Ramesh takes a loan of a certain amount from the bank at $10$% simple
interest. He divides it into two parts and invests one part in a
scheme which gives him $10$% compound interest annually for two years.
He lends the other part to his friend who pays it in two equal
instalments at the rate of $20$% compounded annually to be paid at the
end of each of the two years. If each instalment paid by his friend is
Rs. $7.2$ lakhs and Ramesh has $1.25$ lakhs as profit after two years
in the overall transaction, then find the initial loan amount?
I am overhauling my answer for two reasons:
Initially, I misinterpreted the phrase:
"...to be paid at the end of each of the two years."
That is, I thought that the payments from Ramesh's friend would be at the end of Year-2 and the end of Year-4. Instead, I see now that the payments are at the end of Year-1 and the end of Year-2.
The OP (i.e. original poster) has provided the exact wording of the original problem, which is somewhat different than the excerpt that I worked on.
The actual question is:
If the overall profit is $1.25$, then how much was the original loan amount?
The first thing to do is determine a reasonable interpretation of the word profit, in this situation. I am interpreting it as follows:
Assume that Ramesh initially has no money.
Assume that Ramesh goes to a bank, and borrows $(b)$ at $10\%$ simple interest.
Assume that Ramesh makes no installment payments on the loan, for two years. Instead, Ramesh splits the loan into two parts, and make two investments, as described in the excerpt at the start of this answer.
Assume that one year after the loan, when Ramesh receives the first $7.2$ payment from his friend, he simply puts the money in a drawer. That is, he does not re-invest the money in any savings account.
Assume that at the end of two years, on the same day that Ramesh receives the 2nd payment from his friend, immediately after receiving the second payment, Ramesh goes to the bank and re-pays the loan in full, including the interest that accrued on the loan.
Assume that on the same day, immediately after this repayment, that Ramesh now has $1.25$.
I am interpreting this to mean that Ramesh's bankroll has changed from $(0)$ to $(1.25)$ and thus, his profit is $(1.25)$. So, now the question is:
Based on all of the assumptions above, what is the exact value of $(b)$, the original amount that he borrowed from the bank.
He borrowed $b$, which accrued $10\%$ simple interest at the bank, for two years. This implies that after two years he repaid $(b) + [(b) \times (0.1) \times (2)] = (b \times 1.2).$ Further, it is given that after this repayment, he then had $1.25$.
This means that he borrowed $b$, which then grew to
$$(b \times 1.2) + 1.25. \tag1 $$
Further, Ramesh did the following:
Divided the loan $b$ into two parts.
Denote these parts as $r$ and $s$.
Then $r + s = b.$
The portion $(r)$ was invested at $10\%$, compounded annually, which implies that $(r)$ grew to $[r \times (1.1)^2] = [1.21(r)].$
The portion $(s)$ was loaned to a friend at $20\%$ compounded annually. As the OP reasoned in his posting, because the friend made two payments of $7.2$, you have that
$[s \times (1.2)^2] = [7.2 \times (1.2)] + [7.2]$ ,
which implies that $s = 11.$
Therefore, you can conclude the following:
I emphasize that I am assuming that the $11$ loaned to a friend became $[2 \times 7.2 = 14.4]$, because we have no information to the contrary. An alternate assumption would be that after one year, when he received the first payment of $7.2$, he put this into a savings account that paid $20\%$ interest.
I suspect that this alternative assumption does not represent the problem composer's intent. Therefore, I will go with the first assumption.
Substituting $(b - 11)$ for $(r)$ in the second bullet point, immediately above, gives:
$$(b \times 1.2) + 1.25 = [1.21 \times (b - 11)] + [14.4] \implies $$
$$[11 \times 1.21] + [1.25] - [14.4] = [1.21 \times b] - [1.2 \times b] \implies $$
$$(0.16) = (0.01) \times b \implies b = 16.$$
