Question

Jamie and Imani each play softball. Imani has won 5 fewer games than Jamie. Is it possible for Jamie to have won 11 games if the sum of the games Imani and Jamie have won together is 30?

a Yes; Jamie could have won 11 games because 2x − 5 = 30.
b Yes; Jamie could have won 11 games because 11 − 5 is less than 30.
c No; Jamie could not have won 11 games because 2x − 5 ≠ 30.
d No; Jamie could not have won 11 games because 2x − 11 ≠ 30.

Answer 1

Let's break down the problem using the information given.

Let:
- \( J \) = the number of games Jamie has won
- \( I \) = the number of games Imani has won

We know:
1. Imani has won 5 fewer games than Jamie:
\[
I = J - 5
\]

2. The total number of games they have won together is 30:
\[
I + J = 30
\]

Now, substituting the first equation into the second equation:
\[
(J - 5) + J = 30
\]
\[
2J - 5 = 30
\]
\[
2J = 35
\]
\[
J = 17.5
\]

This means Jamie cannot have won a fractional number of games. Therefore, the possible values for Jamie's wins can only be whole numbers.

Now let's investigate the scenario where Jamie has won 11 games:
If \( J = 11 \):
\[
I = J - 5 = 11 - 5 = 6
\]

Now, checking their total wins:
\[
I + J = 6 + 11 = 17
\]

Since 17 is not equal to 30, Jamie cannot have won 11 games while satisfying both conditions.

Based on this analysis, the correct answer is:
**d No; Jamie could not have won 11 games because 2x − 11 ≠ 30.**