If `a=(1/10)^(2),b=1/5` and `c=sqrt(1/100)`, then which of the following statements is correct?

Options

A. `a lt b lt c`
B. `a lt c lt b`
C. `b lt c lt a`
D. `c lt a lt b`

Answer

To solve the problem step-by-step, we will evaluate the values of \( a \), \( b \), and \( c \) and then compare them.

### Step 1: Calculate \( a \)
Given:
\[ a = \left( \frac{1}{10} \right)^2 \]
Calculating \( a \):
\[ a = \frac{1^2}{10^2} = \frac{1}{100} \]

### Step 2: Calculate \( b \)
Given:
\[ b = \frac{1}{5} \]
To compare with \( a \) and \( c \), we need to express \( b \) with a denominator of 100:
\[ b = \frac{1}{5} = \frac{1 \times 20}{5 \times 20} = \frac{20}{100} \]

### Step 3: Calculate \( c \)
Given:
\[ c = \sqrt{\frac{1}{100}} \]
Calculating \( c \):
\[ c = \frac{\sqrt{1}}{\sqrt{100}} = \frac{1}{10} \]
Now, we need to express \( c \) with a denominator of 100:
\[ c = \frac{1}{10} = \frac{1 \times 10}{10 \times 10} = \frac{10}{100} \]

### Step 4: Compare \( a \), \( b \), and \( c \)
Now we have:
- \( a = \frac{1}{100} \)
- \( b = \frac{20}{100} \)
- \( c = \frac{10}{100} \)

We can compare the numerators:
- \( 1 < 10 < 20 \)

Thus, we can conclude:
\[ a < c < b \]

### Conclusion
The correct statement is:
\[ a < c < b \]

This corresponds to option **b**.

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