If `x=111"...."(20 digits),y=333"...."(10digits) and z=222".....2"(10 digits), then (x-y^(2))/(z)` equals.

Options

A. `(1)/(2)`
B. 1
C. 2
D. 4

Explanation

To solve the problem step by step, we need to evaluate the expression \((x - y^2) / z\) where:

- \(x = 111...1\) (20 digits)
- \(y = 333...3\) (10 digits)
- \(z = 222...2\) (10 digits)

### Step 1: Express \(x\), \(y\), and \(z\) in mathematical form

1. **For \(x\)**:
- \(x\) consists of 20 digits of 1. This can be expressed as:
\[ x = \underbrace{111...1}_{20 \text{ times}} = \frac{10^{20} - 1}{9} \]

2. **For \(y\)**:
- \(y\) consists of 10 digits of 3. This can be expressed as:
\[ y = \underbrace{333...3}_{10 \text{ times}} = 3 \times \frac{10^{10} - 1}{9} \]

3. **For \(z\)**:
- \(z\) consists of 10 digits of 2. This can be expressed as:
\[ z = \underbrace{222...2}_{10 \text{ times}} = 2 \times \frac{10^{10} - 1}{9} \]

### Step 2: Calculate \(y^2\)

Now we need to calculate \(y^2\):
\[ y^2 = \left(3 \times \frac{10^{10} - 1}{9}\right)^2 = 9 \times \left(\frac{10^{10} - 1}{9}\right)^2 = \frac{(10^{10} - 1)^2}{9} \]

### Step 3: Substitute \(x\), \(y^2\), and \(z\) into the expression

Now we substitute \(x\), \(y^2\), and \(z\) into the expression \((x - y^2) / z\):
\[ \frac{x - y^2}{z} = \frac{\frac{10^{20} - 1}{9} - \frac{(10^{10} - 1)^2}{9}}{2 \times \frac{10^{10} - 1}{9}} \]

### Step 4: Simplify the expression

1. Combine the terms in the numerator:
\[ = \frac{1}{9} \left(10^{20} - 1 - (10^{10} - 1)^2\right) \]
2. Expand \((10^{10} - 1)^2\):
\[ (10^{10} - 1)^2 = 10^{20} - 2 \times 10^{10} + 1 \]
3. Substitute back into the expression:
\[ = \frac{1}{9} \left(10^{20} - 1 - (10^{20} - 2 \times 10^{10} + 1)\right) \]
4. Simplify:
\[ = \frac{1}{9} \left(10^{20} - 1 - 10^{20} + 2 \times 10^{10} - 1\right) = \frac{1}{9} \left(2 \times 10^{10} - 2\right) = \frac{2(10^{10} - 1)}{9} \]

### Step 5: Divide by \(z\)

Now we divide by \(z\):
\[ z = 2 \times \frac{10^{10} - 1}{9} \]
Thus:
\[ \frac{\frac{2(10^{10} - 1)}{9}}{2 \times \frac{10^{10} - 1}{9}} = 1 \]

### Final Answer

The final result is:
\[ \frac{x - y^2}{z} = 1 \]