Question

a and b are inversely proportional to each other and are positive. If a increases by \(100%\), then b decreases by:

• A. \(50%\)
• B. \(75%\)
• C. \(100%\)
• D. \(200%\)

Hint:

In inverse proportionality problems, if one variable changes by a certain percentage, the reciprocal factor determines the change in the other variable.

Answer 1

The Correct Option is A

Step 1: Understand inverse proportionality.
If \(a\) and \(b\) are inversely proportional, then \(a \times b = k\), a constant.
Step 2: Represent initial values.
Let initial \(a = a_0\) and \(b = b_0\), so \(a_0 b_0 = k\).
Step 3: Increase \(a\) by \(100%\).
New \(a\) = \(a_0 + 100%\ \text{of }a_0 = 2a_0\).
Step 4: Find new \(b\).
Since \(a \times b\) remains constant: \[ 2a_0 \times b_{\text{new}} = a_0 b_0 \quad \Rightarrow \quad b_{\text{new}} = \frac{b_0}{2}. \] Step 5: Calculate percentage decrease.
Decrease in \(b\) = \(b_0 - \frac{b_0}{2} = \frac{b_0}{2}\). Percentage decrease: \[ \frac{\frac{b_0}{2}}{b_0} \times 100 = 50%. \] \[ \boxed{50%} \]