EXAMPLE.-What principal in 4 yr. 9 mo. at 5% will give $152 interest? yr., EXPLANATION. - 4 being in the divisor, the fraction must be reduced to the form of an improper fraction and inverted. The rate also appears in the inverted form of 100, since .05 cannot be inverted. EXAMPLE. What principal at 33% will, in 2 yr. 3 mo. 15 da., give $374 interest? SOLUTION.-Changing the time to 5 years, writing 33% as 7, and applying the formula, we have 32. Rule. To find the principal when the interest, rate, and time are given, divide 100 times the interest by the product of the time in years and the rate per cent. EXAMPLES FOR PRACTICE. 33. Find the principal, when 1. Interest $96, time = 2 years, rate = 4%. 4. Interest 6. 31%. Interest = $23.75, time = 1 yr. 8 mo. 24 da., rate = 51%. 7. Interest = $43.60, time 2 yr. 11 mo. 12 da., rate = 7%. 8. Interest $124.30, time = 3 mo. 20 da., rate = 33%. Answers.-(1) $1,200; (2) $875; (3) $600; (4) $2,400; (5) $4,500; (6) $249.13-; (7) $211.14-; (8) $11,094.55-. 34. Given the principal, the interest, and the time, to find the rate per cent. EXAMPLE.-At what rate per cent. will $480 in 3 years 10 months give $92 interest? 35. Rule.-Change the time to years, and divide 100 times the interest by the product of the principal and the time. EXAMPLES FOR PRACTICE. 36. Find the rate per cent. when = 1. Principal = 2. Principal 3. Principal 4. Principal = = $2,875, time = 4 yr. 7 mo. 6 da., interest = $529. $144.40. $760, time 3 yr. 9 mo. 18 da., interest = $1,260, time = 2 yr. 1 mo. 10 da., interest = $119.70. $2,340, time 2 yr. 6 mo. 20 da., interest = $328.90. 5. Principal = $4,870, time 3 yr. 5 mo. 24 da., interest $7,200, time = 123 days, interest = $1,500, time 6. Principal 7. Principal = = = 8. Principal $1,600, time = = = = = = $1,017.83. $114.80. 1 yr. 9 mo. 18 da., interest = $99. 5 yr. 7 mo. 6 da., interest = $380.80. Answers.—(1) 4%; (2) 5%; (3) 41%; (4) 51%; (5) 6%; (6) 43%; (7) 3}%; Otherwise expressed, this means that time (in years) = (100 × interest) ÷ (principal X rate per cent.). EXAMPLE.-In what time will $4,480 at 6% give $871.36 interest? 389 EXPLANATION.—The result is obtained in years, and must be reduced to years, months, and days. 38 years 3 years. 29 128 = Since there are 12 months in one year, 2 of a year = 20 29×12 120 = months, or 2 months. Since there are 30 days in one 2 month, of a month = 9 × 30 10 = 27 days. Hence, the required time is 3 yr. 2 mo. 27 da. Ans. 38. Rule.-To find the time, when the principal, interest, and rate are given, divide 100 times the interest by the product of the principal cnd the rate per cent., and reduce the result, which is years, to years, months, and days. 39. EXAMPLES FOR PRACTICE. Find the time, when 1. Principal = $4,800, interest = $652, rate = 6%. = $680, interest = $163.20, rate = 5%. 2. Principal = $26.325, rate = 41%. $35.23, rate = 4%. $1,080, interest = $112.50, rate = 3%. $1,800, interest = $63, rate = 31%. Principal = $1,050, interest = $45.50, rate = 5%. 8. Principal $1,000, interest = $143, rate = 61%. 7. = = 9. Principal = $2,400, interest = $74.80, rate = 51%. Answers.—(1) 2 yr. 3 mo. 5 da.; (2) 4 yr. 9 mo. 18 da.; (3) 1 yr. 7 mo. 15 da.; (4) 2 yr. 7 mo. 6 da. ; (5) 3 yr. 5 mo. 20 da. ; (6) 1 yr.; (7) 10 mo. 12 da.; (8) 2 yr. 2 mo. 12 da. ; (9) 6 mo. 24 da. Given the amount, time, and rate, to find the 40. principal. Since at 6% the interest of $1 for, say, 2 years, is 12 cents, the amount is $1+12 cents $1.12. Each $1 of the principal will, in like manner, amount to $1.12. Hence, if we divide the amount of any principal in a given time at a specified rate by the amount of $1 for that time and rate, it will give the number of times $1 is contained in the principal. Or using the rate per cent, instead of the rate, we may, by means of the following formula, express the process of finding the principal: That is, (100X amount)÷ (100+rate per cent. X time) = principal. EXAMPLE.-What principal will, in 5 yr. 6 mo., at 4%, amount to $591.70 ? 100 X 591.70 59,170 = 122 = $485. Ans. 41. Rule. To find the principal, when the amount, rate, and time are given, divide 100 times the amount by 100 increased by the product of the rate per cent. and the time. NOTE.-In using this rule, and the formula given in Art. 40, care must be taken that the time be expressed in years and fractions of a year. EXAMPLES FOR PRACTICE. 42. Find the principal that will amount 1. To $1,005 in 5 yr. 8 mo. at 6%. 3. To $2,985 in 2 yr. 1 mo. 10 da. at 5%. 5. To $2,353.50 in 1 yr. 6 mo. 12 da. at 3%. Answers.—(1) $750; (2) $3,000; (3) $2,700; (4) $1,200; (5) $2.250. TRUE DISCOUNT. 43. The student has learned that any deduction made from a debt or other obligation is a discount. In making such deductions, the element of time may or may not be considered. When time is considered, we have one of the applications of interest. True discount is discount when time is considered and no interest is allowed on the discount. True discount corresponds exactly to the problems of Interest given in Arts. 40-42—the case in which the amount, rate, and time are given, to find the principal. The terms employed, however, are different. The principal is called the present worth. The interest is called the true discount. The amount is called the debt, or obligation. True discount is so called to distinguish it from bank discount, which will be treated later. 44. The present worth of an obligation is a sum such that, if it be put at interest at a specified rate for a given time, it will amount to the obligation. Thus, if the specified rate is 5%, a debt of $105 due in one year is worth $100 now, since $100 placed at interest at 5% will in one year amount to $105. 45. True discount is the difference between a debt due at a future time and its present worth. Thus, $5 in the illustration given above is the true discount of $105 due in one year, when the rate of discount is 5%. 46. Given the debt, rate of discount, and time, to find the present worth and the discount. The present worth may be found by means of the formula of Art. 40, or by the rule of Art. 41. Thus, |