Problem 82 - College Algebra Essentials

Question

Perform the indicated operations. Simplify the result, if possible. 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 simplified expression is \(\frac{2 r_{1} r_{2}}{ r_{1} + r_{2}}\), and the average speed for the round trip given \(r_{1}=40\) miles per hour and \(r_{2}=30\) miles per hour is approximately 34.29 miles per hour. This result is different from the plain average of the two speeds (35 miles per hour) because it accounts for the unequal travel times at each of the two speeds during the round trip.

Step 1: Simplify the expression

To simplify a complex fraction, one strategy is to multiply the numerator and the denominator by the same number so that the denominator becomes simpler. In this case, \(r_{1} r_{2} (r_{1}^{-1}+r_{2}^{-1})=r_{1}+r_{2}\). Therefore, the original expression can be simplified to \(\frac{2 d r_{1} r_{2}}{ d (r_{1} + r_{2})}\). The term \(d\) can be canceled since it is both in the numerator and the denominator. The simplified expression is: \(\frac{2 r_{1} r_{2}}{ r_{1} + r_{2}}\).

Step 2: Find the average rate

Substitute \(r_{1}=40\) miles per hour and \(r_{2}=30\) miles per hour into the simplified expression: \(\frac{2*40*30}{40+30} = \frac{2400}{70} \approx 34.29\) miles per hour.

Step 3: Explain The Results

The average speed of the round trip commute is not 35 miles per hour because the arithmetic mean of the two speeds does not account for the fact that the travel times for each leg of the trip are not equal. It will take longer time to cover the same distance at a slower speed, so the slower speed has more weight when calculating the average speed of the entire round trip. The harmonic mean gives the correct weight to each speed by taking into account the travel time for each leg of the trip, and thus yields the average speed for the entire round trip.

Key Concepts

Complex Rational Expression

A complex rational expression is a fraction where the numerator, the denominator, or both, are themselves fractions. Some might find them intimidating at first look, but they are nothing more than fractions within fractions. To simplify these expressions, the key is to eliminate the smaller fractions to make the overall fraction easier to work with. This often involves multiplying the numerator and the denominator by the least common denominator of the smaller fractions it contains.

For our example of the round-trip commute rate: the expression given is \( \frac{2d}{\frac{d}{r_1} + \frac{d}{r_2}} \). Here, the numerator \(2d\) is straightforward, but the denominator is a complex expression, containing fractions \(\frac{d}{r_1}\) and \(\frac{d}{r_2}\). By eliminating the smaller fractions, the expression simplifies dramatically, helping to see the real mathematical relationship more clearly.

Recognizing and simplifying complex rational expressions is a valuable skill, especially in topics involving rates, work problems, and mixture problems.

Harmonic Mean

The harmonic mean is a type of average, specifically useful for rates, like speeds or densities, where you are averaging over reciprocal values. Unlike the more familiar arithmetic mean, which simply adds values and divides by the number of values, the harmonic mean is defined as the reciprocal of the arithmetic mean of the reciprocals of the numbers.

In formula terms, for two rates \(r_1\) and \(r_2\), the harmonic mean \(H\) is calculated as:

• \(H = \frac{2}{\frac{1}{r_1} + \frac{1}{r_2}}\)

For a round-trip commute, using the harmonic mean is ideal because it considers the time spent traveling each leg of the trip at different speeds. In our context, this is exactly why the computed average rate of around 34.29 mph works out rather than just averaging \(40\) mph and \(30\) mph to find \(35\) mph.

Harmonic means are critical in travel and communication problems, where the time required to perform tasks is inversely proportional to the rate.

Algebraic Simplification

Algebraic simplification is the process of transforming an algebraic expression into a simpler or a more efficient form. This involves combining like terms, reducing fractions, and applying mathematical operations to make expressions easier to understand and solve. The goal is to transform complex expressions into more manageable forms, which helps in solving equations and understanding underlying relationships.

In our example, the expression \(\frac{2dr_1r_2}{d(r_1 + r_2)}\) simplifies easily by cancelling the \(d\)'s from the numerator and denominator. What's left is \(\frac{2r_1r_2}{r_1 + r_2}\), giving a clearer picture of the relationship between the variables involved.

Mastering algebraic simplification allows for solving more complex problems efficiently. When facing real-world problems, such as calculating trip rates, the ability to simplify expressions aids in drawing practical conclusions and making informed decisions.