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Question

Mahesh borrowed a certain sum for two years at simple interest from Bhim. Mahesh lent this sum to Vishnu at the same rate for two years compound interest. At the end of two years, Mahesh received Rs. $$410$$ as compound interest but paid Rs. $$400$$ as simple interest. Find the sum and rate of interest.


A
400,15%
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B
5000,10%
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C
4000,5%
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D
7000,15%
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Solution

The correct option is C $$4000,5\%$$
Let the sum borrowed by Mahesh be $$RS\ P$$ and rate of interest be $$R\% p.a$$
Simple interest $$(SI)=Rs\ 400, N=2$$ years
$$SI=\dfrac{P\times N\times R}{100}$$
$$\therefore P=\dfrac{SI\times 100}{N\times R}=\dfrac{400\times 100}{2\times R}$$
$$\therefore P=\dfrac{20000}{R}$$.......... $$(1)$$
Mahesh lent this sum to vishnu at the same rate for $$2$$ years at compound interest
$$CI=RS\ 410, N=2$$ years
$$\therefore CI=P\left(1+\dfrac{R}{100}\right)^{N}-P$$
$$\therefore CI=P\left[\left(1+\dfrac{R}{100}\right)^{N}-1\right]$$
$$\therefore 410=\dfrac{20000}{R}\left[\left(\dfrac{100+R}{100}\right)^{2}-1\right]$$ .......... [From $$(1)$$]
$$\therefore 410=\dfrac{20000}{R}\left[\dfrac{10000+R^{2}+200 R-10000}{10000}\right]$$
$$\therefore \dfrac{410\times R}{20000}=\dfrac{R^{2}+200 R}{10000}$$
$$\therefore \dfrac{410\times R}{20000}=\dfrac{R(R+200)}{10000}$$
$$\therefore \dfrac{410\times 10000}{20000}=R+200$$
$$\therefore 205=R+200$$
$$\therefore R=205-200$$
$$\therefore R=5\% p a$$
$$P=\dfrac{20000}{5}=Rs\ 4000$$
$$\therefore$$ The sum borrowed by Mahesh $$=Rs\ 4000$$ 
and rate of interest $$+5\%p a$$

Mathematics

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