# Compound Interest Calculation

In __________ years a sum will double at 5% per annum compound interest.

Options given are:

a. 15 years 3 months

b. 14 years 2 months

c. 14 years 3 months

d. 15 years 2 months

The way to solve is as below:

A = P [1+(r/100)]^n (^ - raised to the power of)

2P = P [1+(5/100)]^n

2 = [1+(5/100)]^n

log 2 = n * log(1.05) 0.3010 = n * 0.02118

Therefore n = 14.2069 years = 14 years and (0.2069 * 12) months = 14 years and 2.48 months.

Now the question is whether 2.48 months should be rounded to 2 months or 3 months?

Edit - The original question, asked and closed at MathEducators.SE contained the wording:

This is a academic question where I need to choose from the 4 options as given above. The book says answer is 14 years 2 months, but conceptually, before 2.48 months, the money does not double, so answer needs to be 14 years 3 months. Just asked this on forum to get to know if I am missing something.

This puts the nature of the question into better perspective.

• Convention is that 2.48 rounds down to 2. An awful problem. You solve, to the day, and then can get it wrong based on rounding? – JTP - Apologise to Monica May 20 '15 at 11:20
• @JoeTaxpayer Unless during the course of the class the teacher made a big deal about proper rounding, in which case it would be a fair exam question. At least in the sense that it would test if you had learned what was taught. Still a dumb real-life question. – Jay May 20 '15 at 13:33
• @Jay - Understood. I imagine this to be on a standardized test, and my rule of rounding is wrong for the reason given in the answer below. If the teacher address how he wants this approached, I'm fine with either method. (I updated the question with a quote from OP's original question. It agrees with me, but I like and agree with base64's answer. – JTP - Apologise to Monica May 20 '15 at 18:44
• Technically, none of the choices is correct. The question is fill in the blank, and none of the options given a sensible result. Fill in (c) and you get, "In 14 years 3 months years a sum will double at 5% per annum compound interest." That makes no sense. What's "3 months years"? – Jay May 20 '15 at 20:57
• As an educator I see this as a poorly designed set of answers. Every wrong answer must be the logical end to a incorrect path: bad decimal location, wrong type of log... – mhoran_psprep May 20 '15 at 21:17

## 1 Answer

Two ways to solve this.

1. Look at the answer. If the answer says 3 months, then using ceiling for similar questions. You have to act according to the exam conventions, not according to own feelings. Whether or not the answer is reasonable and applicable in real life is out of the question.

2. Ask yourself, did the investment double after 14 years 2 months? i.e. FV >= 2PV. Does a person who ran 99.72 meters in a 100-meter dash counted as touching the finish line?

• P.S. FV after 14 years 2 months = 1.9961 < 2. – base64 May 20 '15 at 11:54
• +1 for Does a person who ran 99.72 meters in a 100-meter dash counted as touching the finish line? – AbraCadaver May 20 '15 at 17:24
• (1) is fine, as long as the convention is clear. – JTP - Apologise to Monica May 20 '15 at 18:41
• @AbraCadaver Wwwweeeellll ... If you asked on a math test, "How long does it take to complete a 100 meter dash if the runner travels at 9 meters per second", and the choices are all integer numbers of seconds, I think any math teacher would say that the right answer is "11 seconds", not "12". 12 is the smallest number that is greater than the correct value, but that's not how we normally round. – Jay May 20 '15 at 21:00