The Analysis and Value of the Expression ( b^2 - a^2 ) Given ( frac{7}{22} frac{1}{a/b} )

Introduction

The problem presented involves an algebraic equation and its solution within the context of integer values. The equation in question is ( frac{1}{a frac{1}{b}} frac{7}{22} ), which needs to be solved for positive integers ( a ) and ( b ). This investigation will explore the method to find these values and the value of the expression ( b^2 - a^2 ).

Revisiting and Solving the Equation

Starting with the given condition:

( frac{1}{a frac{1}{b}} frac{7}{22} )

Multiplying both sides by ( a frac{1}{b} ), we obtain:

( 1 frac{7}{22} cdot a frac{1}{b} )

Which simplifies to:

( a frac{1}{b} frac{22}{7} )

Expressing ( a frac{1}{b} ) as a mixed number, we get:

( a frac{1}{b} 3 frac{1}{7} )

From the mixed number representation, we can deduce that:

( a 3 ) and ( b 7 )

Verification and Application to the Expression

To verify, substituting ( a 3 ) and ( b 7 ) back into the original condition:

( frac{1}{3 frac{1}{7}} frac{1}{frac{22}{7}} frac{7}{22} )

This confirms that ( a 3 ) and ( b 7 ) are valid solutions. Thus, the value of the expression ( b^2 - a^2 ) is:

( b^2 - a^2 7^2 - 3^2 49 - 9 40 )

Exploring the Infinite Solutions

The given equation is:

( frac{7}{22} frac{1}{a frac{1}{b}} )

Multiplying both sides by ( a frac{1}{b} ) we get:

( 7 times a frac{1}{b} 22 )

Further simplifying, we find:

( a frac{1}{b} frac{22}{7} 3 frac{1}{7} )

From this, it is evident that there are other possible values for ( a ) and ( b ), but the simplest integer solutions are ( a 3 ) and ( b 7 ).

Conclusion

In conclusion, for positive integers ( a ) and ( b ), the value of the expression ( b^2 - a^2 ) given the equation ( frac{7}{22} frac{1}{a frac{1}{b}} ) evaluates to 40. This is derived through the process of solving the given equation and verifying the integer solutions.

Key Takeaways:

Understanding mixed fraction representation is crucial for solving such problems. The derived value for ( a ) and ( b ) (3, 7) satisfies the given equation. The expression ( b^2 - a^2 ) simplifies to 40 under the given constraints.