Problem 49 - Introductory Algebra for College Students

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.

Step-by-Step Solution

Answer

Thus, the average speed for the round trip, given the outgoing rate of 40 mph and return rate of 30 mph, is approximately 34.28 mph.

Step 1: Simplify the Expression

Begin by replacing \(d\) with 1, because this will not affect the final result due to the ratio and it will simplify the equation. This leaves \(\frac{2}{\frac{1}{r_{1}}+\frac{1}{r_{2}}}\). Multiply the denominator by \(r_{1}r_{2}\) to get rid of the fractions. This gives us \(\frac{2r_{1}r_{2}}{r_{1}+r_{2}}\).

Step 2: Substituting the values of \(r_{1}\) and \(r_{2}\)

The values given in the problem are \(r_{1}=40\) mph (rate going to the campus) and \(r_{2}=30\) mph (rate returning home). Substitute these values into the simplified expression: \(\frac{2*40*30}{40+30}\).

Step 3: Solve

After substitution, proceed by multiplying and adding in the numerator and denominator, respectively. This yields \(\frac{2*40*30}{70}\). That leads us to the expression \(\frac{2400}{70}\). Upon simplification, you get an answer of \(34.28\) mph (rounded to 2 decimal places).

Key Concepts

Complex Rational Expressions

Complex rational expressions can seem a bit intimidating at first, due to their intricate structure. They involve rational expressions where the numerator or the denominator (or both) contain rational expressions themselves. In simple terms, they are fractions within fractions.
When dealing with a complex rational expression like the one presented in the problem,

• Identify the inner fractions.
• Determine a common denominator.
• Multiply to eliminate these fractions.

In our exercise, the expression \( \frac{2d}{\frac{d}{r_{1}}+\frac{d}{r_{2}}} \) consists of a complex fraction where the denominator itself is made up of two fractions, \( \frac{d}{r_{1}} \) and \( \frac{d}{r_{2}} \). Breaking down this complexity step-by-step can make the problem more approachable and easier to understand. Just think of it as solving a puzzle, where you bring all the pieces together.

Algebraic Simplification

Algebraic simplification of expressions involves reducing them to their simplest form. Simplifying a complex rational expression requires careful manipulation of numerators and denominators.
To achieve this, follow these steps:

• Eliminate fractions within the numerator and the denominator by finding a common factor.
• In our specific problem, simplify \( \frac{2}{\frac{1}{r_{1}}+\frac{1}{r_{2}}} \) by multiplying the entire expression by \( r_{1}r_{2} \), effectively getting rid of the smaller fractions.
• This results in a much cleaner expression: \( \frac{2r_{1}r_{2}}{r_{1}+r_{2}} \).

This makes substituting given values considerably easier and efficient. By simplifying complex problems, you can make calculations less cumbersome and more intuitive.

Mathematical Problem Solving

Mathematical problem solving is a skill that extends beyond simply following steps or procedures. It requires understanding the problem, connecting it to known concepts, and applying the correct operations to arrive at a solution.
In this exercise, you're tasked with finding an average rate from provided rates. This involves substituting known values into the simplified expression and solving.

• Substitute the given rates, \( r_{1} = 40 \) mph and \( r_{2} = 30 \) mph, into the expression \( \frac{2r_{1}r_{2}}{r_{1}+r_{2}} \).
• Then, perform the arithmetic: \( \frac{2400}{70} \) leads to a final average rate of \( 34.28 \) mph.

A successful problem-solving approach doesn't just stop at finding the answer. Understanding why this method works and being able to adjust it for different scenarios is what solidifies your mastery over the topic. Practice, patience, and persistence are key in solving mathematical problems efficiently.