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Click here to watch this insightful video. World’s best universities The principle of compounding growth is used extensively in the financial world to transform small savings into a big corpus over time. It’s also the underlying idea behind MBA topics such as time value of money and discounted cash flow (DCF) valuation. Learn about simple and compound interest concepts as you’ll need them not only for entrance exams but in the real world too, especially after you become rich and famous. Simple and Compound interest Problems and SolutionsHere is a list of some basic definition and formulas to solve problems on Interest. Principal: This is the sum of money lent or borrowed. Interest: This is the extra money paid for taking the money as loan. This is often expressed as a percentage. Say, the interest is 10% on a loan of Rs. 100. Then the interest in amount is Rs. 10 and at the end of the year, the amount to be paid is Rs. 110. Time: This is the time period for which the money is lent or the time period in which the money has to be returned with interest. Simple InterestAs the name implies, the calculation of simple interest is pretty simple. Multiply the principal amount with the number of years and the rate of interest. Simple Interest Formula: Abbreviated as SI = PTR/100 Compound InterestIn compound interest, the principal amount with interest after the first unit of time becomes the principal for the next unit. Say, when compounded annually for 2 years, the principal amount with interest accrued at the end of first year becomes the principal for the second year. Compound Interest Formula: Abbreviated as Amount = P * [1 + R/100]t, when compounded annually. Sometimes, the interest is also calculated half-yearly or quarterly. When compounded semi-annually or half-yearly, Amount = P[1 + (R/2)/100]2t When compounded quarterly, Amount = P[1 + (R/4)/100]4t Present worth of Principal P due t years hence is given by: P/[1+ R/100]t Sample problems and solutionsLet us work on some examples to understand the concepts and the differences. Problem 1. A sum of Rs. 25000 becomes Rs. 27250 at the end of 3 years when calculated at simple interest. Find the rate of interest. Solution: Simple interest = 27250 – 25000 = 2250 Time = 3 years. SI = PTR / 100 → R = SI * 100 / PT R = 2250 * 100 / 25000 * 3 → R = 3%. Problem 2. Find the present worth of Rs. 78000 due in 4 years at 5% interest per year. Solution: Amount with interest after 4 years = Rs. 78000 Therefore, simple interest = 78000 – Principal. Let the principal amount be p. 78000 – p = p*4*5/100 → p=13000 Principal = 78000 – 13000 = Rs. 65000 Problem 3. A certain principal amounts to Rs. 15000 in 2.5 years and to Rs. 16500 in 4 years at the same rate of interest. Find the rate of interest. Solution: Amount becomes 15000 in 2.5 years and 16500 in 4 years. Simple interest for (4-2.5) years = 16500 – 15000 Therefore, SI for 1.5 years = Rs. 1500. SI for 2.5 years = 1500/1.5 * 2.5 = 2500 Principal amount = 15000 – 2500 = Rs. 12500. Rate of Interest = 2500 * 100 / 12500 * 2.5 → R = 8%. Problem 4. Find the compound interest on Rs. 3000 at 5% for 2 years, compounded annually. Solution: Amount with CI = 3000 (1+ 5/100)2 = Rs. 3307.5 Therefore, CI = 3307.5 – 3000 = Rs. 307.5 Problem 5. Find the compound interest on Rs. 10000 at 12% rate of interest for 1 year, compounded half-yearly. Solution: Amount with CI = 10000 [1+ (12/2 * 100)]2 = Rs. 11236 Therefore, CI = 11236 – 10000 = Rs. 1236 Problem 6. The difference between SI and CI compounded annually on a certain sum of money for 2 years at 8% per annum is Rs. 12.80. Find the principal. Solution: Let the principal amount be x. SI = x * 2 * 8 / 100 = 4x/25 CI = x[1+ 8/100]2 – x → 104x/625 Therefore, 104x/625 – 4x/25 = 12.80 Solving which gives x, Principal = Rs. 2000. Problem 7. Find the simple interest on Rs. 5000 at a certain rate if the compound interest on the same amount for 2 years is Rs. 253.125. Solution: Let the rate of interest be r. 5000[1+ r/100]2 = 5000+253.125 → [1+r/100]2 = 5253.125/5000 Solving which gives [1+ r/100]2 = 1681/1600 → 1+r/100 = 41/40 → r = 2.5 Therefore, SI = 5000* 2 * 2.5/ 100 = Rs. 250. Problem 8. A certain amount becomes Rs. 5760 in 2 years and Rs. 6912 in 3 years. What is the principal amount and the rate of interest? Solution: Interest on Rs. 5760 for 1 year = 6912 – 5760 = Rs. 1152 Therefore, Rate of interest for 1 year = 100*1152/5760*1 = 20% Let the principal be p. Then, Principal = p[1+ 20/100]2 = 5760 Solving which gives Principal = Rs. 4000 Problem 9. How long will it take a certain amount to increase by 30% at the rate of 15% simple interest? Solution: Let the principal be Rs. x Simple interest = x*30/100 = 3x/10 T = 100*SI/PR = 100*3x/10 / x*15 = 2% Alternatively, this can be solved by considering principal amount to be Rs. 100. Then simple interest becomes Rs. 30. Then, T = 100*30/100*15 = 2% Simple and Compound Interest Quiz: Solve these problemsProblem 1: Click here A money lender lent Rs. 1000 at 3% per year and Rs. 1400 at 5% per year. The amount should be returned to him when the total interest comes to Rs. 350. Find the number of years. A. 3.5 B. 3.75 C. 4 D. 4.5 Answer 1: Click here A. Explanation (1000*t*3/100) + (1400*t*5/100) = 350 → t = 3.5 Problem 2: Click here
