If `sqrt(0.04 xx 0.4 xx a) = 0.04 xx 0.4 xx sqrt(b)` , then `a/b` is equal to

Options

A. `16 xx 10^(-3)`
B. `16 xx 10^(-4)`
C. `16 xx 10^(-5)`
D. `16 xx10^(-2)`

Answer

To solve the equation \( \sqrt{0.04 \times 0.4 \times a} = 0.04 \times 0.4 \times \sqrt{b} \), we will follow these steps:

### Step 1: Square both sides of the equation
To eliminate the square root, we square both sides of the equation:
\[ 0.04 \times 0.4 \times a = (0.04 \times 0.4)^2 \times b \]

### Step 2: Calculate \( (0.04 \times 0.4)^2 \)
First, we calculate \( 0.04 \times 0.4 \):
\[ 0.04 \times 0.4 = 0.016 \]
Now, we square this result:
\[ (0.016)^2 = 0.000256 \]

### Step 3: Substitute back into the equation
Now we substitute back into the equation:
\[ 0.04 \times 0.4 \times a = 0.000256 \times b \]

### Step 4: Isolate \( a \) in terms of \( b \)
We can rearrange the equation to solve for \( a \):
\[ a = \frac{0.000256 \times b}{0.04 \times 0.4} \]

### Step 5: Calculate \( 0.04 \times 0.4 \)
We already calculated \( 0.04 \times 0.4 = 0.016 \). Now we substitute this into the equation:
\[ a = \frac{0.000256 \times b}{0.016} \]

### Step 6: Simplify the expression
Now we simplify the fraction:
\[ a = 0.000256 \div 0.016 \times b \]
Calculating \( 0.000256 \div 0.016 \):
\[ 0.000256 \div 0.016 = 0.016 \]
Thus,
\[ a = 0.016 \times b \]

### Step 7: Find \( \frac{a}{b} \)
Now we can find \( \frac{a}{b} \):
\[ \frac{a}{b} = 0.016 \]

### Step 8: Convert to fraction
To express \( 0.016 \) as a fraction:
\[ 0.016 = \frac{16}{1000} = \frac{2}{125} \]

### Final Answer
Thus, \( \frac{a}{b} = \frac{2}{125} \).

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