A student was supposed to multiplya positive real number $p$ with another positive real number $q$. Instead, the student divided $p$ by $q$. If the percentage error in the student's answer is 80%, the value of $q$ is

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.