If `sqrt(0.05 xx 0.5 xx a) = 0.5 xx 0.05 xx sqrt(b)`, then `(a)/(b)` is equal to

Options

A. 0.0025
B. 0.025
C. 0.25
D. 0.00025

Answer

To solve the equation \( \sqrt{0.05 \times 0.5 \times a} = 0.5 \times 0.05 \times \sqrt{b} \), we will follow these steps:

### Step 1: Simplify both sides of the equation
Starting with the left-hand side:
\[ \sqrt{0.05 \times 0.5 \times a} \]
We can calculate \( 0.05 \times 0.5 \):
\[ 0.05 \times 0.5 = 0.025 \]
So, the left-hand side becomes:
\[ \sqrt{0.025 \times a} \]

Now, for the right-hand side:
\[ 0.5 \times 0.05 \times \sqrt{b} \]
Calculating \( 0.5 \times 0.05 \):
\[ 0.5 \times 0.05 = 0.025 \]
Thus, the right-hand side simplifies to:
\[ 0.025 \times \sqrt{b} \]

### Step 2: Set the simplified expressions equal to each other
Now we have:
\[ \sqrt{0.025 \times a} = 0.025 \times \sqrt{b} \]

### Step 3: Square both sides to eliminate the square root
Squaring both sides gives:
\[ 0.025 \times a = (0.025 \times \sqrt{b})^2 \]
This simplifies to:
\[ 0.025 \times a = 0.025^2 \times b \]

### Step 4: Substitute the value of \( 0.025^2 \)
Calculating \( 0.025^2 \):
\[ 0.025^2 = 0.000625 \]
Thus, we can rewrite the equation as:
\[ 0.025 \times a = 0.000625 \times b \]

### Step 5: Rearranging to find \( \frac{a}{b} \)
Now, we can rearrange the equation to find \( \frac{a}{b} \):
\[ \frac{a}{b} = \frac{0.000625}{0.025} \]

### Step 6: Simplify \( \frac{0.000625}{0.025} \)
To simplify this, we can convert both numbers into fractions:
\[ 0.000625 = \frac{625}{1000000} \quad \text{and} \quad 0.025 = \frac{25}{1000} \]
Now calculating:
\[ \frac{0.000625}{0.025} = \frac{625/1000000}{25/1000} = \frac{625 \times 1000}{25 \times 1000000} \]
This simplifies to:
\[ \frac{625 \times 1000}{25 \times 1000000} = \frac{625}{25 \times 1000} = \frac{625}{25000} = \frac{1}{40} \]

### Final Result
Thus, we find:
\[ \frac{a}{b} = \frac{1}{40} \]