# Ashok borrowed Rs. 16,000 at 10% simple interest. He immediately invested this money at 10% compound interest compounded half-yearly. Calculate Ashok’s gain in 18 months.

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**Hint:**Here, Ashok borrowed Rs. 16,000 at 10% simple interest for 18 months, so calculate interest Ashok has to pay at the end of 18 months, using simple interest formula. Also, he invested the same amount at compound interest compounded half-yearly, so find the interest he gets after 18 months using the compound interest formula and find the difference between the two interests to get the Ashok’s gain in 18 months.

**Complete step by step answer:**

Given, Ashok borrowed Rs. 16,000 at 10% simple interest for 18 months.

Here, Principal amount = Rs. 16000

Rate of interest = 10%

Time = 18 months or \[\dfrac{{18}}{{12}} = \dfrac{3}{2}\] years

Simple Interest = $\dfrac{{P \times R \times T}}{{100}}$ , where P is the principal amount, R is the rate of interest, T is time in years.

Simple interest = $\dfrac{{16000 \times 10 \times \dfrac{3}{2}}}{{100}} = \dfrac{{1600 \times 3}}{2} = 2400$

Therefore, interest Ashok has to pay is Rs. 2400.

Also, he immediately invested this money at 10% compound interest compounded half-yearly for 18 months.

Formula of compound interest, $CI = P\left[ {{{\left( {1 + \dfrac{R}{{100}}} \right)}^T} - 1} \right]$

In case of interest compounded half-yearly, $R \to \dfrac{R}{2}$ and $T \to 2T$

Here, Principal amount = Rs. 16000

Rate of interest = $\dfrac{{10}}{2} = 5\% $ [As compounded half-yearly]

Time = 2 ×18 months = 36 months = 3 years [As compounded half-yearly for 18 months]

Compound interest $ = 16000\left[ {{{\left( {1 + \dfrac{5}{{100}}} \right)}^3} - 1} \right] = 16000\left[ {\dfrac{{{{21}^3}}}{{{{20}^3}}} - 1} \right]$

⇒ Compound interest $ = 16000\left[ {\dfrac{{9261 - 8000}}{{8000}}} \right] = 2 \times 1261 = 2522$

Therefore, interest Ashok will get is Rs. 2522.

Difference between the two interest = Rs. 2522 – Rs. 2400 = Rs. 122

Therefore, Ashok’s gain in 18 months is Rs. 122

**Note:**

In these types of questions, you must know formulas of calculating simple interest and compound interest. Be careful when given is compounded half-yearly, as in that case the rate reduces to half and time becomes double. If you do not apply the formula of compounded half-yearly you will not get the result.

Alternatively, you apply both formulas of Compound interest and simple interest together and take the difference i.e. CI – SI = $ = P\left[ {{{\left( {1 + \dfrac{R}{{100}}} \right)}^T} - 1} \right] - \dfrac{{PRT}}{{100}}$, but formula is applicable when rate of interest and time are equal for both simple interest and compound interest.