Percentage Error Calculation

This problem requires calculating the value of a positive real number q based on a percentage error in a mathematical operation.

Understanding the Calculation Error

Let the two positive real numbers be p and q.

• The intended calculation was multiplication: $P_{correct} = p \times q$.
• The student performed division instead: $P_{actual} = p / q$.
• The percentage error is given as 80%, which equals 0.80 as a decimal.

Solving for q using Percentage Error Formula

The standard formula for percentage error is:

$ \text{Percentage Error} = \frac{| \text{Correct Value} - \text{Actual Value} |}{\text{Correct Value}} \times 100\% $

Substitute the known values and expressions:

$ 0.80 = \frac{|p \times q - p / q|}{p \times q} $

Simplify the equation:

$ 0.80 = \left| \frac{p \times q}{p \times q} - \frac{p / q}{p \times q} \right| $

$ 0.80 = \left| 1 - \frac{1}{q^2} \right| $

We need to solve for q. Consider the two possibilities arising from the absolute value:

1. Case 1: $1 - \frac{1}{q^2} = 0.80$

   Rearranging the terms:

   $ \frac{1}{q^2} = 1 - 0.80 $

   $ \frac{1}{q^2} = 0.20 $

   $ q^2 = \frac{1}{0.20} = 5 $

   Since q is a positive real number, $q = \sqrt{5}$.

2. Case 2: $1 - \frac{1}{q^2} = -0.80$

   Rearranging the terms:

   $ \frac{1}{q^2} = 1 - (-0.80) $

   $ \frac{1}{q^2} = 1.80 $

   $ q^2 = \frac{1}{1.80} = \frac{10}{18} = \frac{5}{9} $

   Since q is a positive real number, $q = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}$.

Comparing the possible values of q ($ \sqrt{5} $ and $ \frac{\sqrt{5}}{3} $) with the given options, $ q = \sqrt{5} $ is present as an option.