Peter gave one-fourth of the amount he had to Michael. Michael in turn gave half of what he received from Peter to Sam. If the difference between the remaining amount with Peter and the amount received by Sam is ₹ 500, how much money did Michael receive from Peter?

### Concept & Logic This problem tracks the distribution of an initial amount through a series of fractional transfers. The strategy is to define Peter's initial money as a variable, track the exact fraction held by each person at the end of the transactions, and use the given absolute difference (₹ 500) to solve for the base amount. ### Step-by-Step Solution * **Given:** Let Peter's initial amount be $x$. Peter gives $\frac{1}{4}x$ to Michael. * **Calculation:** Track the amounts held by everyone: Peter's remaining amount = $x - \frac{1}{4}x = \frac{3}{4}x$ Amount Michael receives = $\frac{1}{4}x$ Amount Sam receives = $\frac{1}{2}$ of Michael's amount = $\frac{1}{2} \times \frac{1}{4}x = \frac{1}{8}x$ * Apply the final condition: The difference between Peter's remaining amount and Sam's amount is 500. $$ \frac{3}{4}x - \frac{1}{8}x = 500 $$ * Find a common denominator to subtract the fractions: $$ \frac{6x}{8} - \frac{1x}{8} = 500 $$ $$ \frac{5x}{8} = 500 $$ $$ x = 500 \times \frac{8}{5} = 800 $$ So, Peter's initial amount was ₹ 800. * The question asks for the amount Michael received from Peter: $$ \text{Michael's amount} = \frac{1}{4}x = \frac{1}{4}(800) = 200 $$ ### Exam Strategy & Shortcut Use the **LCM Assumption Method** to avoid working with fractions. Look at the denominators in the problem: one-fourth (4) and half (2). We also know Sam gets half of a fourth, which is an eighth (8). Assume Peter's starting amount is $8$ units. Peter gives Michael 2 units. Peter has $8 - 2 = 6$ units left. Michael gives Sam 1 unit. Sam has 1 unit. Difference between Peter's remaining (6) and Sam's (1) is 5 units. We are told this difference is ₹ 500. So, $5 \text{ units} = 500 \implies 1 \text{ unit} = 100$. Michael received 2 units from Peter. $2 \times 100 = 200$. ### Common Pitfall Many students successfully calculate $x = 800$ (Peter's initial amount) and mistakenly choose 800 if it were an option, or get confused and mark "None of these" if it isn't. Always highlight the exact entity the question is asking for before selecting an option (in this case, what Michael received). ### Final Answer Therefore, the correct answer is **₹ 200**.