Angela earns a weekly allowance at the start of each week. She started the week with just her weekly allowance. Then, she spent half of her weekly allowance at the movies. To earn more money, her parents let her clean the windows in the house for $5. What is her weekly allowance if she ended the week with $8? Solve using arithmetic or algebra, and explain ALL of your steps!(3 points)

Answer 1

Let \( x \) represent Angela's weekly allowance.

1. Starting Amount: At the start of the week, Angela has \( x \) dollars.

2. Spending Half: Angela spent half of her weekly allowance at the movies. This means she spent \( \frac{x}{2} \) dollars.

   After this spending, her remaining amount is: \[ x - \frac{x}{2} = \frac{2x}{2} - \frac{x}{2} = \frac{x}{2} \]

3. Earning Extra Money: To earn more money, Angela cleaned the windows and earned $5. Therefore, after cleaning the windows, her total amount is: \[ \frac{x}{2} + 5 \]

4. Ending Amount: By the end of the week, Angela has $8. We can set up the following equation: \[ \frac{x}{2} + 5 = 8 \]

5. Solving for \( x \):

   • First, subtract 5 from both sides of the equation: \[ \frac{x}{2} = 8 - 5 \] \[ \frac{x}{2} = 3 \]
   • Next, multiply both sides by 2 to isolate \( x \): \[ x = 3 \times 2 \] \[ x = 6 \]

6. Conclusion: Angela's weekly allowance is \( \boxed{6} \) dollars.

Verification: Let's check if this value makes sense.

• Angela starts the week with $6.
• She spends half, which is \( \frac{6}{2} = 3 \), leaving her with \( 6 - 3 = 3 \).
• She then earns $5 for cleaning the windows, bringing her total to \( 3 + 5 = 8 \).
• Since she ended with $8, our solution is verified to be correct.