Peter, Rene and Andrew were playing a game of marbles. In the game , at a point of time , Peter gave one - fourth of his marbles to Rene , who in turns gave half of what she received to Andrew. If the difference between the marbles left with Peter and the marbles received by Andrew is 30, then how many marbles did Rene receive from Peter ?

Answer

To solve the problem step by step, we can follow this approach:

1. **Define Variables**: Let \( X \) be the total number of marbles Peter has.

2. **Calculate Marbles Given to Rene**: Peter gives one-fourth of his marbles to Rene. Therefore, the number of marbles Rene receives is:
\[ \text{Marbles received by Rene} = \frac{X}{4} \]

3. **Calculate Marbles Left with Peter**: After giving away marbles, the number of marbles left with Peter is:
\[ \text{Marbles left with Peter} = X - \frac{X}{4} = \frac{3X}{4} \]

4. **Calculate Marbles Given to Andrew**: Rene gives half of what she received to Andrew. Therefore, the number of marbles Andrew receives is:
\[ \text{Marbles received by Andrew} = \frac{1}{2} \times \frac{X}{4} = \frac{X}{8} \]

5. **Set Up the Equation**: According to the problem, the difference between the marbles left with Peter and the marbles received by Andrew is 30. This can be expressed as:
\[ \frac{3X}{4} - \frac{X}{8} = 30 \]

6. **Solve the Equation**: To solve the equation, we need a common denominator. The common denominator for 4 and 8 is 8. Rewriting the equation:
\[ \frac{6X}{8} - \frac{X}{8} = 30 \]
Simplifying the left side gives:
\[ \frac{5X}{8} = 30 \]

7. **Isolate \( X \)**: Multiply both sides by 8 to eliminate the fraction:
\[ 5X = 240 \]
Now, divide by 5:
\[ X = 48 \]

8. **Find Marbles Received by Rene**: Now that we have the total number of marbles Peter had, we can find out how many marbles Rene received:
\[ \text{Marbles received by Rene} = \frac{X}{4} = \frac{48}{4} = 12 \]

Thus, the number of marbles Rene received from Peter is **12**.