SECTION LXVII. ANNUITIES AT COMPOUND INTEREST. AN annuity is a certain sum of money to be paid at regular periods, either for a limited time or for ever. The present worth or value of an annuity is that sum which, being improved at compound interest, will be sufficient to pay the annuity. The amount of an annuity is the compound interest of all the payments added to their sum. To find the amount of an annuity at compound interest. RULE.Make $1.00 the first term of a geometrical series, and the amount of $1.00 at the given rate per cent. the ratio. Carry the series to so many terms as the number of years, and find its sum. Multiply the sum thus found by the given annuity, and the product will be the amount. EXAMPLES. 1. What will an annuity of $60 per annum, payable yearly, amount to in 4 years, at 6 per cent. ? 6 1+1.06+1.06+1.06=4.374616. 4.374616 × 60 = $262.47,6+ Answer. Ans. $2818.54,6+. 2. What will an annuity of $500.00 amount to in 5 years, at per cent. ? 3. What will an annuity of $1000.00, payable yearly, amount to in 10 years? Ans. $13180.79,4+-. 4. What will an annuity of $30.00, payable yearly, amount to in 3 years? Ans. $95.50,8+. To find the present worth of an annuity. As the first payment is made at the end of the year, its present worth or value is a sum that will amount in one year to that payment; and as the second payment is made at the end of the second year, its value is a sum that will, at compound interest, amount in two years to that payment; and the same principle is adopted for the third year, fourth year, &c. This may be illustrated in the following question. 5. What is the present worth of an annuity of $1.00, to continue 5 years, at compound interest? The present worth of $ 1.00 for 1 year = $0.943396 = $0.839619 $0.792094 $0.747258 $4.212363 By the above illustration, we perceive that the present worth of an annuity of $ 1, to continue 5 years, is $4.21,2+. Hence, having found the present worth of an annuity of $1 for any given time by Section LXVI., the present worth of any other sum may be found by multiplying it by the present worth of $1 for that time. RULE. Multiply the present worth of the annuity of one dollar for the given time by the given annuity, and the product is the present worth required. Or, find the amount of the annuity by the last rule, and then find its present worth. 6. What is the present worth of an annuity of $ 60, to be continued 4 years, at compound interest? First Method. The present worth of $ 1.00 for 1 year $0.943396 $0.889996 :$0.839619 $0.792093 $3.465104 $3.465104 × 60 = $207.90,6+ Answer. X Second Method. $4.374616 x .792093 = $ 3.465102 × 60 = $207.90,6+ Ans. 7. A gentleman wishes to purchase an annuity, which shall afford him, at 6 per cent. compound interest, $500 a year for What sum must he deposit in the annuity office Ans. $3680.04+. annuity of $1000, to ten years. to produce it? 9. What is the present worth of an continue 3 years? Ans. $7360.08. annuity of $1728, to Ans. $4618.96. By the assistance of the following tables, questions in annuities may be easily performed. TABLE I. Showing the amount of $1 annuity from 1 year to 40. TABLE II. Showing the present value of an annuity of $1 from 1 year to 40. 10. What is the present worth of an annuity of $200, at 5 per cent. compound interest, for 7 years? Ans. $1157.27+. 11. What is the present worth of an annuity of $ 300, to continue 8 years, at 6 per cent. compound interest ? Ans. 1862.93,8+. 12. What is the present value of an annuity of $ 100, at 6 per cent. for 9 years? Ans. 680.16,9+. Questions to be performed by the preceding tables. 13. What will an annuity of $30 amount to in 11 years, at 6 per cent. ? By Table I., the amount of $1 for 11 years is $14.971643; therefore, $ 14.971643 × 30 = $449.14,9+ Answer. 14. What is the present worth of an annuity of $ 80 for 30 years, at 5 per cent. ? By Table II., the present worth of $1 for 30 years is $15.372451, therefore $15.372451 × 80 = $1229.79,6+ Ans. 15. What will an annuity of $800 amount to in 25 years, at 5 per cent. ? Ans. $38181.67,9+. amount to in 30 years, at Ans. $3162.32,7+. 6 16. What will an annuity of $40 cent.? per 17. Required the present worth of continue 40 years, at 6 per cent. an annuity of $ 500, to Ans. $7523.14,8+. 18. A certain parish in the town of B., having neglected for 6 years to pay their minister's salary of $700, what in justice, provided he has preached the truth, should he receive? Ans. $4882.72,3. SECTION LXVIII. ASSESSMENT OF TAXES. A TAX is a duty laid by government, for public purposes, on the property of the inhabitants of a town, county, or State, and also on the polls* of the male citizens liable by law to as sessment. A tax may be either general or particular; that is, it may affect all classes indiscriminately, or only one or more classes. * Poll is said to be a Saxon word, meaning head. In the constitution it means a person; that is, a person who is liable to taxation. Taxes may be either direct or indirect; that is, they may either be imposed on the incomes or property of individuals, or on the articles on which these incomes or property are expended. The method of assessing town taxes is not precisely the same in all the States, yet the principle is virtually the same. In some of the States the poll tax is more than in others. The following is the law regulating taxation in Massachusetts (see Revised Statutes, page 79): — "The assessors shall assess upon the polls, as nearly as the same can be conveniently done, one sixth part of the whole sum to be raised; provided the whole poll tax assessed in any one year upon any individual for town and county purposes, except highway taxes, shall not exceed one dollar and fifty cents; and the residue of said whole sum to be raised shall be apportioned upon property"; that is, on the real and personal estate of individuals which is taxable. RULE FOR ASSESSING TAXES. First take an inventory of all the taxable property, real and personal, in the town or county, and then the number of polls liable to taxation. Multiply the sum assessed on each poll by the number of taxable polls in the town. Subtract this amount from the sum to be raised by the town. Then as the whole valuation of the town is to the sum to be raised, after having deducted the amount to be paid by the polls, so is the amount of each man's real and personal estate to his tax. Or, if the sum to be raised on property be divided by the valuation of the town, the quotient will be the sum to be paid on each dollar of an individual's real or personal estate. Multiply each man's property by this sum, and the product will be the amount of his taxes. The town of B. is to be taxed $4109. The real estate of the town is valued at $493,000, and the personal property at $177,000. There are 506 polls, each of which is taxed $1.50. What is John Smith's tax, whose real estate is valued at $3700 and his personal at $2300, he paying for 6 polls? And what I will be the tax on $1.00 ? OPERATION. $ 1.50 × 506 = $759, amount assessed on the polls. $493,000 +$177,000=$ 670,000, amount of taxable property. $4109 $759 $3350, amount to be assessed on property. $670,000 $3350 :: $1 : $.005, to be assessed on each dollar. $3700 × .005= $18.50, tax on Smith's real estate. $2300 x .005 $11.50, tax on Smith's personal estate. $1.50 X 6: $9.00, tax on 6 polls. $18.50+$11.50 +$9.00 $ 39.00, amount of Smith's tax. |