I am having trouble solving a certain question on a basic Simple Interest. Here's the question
"Person A borrows a sum of 6300 dollars from Person B at the rate of 14% per annum for 3 years. Person A added some more money in it and lent it to Person C at 16% per annum for 3 years. In this process, Person A earns a total profit 618 dollars. Find the amount Person A added to the sum."
My answer for this goes like this:
Lets say that the amount Person A has to pay to Person B after 3 years be $A_1$. Then, using the formula for amount $\big(\text{Amount}=\text{Sum}\big(1+\frac{\text{Rate}\times\text{Time}}{100}\big)\big)$, we have $A_1 = 6300\big(1+\frac{14\times3}{100}\big) = 8946$.
Assume that "$x$" be the amount of money Person A added to the sum (i.e 6300 dollars), which he lent it to Person C at 16% p.a for 3 years. Let $A_2$ be the amount Person C has to pay to Person B. Then $A_2 = (6300+x)\big(1+\frac{16\times3}{100}\big) = 9324+1.48x$.
It is said that Person A earned a profit of 618 dollars. Obviously, Person A earns a profit if Person C pays more amount to Person A than Person A has to pay the amount to Person B. So $A_2-A_1 = 618$ $\implies$ $378+1.48x = 618$ $\implies$ $1.48x = 240$ $\implies$ $x \approx 162.16$ dollars.
This should have been the answer. However the answer was given to be 500 dollars. I don't know how and where I got it wrong. I'm pretty sure I understood the question here, but if that was the case, why I got the answer wrong? I really need some help in clarifying this question and help me point out the mistake that I have made or assumed. I have a suspicion that I got the part where Person A added a certain amount to the sum wrong.
You haven't added the initial investment by A to the profit.
You should have
$$A_2-A_1 = 618+x$$$$ \implies 378 + 1.48x = 618 + x $$$$\implies 0.48x = 240 $$$$\implies x = \$500$$
as we need the return ($378+1.48x$) to equal profit + initial investment ($618+x$).