Problem 82 - College Algebra

Question

The average rate 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 \(r_{1}\) and \(r_{2}\) are the average rates on the outgoing and return trips, respectively. Simplify the expression. Then find your average rate if you drive to campus averaging 40 miles per hour and return home on the same route averaging 30 miles per hour. Explain why the answer is not 35 miles per hour.

Step-by-Step Solution

Answer

The average speed for the round trip, where the outgoing trip speed is 40 mph and the return trip speed is 30 mph, is approximately 34.29 mph, which is less than 35 mph due to the car spending more time on the slower leg of the trip.

Step 1: Simplify the expression

Given formula for average rate is \( \frac{2d}{\frac{d}{r_{1}} + \frac{d}{r_{2}}} \). Notice first that \(d\) is common in the numerator and the denominator, and can be canceled out, so we get: \( \frac{2}{\frac{1}{r_{1}} + \frac{1}{r_{2}}} \). Next, we can simplify this complex fraction by multiplying the numerator and the denominator by \( \frac{r_{1} r_{2}}{r_{1} r_{2}} \), which turns this into: \( \frac{2 r_{1} r_{2}}{r_{1} + r_{2}} \).

Step 2: Substitution

Given that \(r_{1}\) is the average rate for the outgoing trip, 40 miles per hour, and \(r_{2}\) is the average rate for the return trip, 30 miles per hour. Substitute these values into the simplified expression, \( \frac{2 r_{1} r_{2}}{r_{1} + r_{2}} \), which becomes \( \frac{2 \cdot 40 \cdot 30}{40 + 30} = \frac{2400}{70} = 34.2857 \) miles per hour.

Step 3: Explaining the results

Now, one might reasonably ask: why isn't the average speed just the simple average of the two speeds, i.e., 35 miles per hour? That's because average speed is not simply the arithmetic mean. The average speed for the whole journey is the total distance divided by the total time. The time spent at each speed impacts the average, and the car spent more time traveling at 30mph than at 40mph during the round trip, which is why the average speed is slightly less than 35 mph.