Let the number of stamps Arnav has be \( x \). Then, since Matei has 4 times as many stamps as Arnav, we can express the number of stamps Matei has as \( 4x \).
According to the problem, if Matei gives Arnav 8 stamps, the new number of stamps each person has will be:
- Matei: \( 4x - 8 \)
- Arnav: \( x + 8 \)
The problem states that after this exchange, Matei will have twice as many stamps as Arnav. This gives us the equation:
\[ 4x - 8 = 2(x + 8) \]
Now, we can simplify and solve this equation step by step:
- Distribute the 2 on the right side:
\[ 4x - 8 = 2x + 16 \]
- Subtract \( 2x \) from both sides:
\[ 4x - 2x - 8 = 16 \]
This simplifies to:
\[ 2x - 8 = 16 \]
- Add 8 to both sides:
\[ 2x = 24 \]
- Divide by 2:
\[ x = 12 \]
Now, we can find the number of stamps each boy has:
- Arnav has \( x = 12 \) stamps.
- Matei has \( 4x = 4 \times 12 = 48 \) stamps.
Therefore, the number of stamps each boy has is:
- Arnav: 12 stamps
- Matei: 48 stamps
To verify, after Matei gives Arnav 8 stamps:
- Matei will have \( 48 - 8 = 40 \) stamps.
- Arnav will have \( 12 + 8 = 20 \) stamps.
Now checking the relationship: \( 40 \) is indeed twice \( 20 \), thus confirming our solution is correct.
In conclusion, Arnav has 12 stamps and Matei has 48 stamps.
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