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
Find the number of integers between $1$ and $150$ (inclusive) having $7$ as one of the digits but which are not divisible by $7$.
Integers are listed from 700 to 1000. In how many integers is the sum of the digits 10 ?
Using 2, 2, 3, 3, 3 as digits, how many distinct numbers greater than 30000 can be formed ?
Consider the following statements :
1. The sum of 5 consecutive integers can be 100.
2 The product of three consecutive natural numbers can be equal to their sum.
Which of the above statements is/are correct ?
The difference between a 2-digit number and the number obtained by interchanging the positions of the digits is 54.
Consider the following statements:
1. The sum of the two digits of the number can be determined only if the product of the two digits is known.
2. The difference between the two digits of the number can be determined.
Which of the above statements is/are correct?
