Penelope receives the same amount of money each month for her allowance. Each month she spends half of her allowance and puts the rest in a piggy bank. On penelope's 8th birthday, the piggy bank contains $40. If the piggy bank contains $244 after 2 years, what is her monthly allowance? (Ignore the dollar sign when gridding your response.)

Answer

To find Penelope's monthly allowance, we can break down the problem step by step.

### Step 1: Define the variables
Let Penelope's monthly allowance be \( X \) dollars.

### Step 2: Determine the amount saved each month
Each month, Penelope spends half of her allowance and saves the other half in her piggy bank. Therefore, the amount she saves each month is:
\[ \text{Amount saved each month} = \frac{X}{2} \]

### Step 3: Calculate the total time period
Penelope's piggy bank contains $40 on her 8th birthday and $244 after 2 years. Since 2 years equals 24 months, we will use this time frame for our calculations.

### Step 4: Set up the equation
The total amount saved in the piggy bank over the 24 months can be calculated as follows:
\[ \text{Total amount saved} = \text{Initial amount} + \text{Amount saved each month} \times \text{Number of months} \]
Substituting the known values:
\[ 244 = 40 + \left(\frac{X}{2}\right) \times 24 \]

### Step 5: Simplify the equation
First, subtract 40 from both sides:
\[ 244 - 40 = \left(\frac{X}{2}\right) \times 24 \]
\[ 204 = \left(\frac{X}{2}\right) \times 24 \]

### Step 6: Solve for \( X \)
Now, divide both sides by 24:
\[ \frac{204}{24} = \frac{X}{2} \]
To eliminate the fraction, multiply both sides by 2:
\[ \frac{204 \times 2}{24} = X \]
\[ X = \frac{408}{24} \]
Now, simplify \( \frac{408}{24} \):
\[ X = 17 \]

### Conclusion
Penelope's monthly allowance is \( 17 \) dollars.

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