If `X=(a^(3)b^(2))/(sqrtc)` and percentage changes in a, b and c are `2%` increase, `1%` decrease and `2%` decrease respectively then percentage increase or decrease in X is

Options

A. `5%` increase
B. `5%` decrease
C. `9%` increase or decrease
D. `9%` increase

Answer

To solve the problem, we need to find the percentage change in \( X \) given the formula:

\[ X = \frac{a^3 b^2}{\sqrt{c}} \]

We also know the percentage changes in \( a \), \( b \), and \( c \):
- \( a \): 2% increase
- \( b \): 1% decrease
- \( c \): 2% decrease

### Step 1: Write the formula for percentage change in \( X \)

Using the error propagation formula for functions of multiple variables, we can express the percentage change in \( X \) as:

\[ \frac{\Delta X}{X} = 3 \frac{\Delta a}{a} + 2 \frac{\Delta b}{b} - \frac{1}{2} \frac{\Delta c}{c} \]

### Step 2: Substitute the percentage changes

Now we substitute the given percentage changes into the formula:

- For \( a \): \( \frac{\Delta a}{a} = \frac{2}{100} = 0.02 \) (since it's an increase)
- For \( b \): \( \frac{\Delta b}{b} = -\frac{1}{100} = -0.01 \) (since it's a decrease)
- For \( c \): \( \frac{\Delta c}{c} = -\frac{2}{100} = -0.02 \) (since it's a decrease)

Substituting these values into the formula gives:

\[ \frac{\Delta X}{X} = 3(0.02) + 2(-0.01) - \frac{1}{2}(-0.02) \]

### Step 3: Calculate each term

Now, calculate each term:

1. \( 3(0.02) = 0.06 \)
2. \( 2(-0.01) = -0.02 \)
3. \( -\frac{1}{2}(-0.02) = 0.01 \)

### Step 4: Combine the results

Now, combine the results:

\[ \frac{\Delta X}{X} = 0.06 - 0.02 + 0.01 = 0.05 \]

### Step 5: Convert to percentage

To express this as a percentage, we multiply by 100:

\[ \Delta X \% = 0.05 \times 100 = 5\% \]

### Conclusion

Since the result is positive, it indicates a percentage increase in \( X \). Therefore, the final answer is:

\[ \text{Percentage increase in } X = 5\% \]