ANNUITIES AT SIMPLE INTEREST. 852. All problems in annuities at simple interest may be solved by combining the rules in Arithmetical Progression with those in Simple Interest. 5 WRITTEN EXERCISES, 853. 1. What is the amount of an annuity of $300 for years, at 6 per cent. simple interest? These sums form an arithmetical progression, in which the first term is the annuity, $300, the common difference is the interest of the annuity for 1 year, and the number of terms is the number of years. The sum of all the terms of this progression is $1680 (832), which is the amount of the annuity. 2. A father deposits annually for the benefit of his son, beginning with his tenth birthday, such a sum that on his 21st birthday the first deposit, at simple int., amounts to $210, and the sum due his son is $1860. Find the annual deposit, and at what rate per cent. it is deposited. OPERATION. 6 × (1st term + 210) = 1860. Hence, 1st term = 310 (832.) 210 100 = a. (210100) (121) = 10 d. (830.) = 4o ANALYSIS.-Here $210, the first deposit, is the last term; 12, the number of deposits, is the number of terms; and $1860, the final value of the annuity, is the sum of all the terms. Using the principle of 832, we find the first term to be $100, which is the annual deposit. By 830, the common difference is found to be $10; hence 10 per cent. is the required rate. 3. What is the amount of an annuity of $150 for 51 years, payable quarterly, at 14 per cent. per quarter? 4. What is the present worth of an annuity of $300 for 5 years, at 6 per cent.? 5. What is the present worth of an annuity of $500 for 10 years, at 10 per cent.? 6. In what time will an annual pension of $500 amount to $3450, at 6 per cent. simple interest? 7. Find the rate per cent. at which an annuity of $6000 will amount to $59760 in 8 years, at simple interest. 8. A man works for a farmer 1 yr. 6 mo., at $20 per month, payable monthly; and these wages remain unpaid until the expiration of the whole term of service. What is due the workman, allowing simple interest at 6 per cent. per annum ? ANNUITIES AT COMPOUND INTEREST. 854. All problems in annuities at compound interest may be solved by combining the rules in Geometrical Progression with those in Compound Interest. 1. What is the amount of an annuity of $300 for 5 years, at 6 per cent. compound interest? OPERATION. 300 x 1.065-300 =1691.13 .06 ANALYSIS.-At the end of the 5th year the following sums are due: The 5th year's payment The 4th year's payment + interest for 1 year These sums form a geometrical progression, in which the first term is the annuity, $300, the ratio is the amount of $1 for 1 year, and the number of terms is the number of years. The sum of all the terms of this progression is $1691.13 (843), which is the amount of the annuity. 2. What is the present worth of an annuity of $300 for 5 years, at 6 per cent. compound interest? OPERATION. 1691.13 1.338226 = 1263.71 worth of the annuity is ANALYSIS.-The amount of this annuity is $1691.13. The amount of $1 for 5 years, at 6 per cent. compound interest, is $1.338226 (587). Hence, the present $1691.13 or $1263.71. 1.338226' 3. Find the annuity whose amount for 25 years, at 6 per cent. compound interest, is $16459.35. 4. What is the present worth of an annuity of $700 for 7 years, at 6 per cent. compound interest? 5. An annuity of $200 for 12 years is in reversion 6 years. What is its present worth, compound interest at 6%? 6. A man bought a tract of land for $4800, which was to be paid in installments of $600 a year; how much money, at 6 per cent. compound interest, would discharge the debt at the time of the purchase? 7. What is the present value of a reversionary lease of $100, commencing 14 years hence, and to continue 20 years, compound interest at 5 per cent.? ANNUITIES. PROGRESSIONS. EVOLUTION. INVOLUTION. 855. 1. DEFS. SYNOPSIS { 1. A Power. 2. Involution. 3. Base, or Root. 4. Ex. ponent. 5. Square. 6. Cube. 7. Perfect Power. 2. PRINCIPLE. 3. 802. RULE. 1. For Integers. 2. For Fractions. 2. Cube Root, etc. 3. Evolution 4. Radical Sign. 5. Index. 5. 804. 1. DEFS. Principles, 1, 2, 3, 4. Rule, I, II, III. For Fractions. Geometrical Illustration. Principles, 1, 2, 3, 4. Rule, I, II, III, IV, V, VI. For Fractions. Roots of a Higher Degree. Rule. 1. Arithmetical Progression. 2. Terms. 3. Common Difference. 4. Increasing Arithmetical Progression. 5. Decreasing Arithmetical Progression. 2. Quantities considered. 3. 829. Rule, I, II. Formulæ. 4. 830. Rule. Formula. 5. 831. 6. 832. 1. DEFS. 1. Geometrical Progression. 2. Terms. 3. Ratio. 4. Increasing Geom. Prog. 5. Decreasing Geom. Prog. 6. Infinite Decreasing Geom. Prog. 1. Annuity. 2. Certain Annuity. 3. Perpetuity. 2. ANNUITIES AT SIMPLE INTEREST. Problems, how solved. 856. Mensuration is the process of finding the number of units in extension. Vertical. Horizontal. LINES. 857. A Straight Line is a line that does not change its direction. It is the shortest distance between two points. 858. A Curved Line changes its direction at every point. 859. Parallel Lines have the same direction; and being in the same plane and equally distant from each other, they can never meet. 860. A Horizontal Line is a line parallel either to the horizon or water level. 861. A Perpendicular Line is a straight line drawn to meet another straight line, so as to incline no more to the one side than to the other. A perpendicular to a horizontal line is called a verti cal line. ANGLES. 862. An Angle is the difference in the direction of two lines proceeding from a common point, called the vertex. Angles are measured by degrees. (301.) 863. A Right Angle is an angle formed by two lines perpendicular to each other. 864. An Obtuse Angle is greater than a right angle. 865, An Acute Angle is less than a right angle. All angles except right angles are called oblique angles. |