A invested Rs P+500 in compound interest for two years at 10% p.a. and received interest of Rs 840. If he invested Rs P at 15% p.a. for simple interest for the same period, then find the amount he received (in Rs).

The correct option is 4550.

Let's break down the problem into two parts: finding the value of $P$ using the compound interest data, and then calculating the amount received under simple interest.

Step 1: Find the value of $P$ from the compound interest investment
The principal amount invested is $P+500$.
The rate of interest is $R=10\%$ per annum compounded annually.
The time period is $t=2$ years.
The compound interest (CI) received is Rs 840.

The formula for compound interest is:
$\mathrm{CI}=\text{Principal}\times \left[{\left(1+\frac{R}{100}\right)}^t-1\right]$
Substituting the given values:
$840=(P+500)\times \left[{\left(1+\frac{10}{100}\right)}^2-1\right]$
$840=(P+500)\times \left[{\left(1.1\right)}^2-1\right]$
$840=(P+500)\times \left[1.21-1\right]$
$840=(P+500)\times 0.21$

Solving for $P+500$:
$P+500=\frac{840}{0.21}$
$P+500=4000$
$P=4000-500$
$P=3500$

Step 2: Calculate the amount received from the simple interest investment
Now, A invested Rs $P$ (which is Rs 3500) at $15\%$ per annum simple interest for the same period of $2$ years.
The simple interest (SI) is calculated as:
$\mathrm{SI}=\frac{P\times R\times T}{100}$
$\mathrm{SI}=\frac{3500\times 15\times 2}{100}$
$\mathrm{SI}=35\times 30$
$\mathrm{SI}=1050$

The total amount received is the sum of the principal and the simple interest:
$\text{Amount}=P+\mathrm{SI}$
$\text{Amount}=3500+1050=4550$

Thus, the amount received by A is Rs 4550.