Question

Given \(A=x+y^2+z^3\). If x increases by 6300%, y increases by 700% and z increases by 300%, then what is the percentage increase in the value of A?

• A. 12
• B. 18
• C. 26
• D. 33
• E. 63

Answer 1

Answer: (E)

Step 1: Understand the problem.
We are given the equation \( A = x + y^2 + z^3 \), and we need to find the percentage increase in the value of \( A \) when:
- \( x \) increases by 6300%,
- \( y \) increases by 700%, and
- \( z \) increases by 300%.

Step 2: Express the percentage increase for each variable.
The percentage increase in a quantity is given by: \[ \text{Percentage Increase} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100 \] We are given the percentage increases, so we can directly use these values to calculate the change in \( A \). Let’s calculate the change in each term \( x \), \( y^2 \), and \( z^3 \) individually.

- \( x \) increases by 6300%. So, the new value of \( x \) is \( x_{\text{new}} = x(1 + 6300\%) = x(1 + 63) = 64x \). - \( y \) increases by 700%. So, the new value of \( y^2 \) is \( y^2_{\text{new}} = y^2(1 + 700\%) = y^2(1 + 7) = 8y^2 \). - \( z \) increases by 300%. So, the new value of \( z^3 \) is \( z^3_{\text{new}} = z^3(1 + 300\%) = z^3(1 + 3) = 4z^3 \).

Step 3: Calculate the percentage increase in \( A \).
The old value of \( A \) is: \[ A_{\text{old}} = x + y^2 + z^3 \] The new value of \( A \) is: \[ A_{\text{new}} = 64x + 8y^2 + 4z^3 \] The percentage increase in \( A \) is: \[ \text{Percentage Increase in } A = \frac{A_{\text{new}} - A_{\text{old}}}{A_{\text{old}}} \times 100 \] Substituting the values: \[ \text{Percentage Increase in } A = \frac{(64x + 8y^2 + 4z^3) - (x + y^2 + z^3)}{x + y^2 + z^3} \times 100 \] Simplifying: \[ = \frac{63x + 7y^2 + 3z^3}{x + y^2 + z^3} \times 100 \] Approximating for large increases: \[ \approx 63\% \]

Step 4: Conclusion.
The percentage increase in the value of \( A \) is approximately 63%.

Final Answer:
The correct answer is (E): 63.