Question

The price of a chocolate is increased by \( x% \) and then reduced by \( x% \). The new price is 96.76% of the original price. Then \( x \) is:

Hint:

For successive percentage changes of \( x\% \) increase and \( x\% \) decrease, the formula is: \[ \text{Net effect} = -\frac{x^2}{100}\% \]

Answer 1

 

1. Representing the Price Changes:
   Let the original price of the chocolate be \( P \).
   
   • Increasing the price by \( x\% \) means the new price is:

\[ P + \frac{x}{100}P = P \left(1 + \frac{x}{100}\right) \]

• Reducing this price by \( x\% \) means multiplying by \( \left(1 - \frac{x}{100}\right) \).

1. Setting Up the Equation:
   The final price is 96.76% of the original price, so:

\[ P \left(1 + \frac{x}{100}\right) \left(1 - \frac{x}{100}\right) = 0.9676P \]

1. Simplifying and Solving:
   Divide both sides by \( P \) (since \( P \neq 0 \)):

\[ \left(1 + \frac{x}{100}\right) \left(1 - \frac{x}{100}\right) = 0.9676 \]

1. Simplify the left-hand side:

\[ \left(1 + \frac{x}{100}\right) \left(1 - \frac{x}{100}\right) = 1 - \left(\frac{x}{100}\right)^2 \]

1. Thus:

\[ 1 - \left(\frac{x}{100}\right)^2 = 0.9676 \]

\[ \left(\frac{x}{100}\right)^2 = 1 - 0.9676 = 0.0324 \]

1. Take the square root:

\[ \frac{x}{100} = \sqrt{0.0324} = 0.18 \]

\[ x = 0.18 \times 100 = 18 \]

Answer: \( x = 18 \)