If `X= (a^(3)b^(2))/(sqrt(c))` and percentage changes in a,b and c are 2% increase, 1% decreases 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 the variable \( X \) given by the formula:

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

We are given the percentage changes in \( a \), \( b \), and \( c \) as follows:
- \( a \): 2% increase
- \( b \): 1% decrease
- \( c \): 2% decrease

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

The percentage change in \( X \) can be calculated using the following formula:

\[ \frac{\Delta X}{X} \times 100 = n_a \frac{\Delta a}{a} + n_b \frac{\Delta b}{b} - \frac{1}{2} \frac{\Delta c}{c} \]

where \( n_a \), \( n_b \), and \( n_c \) are the powers of \( a \), \( b \), and \( c \) in the expression for \( X \).

### Step 2: Identify the powers and changes

From the formula for \( X \):
- The power of \( a \) is \( 3 \) (i.e., \( n_a = 3 \))
- The power of \( b \) is \( 2 \) (i.e., \( n_b = 2 \))
- The power of \( c \) is \( -\frac{1}{2} \) (since it is in the denominator)

Now, we can express the percentage changes:
- For \( a \): \( \frac{\Delta a}{a} = +2\% \)
- For \( b \): \( \frac{\Delta b}{b} = -1\% \)
- For \( c \): \( \frac{\Delta c}{c} = -2\% \)

### Step 3: Substitute the values into the formula

Now we substitute these values into the formula:

\[ \frac{\Delta X}{X} \times 100 = 3 \times 2 + 2 \times (-1) - \frac{1}{2} \times (-2) \]

### Step 4: Calculate each term

Calculating each term:
- \( 3 \times 2 = 6 \)
- \( 2 \times (-1) = -2 \)
- \( -\frac{1}{2} \times (-2) = 1 \)

### Step 5: Combine the results

Now, we combine the results:

\[ \frac{\Delta X}{X} \times 100 = 6 - 2 + 1 = 5 \]

### Step 6: Conclusion

Thus, the percentage change in \( X \) is:

\[ \Delta X = +5\% \]

This indicates that \( X \) has increased by 5%.