2. A teacher being asked how many scholars he had, said, if I had as many, half as many, and one quarter as many more, I should have 264, how many had he? Ans. 96 scholars. 3. What number is it whose, and make 235? Ans. 300. 4. What is the age of a person who says, that if of the years I have lived be multiplied by 7 and 3 of them be added to the product, the sum will be 292? Ans. 60 years. 5. A's age is double that of B, and B's is triple that of C. and the sum of all their ages is 140, what is each person's age? Ans. A's 84, B's 42, and C's 14 yrs. 6. A certain sum of money is to be divided among 4 persons in such a manner that the first shall have of it, the second, the third, and the fourth which is $28, what is the sum? the remainder, Ans. $112. 7. What sum at 6 per cent. per annum, will amount to $1032 in 12 years? DOUBLE POSITION.* Ans. $600. Double Position teaches to resolve questions, by making two suppositions of false numbers. Those questions in which the results are not proportional to their positions, belong to this rule. * This rule is founded on the suppositon, that the first error is to the second, as the difference between the true and first supposed number is to the difference between the true and second supposed number. When this is not the case, the exact answer to the ques tions cannot be found by this rule. 1. John asked James how much his horse cost, who answered that if he cost him three times as much as he did, and $15 dollars more, he would stand him in $300, what was the price of the horse? Ans. $95. The following rule will be found very concise when the signs of or are alike subtract, when unlike add: Illustration. When both signs are + or both signs subtract, but when one sign is + and the other add. If the errors are alike, divide the difference of the products by the difference of the errors; but, if unlike, divide the sum of the products by the sum of the errors. 1. Divide 15 into 2 such parts, so that when the greater is multiplied by 4, and the less by 16, their products will be equal? Ans. 12 and 3. 2. Divide 21 into 2 such parts, so that when the greater is divided by the lesser, and the lesser by the greater, and afterwards the greater quotient multiplied by 5, and the lesser by 125, their products will be equal? Ans. 17 and 31. 3. A lady being asked her age thus replied: Ans. 28 years. 4. A laborer was hired 80 days upon this condition, that for every day he was idle, he should pay his employer 50 cents, and for every day he was at work he should receive 75 cents, at the expiration of the time, he received $25; now, how many days did he work, and how many days was he idle? Ans. he worked 52 days, and was idle 28. 5. Two persons, A. and B. have the same income, A. saves one-fifth of his every year, but B. by spending $150 per annum more than A. at the end of 8 years finds himself $400 in debt, what is their income, and what does each spend per annum? Ans. Their income is $500 per annum, A. spends $400, and B. $550. 6. A man had two silver cups of unequal weight, having one cover to both, 5 oz.; now if the cover is put on the lesser cup, the whole will be double the weight of the greater cup. Again, if the cover be put on the greater cup, it will be 3 times as heavy as the lesser cup, what is the weight of each cup? Ans. 3 ozs. lesser-greater 4 ozs. 7. Divide 10 into 2 such parts, so that 9 times the lesser number will be equal to 6 times the greater? Ans. 6 and 4. 8. The sum of 2 numbers is 50; now, if you divide the greater part by 7, and multiply the lesser by 3, the sum of the quotient and product will be equal to the given number. Required the parts? Ans. 35 and 15. 9. A young gentleman having asked his father how old he was, received the following reply: seven years ago my age was in a three-fold ratio to yours; but, if we should both happen to live seven years hence, my age shall be just double that of yours. Required their several ages? ́Ans. 49 and 21 years. ANNUITIES. Table First, shewing the amount of $1.00 annuity from One year to Thirty-seven. Yrs. 5 cent. 6 per per cent. Yrs. 5 per cent. 6 per cent. Yrs. 5 cent. 6 per per cent. 1 1,00000 4,310125 1,00000 16 23,65749 5,63709 20 33,06595 36,78559 32 75,29882 39,99272 90,88977 43,39229 33 80,06377 97,34316 46,99582 50,81557 34 85,06696 104,18375 54,86451 ANNUITIES AT COMPOUND INTEREST. DEFINITIONS.-Annuity is a certain sum of money to be paid at regular periods, either for a limited time or for ever. 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 each payment added to their sum. To find the amount of an annuity at compound interest, we adopt the following. RULE.-Make $1 the first term of a geometrical series, and the amount of $1 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. CASE I. Example.-1. What will an annuity of $60 per annum, payable yearly, amount to in 4 years, at 6 per cent? 2 3 1x1.06×1.06 + 1.06 4.37461, the tabular number, answering to 4 years, at 6 per cent. 4 Table 1, 1.06--1 1.06-1x60 $262.47c. 6m. Ans. or 4.374616×60 = $262.47c. 6m. as before. 2. What will an annuity of $30 amount to in 3 years? Ans. $95.50c. 8m. 3. What will an annuity of $200 amount to in 5 years per cent? Ans. $1127.41c. 8m. at 6 CASE II. When the payments are to be made half yearly or quarterly, the amount for the given time, found as before, in table first, multiplied by the tabular number answering |