p and q are positive integers and \(\frac{p}{q}+\frac{q}{p}=3\) then \(\frac{p^2}{q^2} + \frac{q^2}{p^2} =\)
7
To find the value of the given expression, we start by analyzing the provided equation involving positive integers p and q. The problem presents an initial equation: \(\frac{p}{q}+\frac{q}{p}=3\). Our objective is to determine the value of a related expression: \(\frac{p^2}{q^2} + \frac{q^2}{p^2}\).
Expression Identity and Approach
This type of mathematical problem frequently involves recognizing and applying algebraic identities. We are given a sum of two terms, \(\frac{p}{q}\) and \(\frac{q}{p}\), and we need to find the sum of their squares. A fundamental algebraic identity that connects these forms is the square of a sum: \((a+b)^2 = a^2+2ab+b^2\).
Let's simplify the problem by assigning variables to the terms in our given equation. If we let \(a = \frac{p}{q}\) and \(b = \frac{q}{p}\), then the initial equation can be written as \(a+b=3\).
The algebraic expression we need to find is \(a^2+b^2\).
p and q Expression Derivation
We can use the algebraic identity \((a+b)^2 = a^2+2ab+b^2\) to establish a relationship between the given equation and the required expression. By rearranging this identity, we can isolate \(a^2+b^2\):
\(a^2+b^2 = (a+b)^2 - 2ab\)
Now, substitute \(a = \frac{p}{q}\) and \(b = \frac{q}{p}\) back into this rearranged algebraic identity:
\(\frac{p^2}{q^2} + \frac{q^2}{p^2} = \left(\frac{p}{q} + \frac{q}{p}\right)^2 - 2 \left(\frac{p}{q}\right) \left(\frac{q}{p}\right)\)
Observe the product term \(\left(\frac{p}{q}\right) \left(\frac{q}{p}\right)\). This simplifies nicely:
\(\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = \frac{p \cdot q}{q \cdot p} = 1\)
Substituting this simplification back into our expression for the sum of squares:
\(\frac{p^2}{q^2} + \frac{q^2}{p^2} = \left(\frac{p}{q} + \frac{q}{p}\right)^2 - 2(1)\)
\(\frac{p^2}{q^2} + \frac{q^2}{p^2} = \left(\frac{p}{q} + \frac{q}{p}\right)^2 - 2\)
We are given in the problem statement that \(\frac{p}{q}+\frac{q}{p}=3\). Now, we substitute this value into the simplified expression:
\(\frac{p^2}{q^2} + \frac{q^2}{p^2} = (3)^2 - 2\)
\(\frac{p^2}{q^2} + \frac{q^2}{p^2} = 9 - 2\)
\(\frac{p^2}{q^2} + \frac{q^2}{p^2} = 7\)
Final Expression Value
The calculated value for the expression \(\frac{p^2}{q^2} + \frac{q^2}{p^2}\) is 7.
| Given Information | Expression to Find | Algebraic Identity Used | Calculated Value |
|---|---|---|---|
| \(\frac{p}{q}+\frac{q}{p}=3\) | \(\frac{p^2}{q^2} + \frac{q^2}{p^2}\) | \((a+b)^2 = a^2+b^2+2ab\) | 7 |
Important Questions from Integers
- The average of eleven consecutive positive integers is d. If the last two numbers are excluded, by how much will the average increase or decrease?
The numerator of fraction is 3 more than the denominator. When 5 is added to the numerator and 2 is subtracted from the denominator, the fraction becomes 8/3, When the original fraction is divided by \(5 \frac{1}{2}\) , the fraction so obtained is:
The sum of a non - zero number and twenty times its reciprocal is 9. What is the number?
If \(\frac{{45}}{{53}} = \frac{1}{{a + \frac{1}{{b + \frac{1}{{c - \frac{2}{5}}}}}}},\) where a, b and c are positive integers, then what is the value of (4a - b + 3c)
The denominator of a fraction is 4 more than the double of its numerator. When 3 is added to the numerator and 3 is subtracted from denominator the fraction becomes 2/3. Then find the difference between denominator and numerator of the original fration.
