at 20 cents; 40 gal. at 25 cents; 70 gal. at 30 cents, and 80 gal. at 40 cents; what is a gallon of the mixture worth?
Ans. $0.314 3. A farmer mixed 4 bush. of oats at 40 cents; 8 bush. of
QUESTIONS. - Art. 302. What is alligation? What two kinds are there? Art. 303. What is alligation medial? - Art. 304. What is the rule for find. ing the mean price of several articles at different prices? How does it appear that this process will give the mean price of the mixture ?
corn at 85 cents, 12 bush. rye at $ 1.00; and 10 bush. of wheat at $ 1.50
bushel. What will one bushel of the mixture be worth?
Ans. $ 1.0497 4. I wish to mix 30lb. of sugar at 10 cents a pound; 251b. at 12 cents; 41b. at 15 cents; and 50lb. at 20 cents; what is 1 pound of the mixture worth? Ans. $ 0.15.25
Art. 305. Alligation Alternate is the method of finding what quantity of two or more ingredients or articles, whose prices or qualities are given, must be taken, to compose a mixture of any given price or quality.
Art. 306. To find what quantity of each ingredient must be taken to form a mixture of a given price.
Ex. 1. I wish to mix two kinds of spice, one at 21 cents, and the other 26 cents per pound, so that the mixture may be worth 24 cents per pound. How many pounds of each must I take ?
Ans. 21b. at 21 cents; 3lb. at 26 cents.
We connect the price that is
less than the mean price, with 217 2 24
Ans. the price which is greater, and
set the difference between each
price and the mean price opposite the price with which it is connected; these numbers denote the quantity of each ingredient to be taken.
It will be seen that the value of llb. of the spice at 21cts. is 3 cents less than of ilb. of the mixture at the mean price, 24 cents, and that the value of ilb. at 26 cents is 2 cents more than the mean price. Now if one of these prices were as much greater than the mean price as the other is less, the differences would balance each other, and the mixture of the two in equal quantities would be 24 cents per lb., the given mean price. But since the deficiency is more than the excess, we must take more pounds at 26 cents than at 21 cents per lb., to balance the deficiency. If we multiply the 2 cents by some nuinber, as 3, and the 3 cents by some number, as 2, the product of the excess will be just equal to the product of the deficiency; and 31b. at 26 cents and 2lb. at 21 cents per pound, will form a mixture of 5lb., worth
QUESTIONS. Art. 305. What is alligation alternate ? - Art. 306. How do you connect the prices? Where do you set the differences between each price and the mean price? What do these differences denote? How does it appear, from the explanation, that the differences denote the quantity of each kind to be taken?
$ 1.20, of which llb. will be worth 24 cents, the price of the required mixture. Therefore, we must take 3lb. at 26 cents, and 21b. at 21 cents, to make a mixture worth 24 cents per pound, which is the same result as was obtained in the operation. Hence the
RULE. — 1. Place the prices of the ingredients under each other, in the order of their value, and connect the price of each ingredient which is less in value than the price of the mixture, with one that is greater.
2. Then place the difference between the price of the mixture, and that of each of the ingredients, opposite to the price with which it is connected, and the number set opposite to each price is the quantity of the ingredient to be taken at that price.
Note. - There will be as many different answers as there are different ways of connecting the prices, and by multiplying and dividing these answers they may be varied indefinitely.
EXAMPLES FOR PRACTICE.
2. A farmer wishes to mix corn at 75 cents a bushel, with rye at 60 cents a bushel, and oats at 40 cents a bushel, and wheat at 95 cents a bushel; what quantity of each must he take to make a mixture worth 70 cents a bushel ?
60 5 60 25 605 70
70 75 10
75] 10+30=40 95 30 95
30+10=40 3. I have 4 kinds of salt worth 25, 30, 40, and 50 cents per bushel; how much of each kind must be taken, that a mixture might be sold at 42 cents per bushel ?
Ans. 8 bushels at 25, 30, and 40 cents, and 31 bushels at 50 cents.
4. My swamp hay is worth $ 12 per ton, my salt hay $ 15, and
my English hay $ 20; how much of each kind must be taken, that a ton may be sold at $ 18?
Ans. 2 tons of swamp hay, 2 tons of salt hay, and 9 tons of English hay.
Art. 307. When the quantity of one ingredient is given to find the quantity of each of the others.
QUESTIONS. – What is the rule for alligation alternate? How can you obtain different answers ? Are they all true ?
Ex. 1. How much sugar, that is worth 6, 10, and 13 cents a pound, must be mixed with 20lb. worth 15 cents a pound, so that the mixture will be worth 11 cents a pound ?
6 4 10 2
: 20: 4 11
Then, 5:1 13 1
5:2 :: 20: 8 Ans. 15
5:4:: 20 : 16 By the conditions of the question we are to take 2016. at 15 cents a pound ; but by the operation we find the difference at 15 cents a pound to be only 5lb., which is but of the given quantity. Therefore, it we increase the 5lb. to 20, the other differences must be increased in the same proportion. Hence the propriety of the following
RULE. Take the difference between each price and the mean price, as before ; then say, as the difference of that ingredient whose quantity is given is to each of the differences separately, so is the quantity given to the several quantities required.
EXAMPLES FOR PRACTICE.
2. A farmer has oats at 50 cents per bushel, peas at 60 cents, and beans at $ 1.50. These he wishes to mix with 30 bushels of corn at $ 1.70 per bushel, that he may sell the whole at $ 1.25 per bushel ; how much of each kind must he take?
Ans. 18 bushels of oats, 10 bushels of peas, and 26 bushels of beans.
3. A merchant has two kinds of sugar, one of which cost him 10 cents per lb., and the other 12 cents per lb.; he has also 100lb. of an excellent quality which cost him 15 cents per Ib. Now, as he ought to make 25 per cent. on his cost, how much of each quantity must be taken that he may sell the mixture at 14 cents per lb.
Ans. 383}lb. at 10 cents, and 100lb. at 12 cents. Art. 308. When the sum of the ingredients and their mean price are given, to find what quantity of each must be taken.
Ex. 1. I have teas at 25 cents, 35 cents, 50 cents, and 70
QUESTION. – Art. 307. What is the rule for finding the quantity of each of the other ingredients when one is given ?