Question

Which answer expresses the rate 200 miles per 4 hours as a unit rate? A. \(\frac{200 \text { miles }}{4 \text { hours }}\) B. \(50 \frac{\text { miles }}{\text { hours }}\) C. \(\frac{1 \text { hour }}{50 \text { miles }}\) D. \(\frac{4 \text { hours }}{200 \text { miles }}\)

Short answer

B. 50 miles per hour.

Step-by-step solution

Identify the Given Rate

We are given the rate 200 miles per 4 hours. This can be written as \ \(\frac{200 \text{ miles}}{4 \text{ hours}}\).

Understand Unit Rate

A unit rate is a rate with a denominator of 1. Therefore, we need to convert \(\frac{200 \text{ miles}}{4 \text{ hours}}\) into a rate with 1 hour as the denominator.

Divide Numerator and Denominator by 4

To find the unit rate, divide both the numerator and the denominator by 4: \ \(\frac{200 \text{ miles}}{4 \text{ hours}} \div 4 = \frac{200 \div 4 \text{ miles}}{4 \div 4 \text{ hours}} = \frac{50 \text{ miles}}{1 \text{ hour}}\).

Select the Correct Answer

Compare this result with each of the choices given in the problem. The result \(\frac{50 \text{ miles}}{1 \text{ hour}}\) matches option B.

Key concepts

Rate Conversion

Rate conversion involves changing one rate into another form while maintaining the same value. Imagine you have a rate of 200 miles per 4 hours, written as \(\frac{200 \text{ miles}}{4 \text{ hours}}\). The essence here is to transform this into a more useful or understandable form without losing its original meaning. Commonly, you'll find rates transformed to match more practical scenarios.
For example, converting a speed measured over several hours into a unit rate like miles per hour allows you to understand the speed more clearly. The key is ensuring that the numerical value remains equivalent, even though it looks different.
Start by identifying the original rate and then decide what unit you need to convert it to. If the task is to find the speed per 1 hour, you'll divide by the given time in the rate to isolate just 1 unit of time in the denominator.
When converting rates, remember:

• Identify the original rate
• Decide the unit for the new rate
• Use division or multiplication to adjust the units while keeping the rate's value intact

Unit Rate

A unit rate is a rate where the denominator is 1. It simplifies understanding how much of something occurs per single unit of measure. For example, the unit rate for 200 miles per 4 hours is calculated by reducing the rate to miles per 1 hour.
To find a unit rate, perform a division where you adjust your rate so that the denominator equals 1. Let's break it down:
Given: \(\frac{200 \text{ miles}}{4 \text{ hours}}\)
To find miles per 1 hour, divide both the numerator (miles) and denominator (hours) by the denominator (4 hours):\[ \frac{200 \text{ miles}}{4 \text{ hours}} \rightarrow \frac{200 \text{ miles}/ \text{4}}{4 \text{ hours} / \text{4}} = \frac{50 \text{ miles}}{1 \text{ hour}} \]
By simplifying this way, you get 50 miles per hour, giving you a clearer understanding of the speed. Unit rates are essential for comparing different rates and making sense of measurements in real-world contexts.

Division of Fractions

Dividing fractions ensures you correctly calculate scenarios involving parts of a whole, or in this context, breaking down a rate. To divide a fraction by another number, you actually multiply by its reciprocal.
For example, if you have \(\frac{200 \text{ miles}}{4 \text{ hours}}\), and you need to find miles per 1 hour, you divide by the denominator (4 hours). This is done by multiplying by the reciprocal of 4, which is \(\frac{1}{4}\):\[ \frac{200 \text{ miles}}{4 \text{ hours}} \times \frac{1}{4} = \frac{200 \times 1}{4 \times 1} = \frac{200}{4} = 50 \]
This calculation shows that for every 1 hour, there are 50 miles traveled. Understanding the division of fractions helps simplify rates and convert them to unit rates. Whenever you come across a rate conversion, remembering this method will be very helpful.
Key steps include:

• Identify the fraction part of the rate (numerator/denominator)
• Divide by the denominator by multiplying with its reciprocal
• Simplify to find the unit rate