Answer : Option D

Explanation :

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Solution 1
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Let the principal = Rs.x
and time = y years
Principal,x amounts to Rs.400 at 10% per annum in y years
Simple Interest = (400-x)
$MF#%\begin{align}&\text{Simple Interest = }\dfrac{\text{PRT}}{100}\\\\ &\Rightarrow (400-x) = \dfrac{x \times 10 \times y}{100}\\\\ &\Rightarrow (400-x) = \dfrac{xy}{10} \quad \color{#F00}{\text{--- (equation 1)}}\end{align}$MF#%
Principal,x amounts to Rs.200 at 4% per annum in y years
Simple Interest = (200-x)
$MF#%\begin{align}&\text{Simple Interest = }\dfrac{\text{PRT}}{100}\\\\ &\Rightarrow (200-x) = \dfrac{x \times 4 \times y}{100}\\\\ &\Rightarrow (200-x) = \dfrac{xy}{25} \quad \color{#F00}{\text{--- (equation 2)}}\end{align}$MF#%
$MF#%\begin{align}&\color{#F00}{\dfrac{\text{(equation 1)}}{\text{(equation 2)}} \Rightarrow} \dfrac{400-x}{200-x} = \dfrac{\left(\dfrac{xy}{10}\right)}{\left(\dfrac{xy}{25}\right)}\\\\ &\Rightarrow \dfrac{400-x}{200-x} = \dfrac{25}{10}\\\\ &\Rightarrow \dfrac{400-x}{200-x} = \dfrac{5}{2}\\\\ &\Rightarrow 800 - 2x = 1000 - 5x\\\\ &\Rightarrow 200 = 3x\\\\ &\Rightarrow x = \dfrac{200}{3}\\\\ &\color{#F00}{\text{Substituting this value of x in Equation 1, we get,}}\left(400 - \dfrac{200}{3}\right) = \dfrac{\left(\dfrac{200}{3}\right)y}{10}\\\\ &\Rightarrow \left(400 - \dfrac{200}{3}\right) = \dfrac{20y}{3}\\\\ &\Rightarrow 1200 - 200 = 20y\\\\ &\Rightarrow 1000 = 20y\\\\ &y = \dfrac{1000}{20} = 50\text{ years}\end{align}$MF#%
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Solution 2
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If a certain sum of money P lent out for a certain time T amounts to P₁ at R₁% per annum and to P₂ at R₂% per annum, then
$MF#%\text{P = }\dfrac{\text{P}_2\text{R}_1 - \text{P}_1\text{R}_2}{\text{R}_1-\text{R}_2}$MF#%
$MF#%\text{T = }\dfrac{\text{P}_1 - \text{P}_2}{\text{P}_2\text{R}_1 - \text{P}_1\text{R}_2} \times 100 \text { years} $MF#%