Question

Roberta drove 50 miles in 2 hours. Her rate in miles per hour is equivalent to which of the following proportions? Indicate all such proportions. a. 5 to 20 b. 100 to 4 c. 400 to 16 d. 20 to 500 e. 520 to 20

Short answer

The proportions equivalent to Roberta's rate are (b) 100 to 4 and (c) 400 to 16.

Step-by-step solution

Identify the rate

In this case, Roberta's rate is 50 miles per 2 hours, which reduces to 25 miles per 1 hour. This is done by dividing 50 by 2.

Check each proportion

Now, each given proportion should be checked to see if it reduces to the same rate as Roberta's. Begin by converting each proportion to a rate per hour. a. \( \frac{5}{20} = \frac{1}{4} \) miles per hour b. \( \frac{100}{4} = 25 \) miles per hour c. \( \frac{400}{16} = 25 \) miles per hour d. \( \frac{20}{500} = \frac{1}{25} \) miles per hour e. \( \frac{520}{20} = 26 \) miles per hour

Identify the equivalent proportions

b. 100 to 4 and c. 400 to 16 are the only proportions that reduce to a rate of 25 miles per hour. Both of these proportions are equivalent to Roberta's rate per hour. Thus, options b and c are the correct answers.

Key concepts

Proportions

Proportions are mathematical expressions that show the equality between two ratios. Understanding proportions is crucial in solving problems like the one discussed in the exercise, as they allow you to compare different scenarios by equating them to each other. For example, if you have a rate of

• 100 miles in 4 hours, the proportion is expressed as \( \frac{100}{4} \).
• Similarly, if there's another rate of 400 miles in 16 hours, it simplifies to the same proportion \( \frac{400}{16} \).

Both of these are equivalent to each other because they reduce to the same value. The exercise demonstrates this concept by asking you to identify which given proportions equal Roberta’s driving rate.
When dealing with proportions, it's important to focus on simplification. Reduce each ratio to its simplest form to easily compare it to the desired rate.

Rate Conversion

Rate conversion is the process of transforming a given rate into another unit or proportion that makes comparison straightforward. In the context of GRE math problems, rate conversion is essential for quickly solving and contrasting different speeds. Here, we converted Roberta's driving rate into miles per hour for a clear understanding.

• Starting with 50 miles in 2 hours, dividing both the numerator and denominator gives us \( \frac{50}{2} = 25 \) miles per hour.

By converting each of the given proportions into the miles per hour format, it becomes easier to directly compare them with Roberta’s rate of 25 miles per hour. In practice, this simplifies the resolution of complex problems by making direct numerical comparisons possible, sidestepping potential errors in perception linked to non-uniform units.

Miles per Hour

Miles per hour (mph) is a unit of speed commonly used in the United States to express the number of miles traveled in one hour. It's a straightforward measure that allows for easy calculations and comparison of speeds. This unit is particularly useful in everyday movements like driving.
Understanding how to break down a travel distance and time into this unit is key for solving many GRE math problems. Using Roberta's example:

• 50 miles in 2 hours translates to \( \frac{50}{2} = 25 \) miles per hour.

This calculation aligns with her actual traveling speed. Comparing given proportions to this figure allows you to identify which scenarios are similar, making it an essential skill in both academic settings and real-world applications. Grasping this concept will boost your problem-solving skills in various exams and tests.

Problem-Solving Steps

The process of problem-solving in mathematics involves clear systematic steps:

• Start by understanding the problem - In this exercise, it is about identifying equivalent rates.
• Break down the data - Calculate the standard rate first, like Roberta's 25 mph.
• Convert and compare - Simplify each given proportion to see which aligns with the 25 mph rate.
• Identify solutions - Filter options based on simplified proportions that match the calculated rate.

Each step methodically narrows down your choices, making the answer evident and reducing errors. This approach not only applies to GRE math problems but is also useful in solving logic and analytical questions. An efficient problem-solving strategy relies on clarity, simplification, and direct comparison, allowing you to tackle even the most challenging question with confidence.