A student was supposed to multiply a 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 answe... A student was supposed to multiply a 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. Read more

Steps to Solve

1. Understanding the percentage error formula

The percentage error is given by the formula:

$$ \text{Percentage Error} = \left| \frac{\text{Actual Value} - \text{Error Value}}{\text{Actual Value}} \right| \times 100 $$

In this case, the actual value (intended operation) is ( p \times q ), and the error value (mistaken operation) is ( \frac{p}{q} ).

2. Setting up the equation

Using the formula, we set up the equation based on the given percentage error of 80%:

$$ 80 = \left| \frac{pq - \frac{p}{q}}{pq} \right| \times 100 $$

Dividing both sides by 100, we have:

$$ 0.8 = \left| \frac{pq - \frac{p}{q}}{pq} \right| $$

3. Simplifying the equation

We can simplify the equation:

$$ 0.8pq = pq - \frac{p}{q} $$

Rearranging gives:

$$ pq - 0.8pq = \frac{p}{q} \implies 0.2pq = \frac{p}{q} $$

4. Finding ( q )

Dividing both sides by ( p ) (since ( p ) is positive):

$$ 0.2q = \frac{1}{q} \implies 0.2q^2 = 1 $$

Multiplying both sides by 5 to eliminate the decimal:

$$ q^2 = 5 $$

5. Finding the final answer

Taking the positive square root (since ( q ) is a positive real number):

$$ q = \sqrt{5} $$