# RE7 | google/gemini-3.6-flash | run 1

## Prompt

A bat and a ball cost $1.10 together. The bat costs $1.00 more than the ball. How much does the ball cost?

## Response

The ball costs $0.05 (5 cents), while the bat costs $1.05. Together, their combined price equals $1.10, satisfying the condition that the bat costs exactly $1.00 more than the ball. The intuitive answer of $0.10 is incorrect because a $0.10 ball would make the bat cost $1.10, resulting in a total of $1.20.

## Mathematical Breakdown

Let \( x \) represent the cost of the ball in dollars. Since the bat costs $1.00 more than the ball, the cost of the bat is \( x + 1.00 \).

\[ x + (x + 1.00) = 1.10 \]

\[ 2x + 1.00 = 1.10 \]

\[ 2x = 0.10 \]

\[ x = 0.05 \]

Substituting \( x = 0.05 \) back into the equations confirms that the ball costs $0.05 and the bat costs $1.05.

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