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.
Understand the Problem
The question involves a scenario where a student intended to multiply two positive real numbers but mistakenly divided them instead. It asks for the value of one of those numbers, given that the resulting percentage error is 80%. This requires understanding the relationship between multiplication, division, and percentage error to find the value of q.
Answer
The value of \( q \) is $q = \sqrt{5}$.
Answer for screen readers
The value of ( q ) is $q = \sqrt{5}$.
Steps to Solve
- 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} ).
- 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:
$$ 0.8 = \left| \frac{pq - \frac{p}{q}}{pq} \right| $$
- Simplifying the equation
We can simplify the equation:
$$ 0.8pq = pq - \frac{p}{q} $$
Rearranging gives:
$$ pq - 0.8pq = \frac{p}{q} \implies 0.2pq = \frac{p}{q} $$
- 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:
$$ q^2 = 5 $$
- Finding the final answer
Taking the positive square root (since ( q ) is a positive real number):
$$ q = \sqrt{5} $$
The value of ( q ) is $q = \sqrt{5}$.
More Information
The problem revolves around understanding how division can lead to an error when one intends to perform multiplication. The percentage error gives insight into how significantly the error affects the result, which is a crucial concept in various fields, particularly engineering and science.
Tips
- Confusing the actual value with the erroneous value can lead to incorrect percentage error computation. Ensure to clearly identify the intended and mistaken operations.
- Neglecting to take the absolute value when setting up the percentage error formula can cause sign errors.
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