If `sqrt(0.04xx0.4xxa)=0.004xx0.4xxsqrt(b)`, then `a/b` is

Options

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

Answer

To solve the equation \( \sqrt{0.04 \times 0.4 \times a} = 0.004 \times 0.4 \times \sqrt{b} \), we will follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the equation for clarity:
\[ \sqrt{0.04 \times 0.4 \times a} = 0.004 \times 0.4 \times \sqrt{b} \]

### Step 2: Simplify the numbers
Convert the decimal numbers into fractions:
\[ 0.04 = \frac{4}{100} = \frac{1}{25}, \quad 0.4 = \frac{4}{10} = \frac{2}{5}, \quad 0.004 = \frac{4}{1000} = \frac{1}{250} \]
Now substitute these values back into the equation:
\[ \sqrt{\frac{1}{25} \times \frac{2}{5} \times a} = \frac{1}{250} \times \frac{2}{5} \times \sqrt{b} \]

### Step 3: Multiply the fractions
Calculate the left side:
\[ \frac{1}{25} \times \frac{2}{5} = \frac{2}{125} \]
So, we have:
\[ \sqrt{\frac{2}{125} \times a} = \frac{2}{1250} \times \sqrt{b} \]

### Step 4: Square both sides
Square both sides to eliminate the square root:
\[ \frac{2}{125} \times a = \left(\frac{2}{1250}\right)^2 \times b \]
Calculating the right side:
\[ \left(\frac{2}{1250}\right)^2 = \frac{4}{1562500} \]
Thus, we have:
\[ \frac{2}{125} \times a = \frac{4}{1562500} \times b \]

### Step 5: Cross-multiply
Cross-multiply to isolate \( a \) and \( b \):
\[ 2 \times 1562500 \times a = 4 \times 125 \times b \]
This simplifies to:
\[ 3125000a = 500b \]

### Step 6: Solve for \( \frac{a}{b} \)
Now, divide both sides by \( b \) and \( 3125000 \):
\[ \frac{a}{b} = \frac{500}{3125000} \]
This can be simplified:
\[ \frac{a}{b} = \frac{1}{6250} \]

### Final Result
Thus, the value of \( \frac{a}{b} \) is:
\[ \frac{a}{b} = \frac{1}{6250} \]

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