Problem 41 - Prealgebra

Question

Determine whether the set of numbers in the table is proportional. $$\begin{array}{|lc|c|c|c|} \hline \text { Time (hours) } & 1 & 2 & 3 & 4 \\ \hline \text { Rental Charge } & \$ 13 & \$ 23 & \$ 33 & \$ 43 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The set of numbers is not proportional.

Step 1: Determine Unit Ratios

Identify the unit ratio for each pair of time (in hours) and rental charge. This is done by dividing the rental charge by the time to find the rate per hour for each point. For the first pair, the unit ratio is \( \frac{13}{1} = 13 \). For the second pair, \( \frac{23}{2} = 11.5 \). For the third pair, \( \frac{33}{3} = 11 \). For the fourth pair, \( \frac{43}{4} = 10.75 \).

Step 2: Compare Unit Ratios

Compare the unit ratios obtained from Step 1. For the set of numbers to be proportional, all the unit ratios should be equal. The calculated unit ratios are 13, 11.5, 11, and 10.75, which are not equal.

Step 3: Conclusion on Proportionality

Since the unit ratios from Step 2 are not all equal, the set of numbers in the table is not proportional. Each increase in time does not result in a consistent increase in rental charge proportional to time.

Key Concepts

Understanding Unit Ratios

A unit ratio is a fundamental concept in mathematics that helps us understand the relationship between two measurements. It's like finding out how much of one thing exists in terms of one unit of another.
For example, in real-world scenarios, we often use a unit ratio to express distances in terms of time, such as miles per hour, or in our example, rental charges per hour.
To calculate a unit ratio, you perform a simple division: divide one measurement by another.

• In our exercise, the rental charges are divided by the time in hours.
• This gives us the rental charge per hour for each timeframe.

Each resulting number is a unit ratio, where we're seeing how rental charge changes for every additional hour. This helps us determine if the rental charge increases uniformly as time increases. It's an essential step to verify if a relationship is proportional.

Exploring Proportional Relationships

Proportional relationships are everywhere! They're the idea that two quantities maintain a consistent ratio to each other. This means if one quantity doubles, the other doubles too - so their ratio stays the same.
In a unique proportional relationship, every pair of numbers you compare should have the same unit ratio.

• Imagine buying pencils: if 3 pencils cost $1, then 6 should cost $2 if the price is proportional.
• Similarly, more pencils should increase in cost at the same rate.

In our exercise, if the rental costs were truly proportional to time, we'd expect the unit ratios to be identical across the board. A steady unit ratio suggests that the relationship between time and rental charge is consistent and proportional. But since our ratios (13, 11.5, 11, and 10.75) differ, the rental cost does not increase uniformly with time. Thus, it confirms a non-proportional relationship.

Interpreting Rate Per Hour in Context

Rate per hour is a practical illustration of unit ratios. It lets us see how something changes as we add one unit of time. In our example, the rate per hour tells us exactly how much rental charge is added with each passing hour.
This concept isn't just about numbers; it's about understanding how things scale:

• If today you work for $20 per hour, knowing this rate lets you calculate your earnings over different time frames, like 3 hours or 8 hours.
• Let's look at the exercise: one rental rate per hour is $13 (on the first calculation), while another is only $10.75 (on the last one).

Since the rates per hour change and are not the same, it implies inconsistency in how charges are determined relative to time. In life, when analyzing bills, wages, or costs, understanding the rate per hour is pivotal in planning and decision-making, helping ensure fairness and predictability.