Find how long each of the following sums at the given rate must draw interest to amount to $1000: 8. In how many years will any principal double itself at 4%? at 5%? at 2%? at 6%? at 41%? SUGGESTION. The interest to be earned is equal to the principal. 9. In how many years will any principal treble itself at 5% ? 406. To find the principal. WRITTEN EXERCISES 1. What principal invested at 5% per annum will yield a yearly income of $1000? SOLUTIONS 1. Since $1 at 5% yields $.05 interest per year, as many dollars must be invested to yield $1000 per year as $.05 is contained times in $1000. $1000 $.05 20,000. Hence $20,000 must be invested. = 2. Let x number of dollars that must be invested. The principal equals the given interest divided by the interest on $1 for the given time at the given rate. What principal will yield an annual income of What principal will yield an interest of 8. $375 in 4 yr. at 5% ? 9. $900 in 2 yr. at 6% ? 10. $840 in 7 yr. at 3%? 11. $500 in 2 yr. 6 mo. at 4%? 12. $480 in 3 yr. 4 mo. at 6% ? 13. $1000 in 60 da. at 5%? 14. A man wishes to insure his life for an amount that invested at 4% will provide an annual income of $3000 after his death. For what amount shall he insure his life? 15. A man bequeathed to a college enough money to provide for 10 scholarships of $350 each, reckoning interest at 31%. Find the amount of the gift or endowment. Present Worth and True Discount 407. 1. What will be the amount of $100 in 1 year, if loaned at 6% interest? in 2 years? in 3 years? 2. If money is loaned at 6%, what is the value now of $106 due 1 year hence? What is the present value, or present worth, of $112 due 2 years hence? of $118 due in 3 years? 3. When money is worth 6 %, what sum should be deducted from a debt of $106 paid 1 year before it is due? What dis- ĥ count should be allowed on $1060 paid 1 year before it is due? 408. The present worth of a sum due at a future time is its cash value; or it is the sum that loaned at the current market rate would amount to the given sum in the given time. 409. The difference between the face value of a sum due at a future time and its present worth, or the sum that should be deducted for immediate payment, is sometimes called the true discount. The present worth may be regarded as the principal, the true discount as the interest, and the sum due at a future time as the amount. 410. 1. A man to pay me now. being worth 5 %? WRITTEN EXERCISES owing me $ 500 due in 1 yr. 3 mo. wishes How much should I accept in payment, money Find the true discount. SOLUTIONS 1. Since every dollar put at interest at 5% would amount to $1.0625 in 1 yr. 3 mo., it will require as many dollars now to amount to $500 in 1 yr. 3 mo. as $ 1.0625 is contained times in $500. $500 ÷ $1.0625 = 470.59 (to the nearest .01). Hence I should accept a cash payment of $470.59. The true discount is $500 - $470.59, or $ 29.41. 2. In 1 yr. 3 mo., x dollars at 5% will amount to 1.0625 x dollars, or to the sum due at that time, $500. Hence, Hence present worth = $470.59; true discount = $500 - $470.59, or $29.41. The present worth equals the given amount divided by the amount of $1 for the given time at the given rate. 2. A merchant bought a bill of goods amounting to $800 on 60 days' credit. Find the equivalent cash value of the goods, if money was worth 6%. 3. What is the present worth of $1000 due 2 yr. 6 mo. from date, if money can be borrowed at 6% ? 4. Find the true discount on $5000 paid 6 months before it is due, if money is worth 5% per annum. 5. What sum of money put at interest for 9 months at 31% will amount to $1642 ? 6. A person just 20 years old is named in a will as heir to $5000 due with interest when he is 21 years old. What is the present worth of his inheritance, money being worth 5% ? 7. I have an endowment policy for $ 5000 due in 10 months. Find its present value, money being worth 6%. 8. I am offered a credit of 90 days on a bill for $ 2500, or 1% off for cash. Which is the better offer, credit or cash, and how much, if money can be borrowed at 4 %? 9. Mr. Brown offers me $8800 in cash for a house and lot, and Mr. White offers $ 6000 in cash and $3000 payable in 1 year without interest. Which is the better offer, and how much better, if money is worth 6 % ? if money is worth 8 % ? 411. In some states when a written agreement contains the expression "with interest payable annually," simple interest may be collected upon the principal and upon each year's interest which has not been paid when due. This is called annual interest. WRITTEN EXERCISES 412. 1. No interest having been paid, find the amount due in 3 years 6 months 6 days on $1000, with interest payable annually at 6%. Principal. SOLUTION Simple interest on $1000, at 6%, for 3 yr. 6 mo. 6 da. The interest for each year is $60 The 1st annual int., $60, remains unpaid for 2 yr. 6 mo. 6 da. The amount equals the principal, plus the simple interest for the entire time, plus the interest on each year's interest for the time it remains unpaid. 2. Find the amount of $5850 in 4 years 3 months, with interest payable annually at 4%. 3. Find the amount of $24,000 in 3 years 3 months 12 days, with interest payable annually at 5%. COMPOUND INTEREST 413. Interest on the principal the principal and its unpaid interest, combined at regular intervals, is called compound interest. Interest may be compounded with the principal annually, or semiannually, or quarterly, etc. Compound interest cannot usually be enforced by law, even though it is specified in the contract. Nevertheless the subject is of importance since it is at the basis of computations concerning investments, for there is nothing to hinder a man from drawing his interest and investing it. WRITTEN EXERCISES 414. 1. Find the amount of $400 for 2 yr. 4 mo. at 6%, interest compounded annually; also find the compound interest. Since the amount of $400 at compound interest is $458.43, the compound interest is $458.43 - $ 400, or $58.43. NOTE. Unless otherwise specified, interest is understood to be compounded annually. If compounded semiannually, the rate must be considered one half the annual rate mentioned; if quarterly, one fourth, etc. When the time consists of years, months, and days, the amount is to be found for the greatest number of entire periods, as years, half years, quarter years, etc., and the simple interest upon this for the rest of the time. |