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[Smart Math] Percentages Problem 5

Here’s and example of a SMART MATH problem for 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

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}-8000$$160r$ = 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

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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