Find the present worth of Rs. 20872.5 due 2 years hence at 10% rate of interest. A. 17500 B. 17520 C. 17750 D. 17250 Answer 2: Click here
D. Explanation: Present worth = 20872.5/[1+ 10/100]2
Grandpa Jack wants to help his grandson, Little Jack, with college expenses. Little Jack is currently 3 years old. If Grandpa Jack invests $5,000 in a college savings account earning 5% compounded yearly, how much money will he have in 15 years when Little Jack is 18? Possible Answers:
Correct answer: Between $10,000-$10,500 Explanation: To solve this, we can create an equation for the value based on time. So if we let t be the nmbers of years that have passed, we can create a function f(t) for the value in the savings account. We note that f(0) =5000. (We invest 5000 at time 0.) Next year, he will have 5% more than that. To find our total value at the end of the year, we multiply 5,000 * 1.05 = 5,250. f(1) = 5000(1.05)=5,250. At the end of year 2, we will have a 5% growth rate. In other words, f(2) = (1.05)* f(1). We can rewrite this as
Cherry invested Possible Answers:
Correct answer:
Explanation: The monthly rate is 3 years = 36 months According to the compound interest formula and here
Scott wants to invest $1000 for 1 year. At Bank A, his investment will collect 3% interest compounded daily while at Bank B, his investment will collect 3.50% interest compounded monthly. Which bank offers a better return? How much more will he receive by choosing that bank over the other? Possible Answers:
Correct answer: Explanation: Calculate the total amount from each bank using the following formula: Bank A: Bank B:
Bryan invests $8,000 in both a savings account that pays 3% simple interest annually and a certificate of deposit that pays 8% simple interest anually. After the first year, Bryan has earned a total of $365.00 from these investments. How much did Bryan invest in the certificate of deposit? Possible Answers:
Correct answer: Explanation: Let
Barry invests $9000 in corporate bonds at 8% annual interest, compounded quarterly. At the end of the year, how much interest has his investment earned? Possible Answers:
Correct answer: Explanation: Use the compound interest formula substituting Subtract 9,000 from this figure - the interest earned is $741.89
Tom deposits his $10,000 inheritance in a savings account with a 4% annual interest rate, compounded quarterly. He leaves it there untouched for six months, after which he withdraws $5,000. He leaves the remainder untouched for another six months. How much interest has Tom earned on the inheritance after one year? Possible Answers:
Correct answer: Explanation: Since in each case the interest is compounded quarterly, the annual interest rate of 4% is divided by 4 to get 1%, the effective quarterly interest rate. The $10,000 remains in the savings account six months, or two quarters, so 1% is added twice - equivalently, the $10,000 is multiplied by 1.01 twice: $5,000 is withdrawn from the savings account, leaving This money is untouched for six months, or two quarters, so again, we multiply by 1.01 twice: Subtract $5,000 to get the interest:
On January 1, Gary borrows $10,000 to purchase an automobile at 12% annual interest, compounded quarterly beginning on April 1. He agrees to pay $800 per month on the last day of the month, beginning on January 31, over twelve months; his thirteenth payment, on the following January 31, will be the unpaid balance. How much will that thirteenth payment be? Possible Answers:
Correct answer: Explanation: 12% annual interest compounded quarterly is, effectively, 3% interest per quarter. Over the course of one quarter, Gary pays off First quarter: Second quarter: Third quarter: Fourth quarter: The thriteenth payment, with which Gary will pay off the loan, will be $913.16.
Jessica deposits $5,000 in a savings account at 6% interest. The interest is compounded monthly. How much will she have in her savings account after 5 years? Possible Answers:
None of the other answers are correct.
Correct answer: Explanation: where
A real estate company is considering whether to accept a loan offer in order to develop property. The principal amount of the loan is $400,000, and the annual interest rate is 7% compounded semi-anually. If the company accepts the loan, what will be the balance after 4 years? Possible Answers:
Correct answer: Explanation: Recall the formula for compound interest: Plug in the values given in the question:
Nick found a once-in-a-lifetime opportunity to buy a rare arcade game being sold at a garage sale for $5730. However, Nick can't afford that right now, and decides to take out a loan for $1000. Nick didn't really read the fine print on the loan, and later figures out that the loan has a 30% annualy compounded interest rate! (A very dangerous rate). How much does Nick owe on the loan 2 years from the time he takes out the loan? (Assume he's lazy and doesn't pay anything back over those 2 years.) Possible Answers:
Correct answer: Explanation: For compound interest, the amount Nick owes is where We have Hence our answer is $1690.
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A sum of p500 was lent for two years at 2% compound interest. the interest for two years will be:
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