Problem 1 - Prealgebra

Question

Determine whether the set of numbers in each table is proportional. $$\begin{array}{l|c|c|c|c|}\hline \text { Blue Paint (quarts) } & 1 & 2 & 3 & 4 \\ \hline \text { Yellow Paint (quarts) } & 5 & 6 & 7 & 8 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The set is not proportional because the ratios are not constant.

Step 1: Understanding Proportions

A proportional relationship means that two quantities have a constant ratio. In this case, we want to see if the ratio of blue paint to yellow paint is constant for all pairs.

Step 2: Calculate Ratios

Calculate the ratio of blue paint to yellow paint for each pair. For 1 blue paint to 5 yellow paint, the ratio is \( \frac{1}{5} \). For 2 blue paint to 6 yellow paint, the ratio is \( \frac{2}{6} \). For 3 blue paint to 7 yellow paint, the ratio is \( \frac{3}{7} \). For 4 blue paint to 8 yellow paint, the ratio is \( \frac{4}{8} \).

Step 3: Simplify Ratios

Simplify the calculated ratios: \( \frac{1}{5} \) is already in simplest form. Simplify \( \frac{2}{6} \) to \( \frac{1}{3} \), and \( \frac{4}{8} \) to \( \frac{1}{2} \). \( \frac{3}{7} \) is already in simplest form.

Step 4: Compare Ratios

Compare all the simplified ratios to determine if they are the same: \( \frac{1}{5} \), \( \frac{1}{3} \), \( \frac{3}{7} \), and \( \frac{1}{2} \). Since none of these ratios are equal, the relationship is not proportional.

Key Concepts

Understanding Ratios

Ratios are fundamental in understanding mathematical relationships between quantities. A ratio is simply a way of comparing two amounts, showing how many times one number is contained within the other. For example, if you have 1 quart of blue paint and 5 quarts of yellow paint, the ratio is written as \( \frac{1}{5} \). This means for every 1 quart of blue paint, there are 5 quarts of yellow paint.

Ratios can help you understand relationships between different quantities with ease. A common way to express ratios is in fractions, allowing you to see the exact relationship between the two measurements. It's important to note that ratios should always be simplified to their simplest form, as it helps make comparisons more straightforward.

Simplifying Fractions

Simplifying fractions is a key mathematical skill that makes handling ratios much easier. When you simplify a fraction, you're reducing it so that its numerator and denominator no longer have any common factors. This means dividing both the top and bottom by their greatest common divisor.

• For instance, in the ratio \( \frac{2}{6} \), both the numerator and denominator can be divided by 2, simplifying the ratio to \( \frac{1}{3} \).
• Another example is \( \frac{4}{8} \), which can be simplified to \( \frac{1}{2} \) since both 4 and 8 are divisible by 4.

Simplifying fractions makes it easier to compare and understand the relationships they represent. Always check if the numbers in your ratios can be reduced, as this will lead to clearer insights into the nature of proportional relationships.

The Concept of Constant Ratio

For two quantities to have a constant ratio, their comparison must yield the same value every time. This constancy suggests a proportional relationship, where one quantity consistently varies with the other at the same rate.

• Think of a scenario where doubling one quantity precisely doubles the other, preserving the original ratio.
• For instance, traveling at a constant speed is akin to maintaining a constant ratio of distance to time.

In the provided exercise, each pair of paints should ideally give the same simplified ratio to indicate that the relationship between blue and yellow paints is proportional. For a valid proportional relationship, the fractions representing each pair must simplify to the same value.

Comparing Fractions for Proportionality

To ascertain whether two sets of quantities are proportional, it's vital to compare their simplified ratios. If these ratios are equal, then the quantities have a proportional relationship. If they differ, they do not.

• After simplifying, ratios like \( \frac{1}{5} \), \( \frac{1}{3} \), \( \frac{3}{7} \), and \( \frac{1}{2} \) each represent unique relationships. None of these are equal, indicating no proportionality.
• Comparing fractions is therefore a direct pathway to understanding the nature of relationships between different quantities.

This approach relies on the mathematical principle that only equal fractions reflect a true proportional relationship. In our exercise, because the fractions are not equal, the relationship between the blue and yellow paints is deemed not proportional.