2. What is the present value of an annuity of $500 for 10 yr., at 6% ? Ans. $3968.75. 3. An author has an annual income of $1200 as copyright. Suppose his copyright is still good for 25 yr., what is the present value, interest at 6% ? Ans. $20640.

4. A widow has a dower of $300 to continue 16 yr., at 6% : what is the present value? Ans. $3551.02. 5. I buy a house for $18000, to be paid in instalments of $1500 annually, without interest: what cash payment will cancel the debt, money being worth 6% interest?

6. I rent a house for $300 a year, the monthly in advance: what amount of cash of the year will pay one year's rent?

SECTION II.

Ans. $13918.60.

rent to be paid at the beginning Ans. $292.22.

ANNUITIES AT COMPOUND INTEREST. All problems in annuities at compound interest may be solved by applying the rules and principles pertaining to Geometrical Progression and Compound Interest.

CASE I.

To find the Amount of an Annuity at Compound Interest. 1. What is the amount of an annuity of $200 for 5 yr., at 5%, compound interest ? Ans. $1105.128.

Explanation.-Taking 1 as the first term of the geometrical series, and 1.05, the amount of $1 for 1 yr., as the rate, the 5th term is 1 × 1.05*; and

final value of $1 for the given time and rate per cent., and for $200,

200 $5.52564, or $1105.128.

NOTE.-The rate to any power may be found by consulting the Compound Interest Table, page 235, in which the required power of the rate is given as the amount of $1 for the given time at the given rate per

cent.

The following is the rule for finding the amount of an annuity at compound interest:

RULE.

Find the compound interest of $1 for the given rate and time; divide this by the rate, and multiply this result by the annuity. 2. What is the final value of an annuity of $300 for 6 yr., at 4%? Ans. $1989.75. 3. If I deposit $600 a year in a bank for 10 yr. at 5%, compound interest, what is the value at the expiration of the period? Ans. $7546.80. 4. If it cost me $123.90 yearly to have my life insured, what ought my policy to be worth at the end of 10 yr., money being worth 7% ? Ans. $1711.94.

5. A gentleman deposited $50 every six months in a savings bank, the amount to draw interest at 6% semi-annually: what is it worth at the end of 10 yr.? Ans. $1343.50.

CASE II.

To find the Present Value of an Annuity at Compound Interest, first find the Final Value as above, and Divide by the Amount of $1 for the given Rate and Time, at Compound Interest. 1. What is the present value of an annuity of $200 at compound interest for 7 yr., at 6% ? Ans. $1116.43.

2. What amount of cash deposited in a savings bank at 5% compound interest would amount to as much as $500 annually for 15 yr.? Ans. $5189.76.

3. What amount of cash deposited in a savings bank at 4% compound interest would amount to as much as $100 deposited annually for 12 yr.? Ans. $938.48.

4. What is the present value of a semi-annual deposit of $600 at compound interest for 7 yr. at 8%, payable semiannually? Ans. $6337.99.

CHAPTER XV.

CIRCULATING DECIMALS.

321. A Circulating Decimal is a decimal in which one or more of the figures are continually repeated.

322. The repeating part is called a Repetend. The repetend is indicated by placing a dot over the first and the last of the repeating figures.

323. A Pure Circulating Decimal is one which consists of repeating figures only; as, .3636, etc., written .36.

324. A Mixed Circulating Decimal is one in which the repetend is preceded by some other figures, called finite decimals; as, .1766, etc., written .176.

325. A circulating decimal is read as any other decimal, except that the word repetend is used before the repeating part.

326. A pure circulating decimal is reduced to a common fraction by writing as many 9's for a denominator as there are repeating figures in the repetend or numerator.

This will be evident from the following proof:

Let .632 be a pure circulating decimal.

1000 R, the circulate=632.632

it is evident that the reverse is true, or that every pure circulating decimal equals the repetend with as many 9's in the denominator as there are repeating figures in the numerator.

A mixed circulating decimal may be reduced to a common fraction by subtracting the finite part from the entire decimal, and writing for the denominator as many 9's as there are figures in the repetend, and annexing as many ciphers as there are figures in the finite part.

Ex. Reduce the decimal .7543 to a common fraction.

327. Duodecimals are fractions of which 12 of any order equal 1 of the next higher order.

328. The unit is the foot, which is divided into 12 equal parts called primes ('), each prime (') being divided into 12 equal parts called seconds ("), each second (") in the same manner into 12 thirds (""), and each "" into twelve fourths ('''').

1. ADDITION AND SUBTRACTION OF DUODECIMALS.

Problems in Addition and Subtraction of Duodecimals are solved like those in Denominate Numbers.

1. Add 6 ft. 4' 2"; 15 ft. 4" 3""; 8' 9" 4"" 3""", and 7""".

PROCESS.

6 ft. 4/2//

3. Add 7 ft. 1'′ 3′′ 6""; 1 ft. 3" 6" 1"""; 7 ft. 8' 7" 9"""; 8 ft.

10' 6".

Ans. 24 ft. 8' 8" 10""".

4. From 18 ft. 3" 9""" take 10 ft. 2′ 2′′ 6"".

Ans. 7 ft. 10′ 6′′"' 9""".

2. MULTIPLICATION OF DUODECIMALS.

1. What is the product of 9 ft. 6′ 5′′ and 6 ft. 4'?

SOLUTION.

9 ft. 6' 5'

6 ft. 4'

3 ft. 2/1/8!!! 57 ft. 2/6/

Explanation.-5'', or 14, multiplied by 4', or , equals 1, or 20/1 8''; in the same manner, 6′ multiplied by 4'-24'', which added to 1' is 25′, or 2′ 1′′; 9 ft. multiplied by 4′ = 36′, which added to 2' is 38', or 3 ft. 2. Multiply by 6 ft. in the same manner, and we have for the second product 57 ft. 2′ 6', which being added to the first product, the result is 60 ft. 4' 7" 8.

60 ft. 4' 7" 8!!!

2. What is the entire surface of a floor 15 ft. 8' long and 16 ft. 3' wide? Ans. 254 sq. ft. 7'. 3. What is the amount of surface to be plastered in a room 18 ft. 3′ long, 15 ft. 2′ wide, and 10 ft. 3' high, allowing 8' for the width of the base-board? Ans. 917 sq. ft. 3′ 4′′.