In the expression `x^(3)y^(2)` , the value of x and y decrease by `25%` and `20%` respectively then find the percent decrease in the expression ?

Options

A. `63%`
B. `66%`
C. `71%`
D. `73%`

Answer

To solve the problem step by step, we need to find the percent decrease in the expression \( x^3 y^2 \) when \( x \) decreases by 25% and \( y \) decreases by 20%.

### Step 1: Determine the new values of \( x \) and \( y \)
- The original value of \( x \) is taken as 1.
- Since \( x \) decreases by 25%, the new value of \( x \) can be calculated as:
\[ x' = x - 0.25x = 1 - 0.25 = 0.75 \]

- The original value of \( y \) is also taken as 1.
- Since \( y \) decreases by 20%, the new value of \( y \) can be calculated as:
\[ y' = y - 0.20y = 1 - 0.20 = 0.80 \]

### Step 2: Calculate the original expression \( x^3 y^2 \)
- The original expression is:
\[ E = x^3 y^2 = 1^3 \cdot 1^2 = 1 \]

### Step 3: Calculate the new expression with the decreased values of \( x \) and \( y \)
- Substitute the new values of \( x \) and \( y \) into the expression:
\[ E' = (0.75)^3 \cdot (0.80)^2 \]

- Calculate \( (0.75)^3 \):
\[ (0.75)^3 = 0.75 \cdot 0.75 \cdot 0.75 = 0.421875 \]

- Calculate \( (0.80)^2 \):
\[ (0.80)^2 = 0.80 \cdot 0.80 = 0.64 \]

- Now multiply these two results:
\[ E' = 0.421875 \cdot 0.64 = 0.26984375 \]

### Step 4: Calculate the percent decrease in the expression
- The percent decrease can be calculated using the formula:
\[ \text{Percent Decrease} = \left( \frac{E - E'}{E} \right) \times 100 \]

- Substitute the values:
\[ \text{Percent Decrease} = \left( \frac{1 - 0.26984375}{1} \right) \times 100 = (0.73015625) \times 100 = 73.015625\% \]

- Rounding this to two decimal places gives us approximately:
\[ \text{Percent Decrease} \approx 73\% \]

### Final Answer
The percent decrease in the expression \( x^3 y^2 \) is approximately **73%**.