A sum of money becomes Rs. 8000 after 3 years and Rs. 12167 after 6 years at compound interest. The sum is approximately
Solution
Correct Answer: Option C
Difference in amount after 3rd year and 6th year, i.e., for a tenure of 3 years = Rs.(12167 – 8000) = Rs. 4167
Thus, Compound Interest for 3 years = Rs. 4167
Let the principal be Rs. 8000 and CI be Rs. 4167 for a time period of 3 years.
We know the formula for compound interest
\(\Rightarrow {\rm{CI}} = \left[ {{\rm{P}}\left\{ {{{\left( {1 + \frac{{\rm{R}}}{{100}}} \right)}^t} - 1} \right\}} \right]\)
Where,
CI = Compound interest
P = Principal
R = Rate of interest
T = Time period
\(\begin{array}{l} \Rightarrow 4167 = \left[ {8000\left\{ {{{\left( {1 + \frac{{\rm{R}}}{{100}}} \right)}^3} - 1} \right\}} \right]\\ \Rightarrow \frac{{4167}}{{8000}} = \left[ {\left\{ {{{\left( {1 + \frac{{\rm{R}}}{{100}}} \right)}^3} - 1} \right\}} \right]\\ \Rightarrow \frac{{4167}}{{8000}} + 1 = \left[ {\left\{ {{{\left( {1 + \frac{{\rm{R}}}{{100}}} \right)}^3}} \right\}} \right]\\ \Rightarrow \frac{{12167}}{{8000}} = \left[ {\left\{ {{{\left( {1 + \frac{{\rm{R}}}{{100}}} \right)}^3}} \right\}} \right]\\ \Rightarrow {\left( {\frac{{23}}{{20}}} \right)^3} = {\left( {1 + \frac{{\rm{R}}}{{100}}} \right)^3}\\ \Rightarrow \frac{{23}}{{20}} = \;1 + \frac{{\rm{R}}}{{100}}\; \end{array}\)
⇒ R = (3 × 100)/20 = 15% p.a.
Let the original sum be Rs. x
Amount after 3 years = Rs. 8000
We know the formula for amount-
\(\Rightarrow {\rm{A}} = \left[ {{\rm{P}}\left\{ {{{\left( {1 + \frac{{\rm{r}}}{{100}}} \right)}^t}} \right\}} \right]\)
Where,
A =Amount
P = Principal
R = Rate of interest
T = Time period
\(\begin{array}{l} \Rightarrow 8000 = \left[ {{\rm{x}}\left\{ {{{\left( {1 + \frac{{15}}{{100}}} \right)}^3}} \right\}} \right]\;\\ \Rightarrow 8000 = \left[ {{\rm{x}}\left\{ {{{\left( {1 + \frac{3}{{20}}} \right)}^3}} \right\}} \right]\\ \Rightarrow 8000 = \left[ {{\rm{x}}\left\{ {{{\left( {\frac{{23}}{{20}}} \right)}^3}} \right\}} \right]\\ \Rightarrow 8000 = \left[ {{\rm{x\;}} \times {\rm{\;}}\frac{{12167}}{{8000}}} \right] \end{array}\)
⇒ x = (8000 × 8000)/ 12167
⇒ x = 5260 (approx)
Thus, the original sum is approximately Rs. 5,260.
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