Mohan ate half a pizza on Monday. He ate half of what was left on Tuesday and so on. He followed this pattern for one week. How much of the pizza would he have eaten during the week?

Options

A. 0.9922
B. 0.95
C. 0.9822
D. 1

Answer

To find out how much pizza Mohan ate during the week, we can break it down step by step:

### Step 1: Calculate the amount eaten on each day
- **Monday**: Mohan ate half of the pizza.
\[ \text{Amount eaten on Monday} = \frac{1}{2} \]

- **Tuesday**: He ate half of what was left. After Monday, half of the pizza is left, so he eats:
\[ \text{Amount eaten on Tuesday} = \frac{1}{2} \times \left(1 - \frac{1}{2}\right) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]

- **Wednesday**: He again eats half of what is left after Tuesday. The amount left after Tuesday is:
\[ 1 - \left(\frac{1}{2} + \frac{1}{4}\right) = 1 - \frac{3}{4} = \frac{1}{4} \]
So, he eats:
\[ \text{Amount eaten on Wednesday} = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} \]

- **Thursday**: The amount left after Wednesday is:
\[ 1 - \left(\frac{1}{2} + \frac{1}{4} + \frac{1}{8}\right) = 1 - \frac{7}{8} = \frac{1}{8} \]
Thus, he eats:
\[ \text{Amount eaten on Thursday} = \frac{1}{2} \times \frac{1}{8} = \frac{1}{16} \]

- **Friday**: The amount left after Thursday is:
\[ 1 - \left(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}\right) = 1 - \frac{15}{16} = \frac{1}{16} \]
So, he eats:
\[ \text{Amount eaten on Friday} = \frac{1}{2} \times \frac{1}{16} = \frac{1}{32} \]

- **Saturday**: The amount left after Friday is:
\[ 1 - \left(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32}\right) = 1 - \frac{31}{32} = \frac{1}{32} \]
Thus, he eats:
\[ \text{Amount eaten on Saturday} = \frac{1}{2} \times \frac{1}{32} = \frac{1}{64} \]

- **Sunday**: The amount left after Saturday is:
\[ 1 - \left(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}\right) = 1 - \frac{63}{64} = \frac{1}{64} \]
So, he eats:
\[ \text{Amount eaten on Sunday} = \frac{1}{2} \times \frac{1}{64} = \frac{1}{128} \]

### Step 2: Sum the amounts eaten over the week
Now, we can sum all the amounts eaten from Monday to Sunday:
\[ \text{Total amount eaten} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64} + \frac{1}{128} \]

### Step 3: Convert to a common denominator
The common denominator for these fractions is 128. We can convert each fraction:
- \(\frac{1}{2} = \frac{64}{128}\)
- \(\frac{1}{4} = \frac{32}{128}\)
- \(\frac{1}{8} = \frac{16}{128}\)
- \(\frac{1}{16} = \frac{8}{128}\)
- \(\frac{1}{32} = \frac{4}{128}\)
- \(\frac{1}{64} = \frac{2}{128}\)
- \(\frac{1}{128} = \frac{1}{128}\)

Now, adding these together:
\[ \frac{64 + 32 + 16 + 8 + 4 + 2 + 1}{128} = \frac{127}{128} \]

### Conclusion
Thus, the total amount of pizza Mohan ate during the week is:
\[ \frac{127}{128} \]