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Percentages Smart Math

[Smart Math] Percentages Problem 5

Here’s and example of a SMART MATH problem for PERCENTAGES.

Percentages

Problem

If the rate of Compound interest and Simple interest is the same and the difference between the interest earned on a principal of $8000 over 2 years is $180, find the interest rate.

  1. 12%
  2. 15%
  3. 10%
  4. 18%
  5. 20%

The Usual Method

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Let the rate of interest be ‘r’%.

Simple interest earned over two years

= \frac{2\times r\times 8000}{100}=160r

Compound interest earned over two years

= 8000\left( 1+\frac{r}{100} \right)^{2}-8000

Hence, difference is given by

8000\left( 1+\frac{r}{100} \right)^{2}-8000160r = 180

\therefore 8000\left[ \left( 1+\frac{r}{100} \right)^{2}-1 \right]-160r=180

\therefore 50\left( 1+\frac{r}{50}+\frac{r^{2}}{10000}-1 \right)-r=\frac{180}{160}

\therefore r+\frac{r^{2}}{200}-r=\frac{180}{160}=\frac{9}{8}

\therefore r^{2}=\frac{9\times 200}{8}=225

\therefore r=15%

(Ans: 2)

Estimated Time to arrive at the answer = 90 seconds.

Using Technique

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Remember that the difference of $180 is nothing but the interest on interest earned in the 2nd year.

Interest earned in 1st year by Compound as well as Simple is the same = 80r

Interest earned on this interest in the next year = \frac{80r^{2}}{100}=180

\therefore r^{2}=\frac{18000}{80}=225

\therefore r=15% i.e option ‘2’.

(Ans: 2)

Estimated Time to arrive at the answer = 10 seconds.
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