Question

Two equal sums of money are lent at the same time at $8\%$ and $7\%$ per annum simple interest. The former is recovered $6$ months earlier than the later and the amount in each case is Rs. $2560$. The sum and time for which the sums of money are lent out are

• A. Rs. $1500,3.5$ year and $4$ year
• B. Rs. $2000,3.5$ years and $4$ years
• C. Rs. $2000,4$ years and $5.5$ years
• D. Rs. $3000,4$ years and $4.5$ years

Answer 1

Answer: (B)

Rs. $2000,3.5$ years and $4$ years

Let the sum be Rs. $x$.
Therefore, Rs. $x$ are lent at $8\%$ for $t$ years and Rs. $x$ are lent at $7\%$ for $(t+\frac{1}{2})$ years.
$\therefore \frac{x\times t\times 8}{100}+x=2560$
$\Rightarrow 8tx+100x=256000$ .....(i)
and $\frac{x\times (2t+1)\times 7}{2\times 100}+x=2560$
$\Rightarrow \frac{14tx+7x+200x}{200}=2560$
$\Rightarrow 14tx+207x=512000$ .....(ii)
Performing (i) $\times 7-$ (ii) $\times 4$, we get
$(56tx+700x)-(56tx+828x)=256000\times 7-512000\times 4$
$\Rightarrow 700x-828x=1792000-2048000$
$\Rightarrow 120x=256000$

$\Rightarrow x=2000$
Putting the value of $x$ in (i), we get
$8\times t\times 2000+100\times 2000=256000$
$\Rightarrow 16000t=256000-200000$
$\Rightarrow 16000t=56000$

$\Rightarrow t=\frac{56000}{16000}=3.5$ years