This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

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Answer
244
Hey EBWILS, good to see you again.
Step 1: Set up the inequality based on the problem statement. Joy starts with 457 and adds a number (let's call it ) to get an answer more than 700.
Step 2: Solve the inequality for . Subtract 457 from both sides of the inequality. This means the number added must be greater than 243.
Step 3: Consider the second condition given in the problem. The number she added () was less than 300.
Step 4: Combine both conditions and find the smallest integer. We have two conditions for : and . This means must be an integer between 243 and 300 (exclusive). The integers satisfying are . The smallest integer that is greater than 243 is 244. This number, 244, also satisfies the condition .
The smallest number it could be is .
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Hey EBWILS, good to see you again. Step 1: Set up the inequality based on the problem statement.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.