### Question 9

SSC-CGL 2020 Mar 4 Shift 2

The difference between the compound interest compounded half-yearly and compound interest compounded yearly is Rs 88.50 at the rate of 10% in 1 year. What is the SI on the same sum at the same rate per annum for 1-2/3 year?

### Solution in Short

Remember: CI at R% for 2 years __annual__ compounding is by a shortcut formula R(R + 2)P.

By the above shortcut, CI at 10% half-yearly compounding for 2 half-years will be = 0.05(0.05 + 2)P. Next, CI compounded annually for 1 year [same as SI, if T is 1] = P x 10% x 1 = 0.1P. The difference is 0.05(0.05 + 2)P - 0.1P = 0.0025P, which is given = 88.50, whence P = 35400, and finally SI for 1-2/3, i.e., 5/3 = (35400 x 10 x 5/3)/100 = 5900 Rs. ans!

### Solution in Detail

Take P = 100

CI @10% compounded 1/2 yearly

$\displaystyle = 100 \bigg(1 + \frac{10\%}{2}\bigg)^2 - 100$

$\displaystyle = 10.25$ Rs.

CI @10% compounded annually for 1 year is same as SI at 10%:

$\displaystyle = \frac{100 \times 10 \times 1}{100} = 10$ Rs.

Difference is $\displaystyle 10.25 - 10 = 0.25$

If difference is $\displaystyle 0.25$, P $\displaystyle = 100$

if $\displaystyle 88.50, P = \frac{100}{0.25} \times 88.50$

$\displaystyle \therefore P = 35400$ Rs.

To calculate SI for $\displaystyle T = 1\frac23 = 5/3$

$\displaystyle \therefore$ SI = $\displaystyle \frac{35400 \times 10 \times 5/3}{100}$

$\displaystyle = 5900\text{ Rs. }\underline{Ans}$

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