Write a pair of fractions whose sum is \(\dfrac{7}{11}\) and difference is \(\dfrac{2}{11}\).
\(\dfrac{9}{22}\) and \(\dfrac{5}{22}\)
Let the two fractions be \(a\) and \(b\).
Given:
\(a + b = \dfrac{7}{11}\)
\(a - b = \dfrac{2}{11}\)
Step 1: Find \(a\) by adding the two equations.
\((a + b) + (a - b) = \dfrac{7}{11} + \dfrac{2}{11}\)
Left side simplifies to \(2a\).
Right side is \(\dfrac{9}{11}\).
So, \(2a = \dfrac{9}{11}\).
Divide both sides by \(2\): \(a = \dfrac{9}{11} \times \dfrac{1}{2} = \dfrac{9}{22}\).
Step 2: Find \(b\) by subtracting the second equation from the first.
\((a + b) - (a - b) = \dfrac{7}{11} - \dfrac{2}{11}\)
Left side simplifies to \(2b\).
Right side is \(\dfrac{5}{11}\).
So, \(2b = \dfrac{5}{11}\).
Divide both sides by \(2\): \(b = \dfrac{5}{11} \times \dfrac{1}{2} = \dfrac{5}{22}\).
Answer: The fractions are \(\dfrac{9}{22}\) and \(\dfrac{5}{22}\).
Quick Check:
Sum: \(\dfrac{9}{22} + \dfrac{5}{22} = \dfrac{14}{22} = \dfrac{7}{11}\)
Difference: \(\dfrac{9}{22} - \dfrac{5}{22} = \dfrac{4}{22} = \dfrac{2}{11}\)