Simplify Complex Rational Expressions
-The average speed on a round-trip commute having a one-way distance d is given by the complex rational expression $\frac{2 d}{\frac{d}{r_{1}}+\frac{d}{r_{2}}}$
in which r1 and r2 are the speeds on the outgoing and return trips, respectively. Fred and Michael both drove to campus averaging 35 miles per hour and each returned home on the same route he used going and averaged 50 miles per hour. Fred's one-way route was 6 miles longer than Michael's. Simplify the complex rational expression and answer the question: How does Fred's overall average speed compare with Michael's?

Question

Simplify Complex Rational Expressions
-The average speed on a round-trip commute having a one-way distance d is given by the complex rational expression $\frac{2 d}{\frac{d}{r_{1}}+\frac{d}{r_{2}}}$
 in which r1 and r2 are the speeds on the outgoing and return trips, respectively. Fred and Michael both drove to campus averaging 35 miles per hour and each returned home on the same route he used going and averaged 50 miles per hour. Fred's one-way route was 6 miles longer than Michael's. Simplify the complex rational expression and answer the question: How does Fred's overall average speed compare with Michael's?

Quizplus · Accepted Answer

The expression simplifies to the harmonic mean of $r_1$ and $r_2$, which is independent of the distance $d$. Therefore, Fred's and Michael's overall average speeds are the same since they traveled at the same speeds, just over different distances.

Simplify Complex Rational Expressions -The Average Speed on a Round-Trip 2ddr1+dr2\frac{2 d}{\frac{d}{r_{1}}+\frac{d}{r_{2}}}r1​d​+r2​d​2d​

Simplify Complex Rational Expressions
-The Average Speed on a Round-Trip $\frac{2 d}{\frac{d}{r_{1}}+\frac{d}{r_{2}}}$