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The difference between the Compound Interest and Simple Interest on a certain sum at the same rate of interest for the second year is Rs. 32 and for the third year is Rs. 66.56. Find the rate of interest (in %).

Solution:-

The difference between SI and CI in the third year 2 * The difference between SI and CI in the second year + Interest on the difference between the interests for the second year

66.56 = 2 x 32 + D, where D is the interest on the difference between the interests for the second year.

D = 2.56

2.56 is 8% of 32

I am not able to understand the bolded part in the solution given

Fin27
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  • Sorry for the confusion. As I expected, I was misreading the problem. I will correct my inaccurate posted solution in a few minutes. – lulu Jun 02 '23 at 00:17
  • I have posted what I believe to be the correct solution to the problem. Note that it matches the official solution. My prior (incorrect) solution assumed that they were talking about the difference in balances. However, the question clearly refers to the difference in paid interest and these are not the same since the balances are not the same in later years. – lulu Jun 02 '23 at 00:28

1 Answers1

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Here is my algebraic description:

Let $B$ be the (unknown) starting balance, and let $r$ be the (unknown) interest rate.

Let $S_i$ be the balance with simple interest in year $i$, So $$S_i=B\times (1+i\times r)$$

Let $C_i$ be the balance with compound interest in year $i$. So $$C_i=B\times (1+r)^i$$

Then the interest paid to the simple account in year $i$ is $S_i-S_{i-1}$, with a similar result for the interest paid to the compound account.

The balances coincide at the end of year $1$ (both are $(1+r)B$. In year $2$, the simple account of course receives $rB$. The compound account receive $rC_1=r(1+r)B$ Thus the difference in interest over year $2$ is $$\Delta_2=r(1+r)B-rB=r^2B$$

Now we move to year $3$. The simple account receives $rB$ as always. The compound account receives $rC_2=r(1+r)^2B$. Thus $$\Delta_3=r(1+r)^2B-rB=(r+2r^2+r^3)B-rB=(2r^2+r^3)B$$

We remark that we have shown that $$\Delta_3=2\Delta_2+r\Delta_2$$ which is the bolded part of the official solution.

In any case, solving the pair of equations, $\Delta_2=32, \Delta_3=66.56$ yields $$\boxed {r=.08\quad \&\quad B=5000}$$

lulu
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