NOTE 2.If the second rule is employed, when the ratio is less than one, its power, denoted by the number of terms, must be subtracted from 1, and the remainder divided by the difference between 1 and the ratio. EXAMPLES FOR PRACTICE. 1. If the first term of a series is 12, the ratio 3, and the number of terms 8, what is the sum of the series? Ans. 39360. OPERATION. Ratio 37 × 12=26244, the last term; 26244 × 3=78732; 78732—1278720; 78720 ÷ (3 — 1) : 39360, the sum of the series. 2. The first term of a series is 5, the ratio, and the number of terms 6; required the sum of the series. Ans. 131. OPERATION. Ratio (3) 5-6, the last term; 18 X 3 = 733; × 5= 160 5-438-3325; 3325 ÷ (1 − 3) = 3325 = 1319, the sum of the series. 729 3. If the first term of a series is 8, the ratio 4, and the number of terms 7, required the sum of the series. 4. If the first term is 10, the ratio, and the number of terms 5, what is the sum of the series? 5. If the first term is 18, the ratio 1.06, and the number of terms 4, what is the sum of the series? 6. When the first term is $144, the ratio $1.05, and the number of terms 5, what is the sum of the series? 7. D. Baldwin agreed to labor for E. Thayer for 6 months. For the first month he was to receive $3, and each succeeding month's wages were to be increased by of his wages for the month next preceding; required the sum he received for his 6 months' labor. 8. A lady, wishing to purchase 10 yards of silk for a new dress, thought $1.00 per yard too high a price; she, however, agreed to give 1 cent for the first yard, 4 for the second, 16 for the third, and so on, in a four-fold ratio; what was the cost of the dress? 25 ANNUITIES AT COMPOUND INTEREST BY GEOMETRICAL PROGRESSION. ART. 300. WHEN compound interest is reckoned on an annuity in arrears, the annuity is said to be at compound interest; and the amounts of the several payments form a geometrical series, of which the annuity is the first term, the amount of $1.00 for one year the ratio, the years the number of terms, and the amount of the annuity, the sum of the series. Hence, ART. 301. To find the amount of an annuity at compound interest, we have the following RULE I. — Find the sum of the series by either of the preceding rules, (Art. 299.) Or, RULE IÍ. Multiply the amount of $1.00, for the given time, found in the table, by the annuity, and the product will be the required amount. TABLE, Showing the amount of $1 annuity from 1 year to 40. QUESTIONS. Art. 300. When is an annuity said to be at compound interest? What do the amounts of the several payments form? What is the first term of the series? What the ratio? What the number of terms? What the sum of the series?- Art. 301. What is the first rule for finding the amount of an annuity? What the second? What does the table show? EXAMPLES FOR PRACTICE. 1. What will an annuity of $378 amount to in 5 years, at 6 per cent. compound interest? Ans. $2130.821+. 2. What will an annuity of $ 1728 amount to in 4 years, at 5 per cent. compound interest? 3. What will an annuity of $87 amount to in 7 years, at 6 per cent. compound interest? 4. What will an annuity of $500 amount to in 6 years, at 6 per cent. compound interest? 5. What will an annuity of $ 96 amount to in 10 years at 6 per cent. compound interest? 6. What will an annuity of $1000 amount to in 3 years at 6 per cent. compound interest? 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. On 8. C. Greenleaf has two sons, Samuel and William. 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 years of age? 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 was 21 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, Alligation Medial and Alligation Alternate. ALLIGATION MEDIAL. 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 20lb. 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. $55.00, value. = $16.50 = $55.00. $27.50 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? 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 finding 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 per bushel. What will one bushel of the mixture be worth? 4. I wish to mix 30lb. of sugar at 10 cents a pound; 25lb. at 12 cents; 4lb. at 15 cents; and 50lb. at 20 cents; what is 1 pound of the mixture worth? ALLIGATION ALTERNATE. 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? We connect the price that is less than the mean price, with the price which is greater, and set the difference between each price and the mean price oppo site 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 1lb. of the spice at 21cts. is 3 cents less than of 1lb. of the mixture at the mean price, 24 cents, and that the value of 1lb. 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 number, 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 3lb. at 26 cents and 2lb. at 21 cents per pound, will form a mixture of 5lb., worth QUESTIONS. Art. 305. do you connect the prices? price and the mean price? What is alligation alternate? - Art. 306. How Where do you set the differences between each What do these differences denote? How does it appear, from the explanation, that the differences denote the quantity of each kind to be taken? |