Zoe won the raffle at a fair. She will receive the prize money in 5 monthly payments. If each payments is half as much as the previous month's payments and the total of the payments is $496, what is the amount of the first payment?

Options

A. 256
B. 96
C. 84
D. 16

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the first payment
Let the amount of the first payment be \( X \).

### Step 2: Define the subsequent payments
The payments for the subsequent months will be:
- Second payment: \( \frac{X}{2} \)
- Third payment: \( \frac{X}{4} \)
- Fourth payment: \( \frac{X}{8} \)
- Fifth payment: \( \frac{X}{16} \)

### Step 3: Write the equation for the total payments
The total of all the payments is given as $496. Therefore, we can write the equation:
\[ X + \frac{X}{2} + \frac{X}{4} + \frac{X}{8} + \frac{X}{16} = 496 \]

### Step 4: Find a common denominator
The common denominator for the fractions is 16. We can rewrite the equation as:
\[ \frac{16X}{16} + \frac{8X}{16} + \frac{4X}{16} + \frac{2X}{16} + \frac{X}{16} = 496 \]

### Step 5: Combine the fractions
Now, combining the fractions gives:
\[ \frac{16X + 8X + 4X + 2X + X}{16} = 496 \]
This simplifies to:
\[ \frac{31X}{16} = 496 \]

### Step 6: Solve for \( X \)
To find \( X \), multiply both sides by 16:
\[ 31X = 496 \times 16 \]
Now, calculate \( 496 \times 16 \):
\[ 496 \times 16 = 7936 \]
So, we have:
\[ 31X = 7936 \]
Now, divide both sides by 31:
\[ X = \frac{7936}{31} \]

### Step 7: Calculate \( X \)
Calculating \( \frac{7936}{31} \):
\[ X = 256 \]

### Conclusion
The amount of the first payment is \( \boxed{256} \).