Problem 9 - Prealgebra

Question

Determine whether the set of numbers in each table are proportional. $$\begin{array}{|l|c|c|c|c|}\hline \text { Jars } & 3 & 9 & 12 & 15 \\\\\hline \text { Jelly beans } & 18 & 36 & 54 & 72 \\ \hline\end{array}$$

Step-by-Step Solution

Answer

The sets of numbers are not proportional because the ratios are not constant.

Step 1: Understanding Proportionality

Two sets of quantities are proportional if their ratio is constant. This means for two variables X and Y, they are proportional if \( \frac{X}{Y} \) is constant for all corresponding values.

Step 2: Calculate Ratios

Calculate the ratio of jelly beans to jars for each pair:1. For 3 jars and 18 jelly beans, the ratio is \( \frac{18}{3} = 6 \).2. For 9 jars and 36 jelly beans, the ratio is \( \frac{36}{9} = 4 \).3. For 12 jars and 54 jelly beans, the ratio is \( \frac{54}{12} = 4.5 \).4. For 15 jars and 72 jelly beans, the ratio is \( \frac{72}{15} = 4.8 \).

Step 3: Evaluate Constant Ratio

Examine the calculated ratios: 6, 4, 4.5, and 4.8. Since these values are not the same, the ratio is not constant across the pairs.

Step 4: Conclusion on Proportionality

Since the ratios are not constant, the sets of numbers in the table are not proportional.

Key Concepts

Ratios

In mathematics, a ratio is a way to compare two quantities by showing how many times one value contains or is contained within the other. Ratios are often written as fractions, such as \( \frac{a}{b} \), but they can also be expressed using a colon, like \( a:b \). When comparing quantities using ratios, it's essential to ensure the units are the same to avoid incorrect comparisons.

In our exercise, we compare jelly beans to jars to see how many jelly beans there are per jar. By calculating these ratios, we find out if the two sets of numbers (jars and jelly beans) share a consistent relationship. Though it might seem straightforward, properly understanding and using ratios can reveal vital information about the relationship between different quantities.

Constant Ratio

The concept of a constant ratio is key to understanding proportionality. A constant ratio means the relationship between two variable quantities remains the same across varying instances. It's a fundamental part of identifying proportionality between sets of numbers.

• If you calculate the ratio between two sets of numbers and find that this ratio remains the same, the quantities are considered to be proportional.
• For example, if every time you have 3 jars you have 18 jelly beans, then for any other number of jars, the ratio must remain consistently \( \frac{18}{3} = 6 \).

In the given exercise, the ratios of jelly beans to jars were not constant (6, 4, 4.5, and 4.8), which confirmed the sets were not proportional.

Proportions

A proportion is an equation that states that two ratios are equal. Proportions are commonly used to solve problems involving scaling up or down and predicting unknown quantities. To determine if two sets of numbers form a proportion, you check if their ratios are equal.

In our exercise, if the ratio of jelly beans in each jar was constant and equal, each pair of numbers would form a proportion. For example, 3 jars with 18 jelly beans per jar would proportionally relate to 15 jars with 90 jelly beans, as both would have the same ratio. Proportions are powerful for understanding and working through practical problems related to equivalent relationships.

Mathematics Education

Learning and understanding ratios and proportions is a critical aspect of mathematics education. These concepts are not only foundational for advanced mathematical topics but are also essential for solving real-life problems.

• They help students grasp the concept of multiplicative relationships between quantities.
• This understanding is essential for topics such as scaling, probability, and even financial literacy.
• Educators emphasize the importance of identifying and working with constant ratios to solve practical issues, from adjusting recipes to mixing chemicals accurately.

By developing a strong foundation in these topics, students can better appreciate and apply mathematical principles in diverse fields.