Problem 19 - Prealgebra

Question

Write a proportion that could be used to solve for each variable. Then solve. 12 glasses in 3 crates 72 glasses in \(m\) crates

Step-by-Step Solution

Answer

The proportion is \( \frac{12}{3} = \frac{72}{m} \), and \( m \) is 18 crates.

Step 1: Understanding the Proportion

A proportion is a statement that two ratios are equal. Here, we are comparing the ratio of glasses to crates in two scenarios. The first scenario is 12 glasses in 3 crates, and the second is 72 glasses in \( m \) crates.

Step 2: Setting Up the Proportion

Set up the proportion by equating the two ratios: \( \frac{12}{3} = \frac{72}{m} \). This equation expresses that the ratio of glasses to crates in both scenarios is the same.

Step 3: Cross Multiplying

Use cross-multiplication to solve for \( m \). Multiply 12 by \( m \) and 3 by 72: \( 12 \times m = 3 \times 72 \). This simplifies to \( 12m = 216 \).

Step 4: Solving for m

Solve for \( m \) by dividing both sides of the equation by 12:\[ m = \frac{216}{12} = 18 \]. Therefore, \( m \) equals 18 crates.

Key Concepts

Understanding Proportions

In mathematics, a proportion is a statement where two ratios are equal. This concept is crucial for comparing relationships between different groups of items. For example, in our current problem, we have two groups: glasses and crates. A proportion allows us to set the ratio of glasses to crates in one scenario equal to another scenario. By understanding proportions, you can easily find the missing value in a mathematical relationship using known quantities. Proportions are particularly helpful in various real-world applications like cooking, budgeting, and any situation where you need to keep a consistent relationship between quantities.

Exploring Ratios

A ratio is a comparison between two numbers or quantities. It can be expressed in several ways: as a fraction, with a colon, or in words. In our exercise, we compared two ratios: 12 glasses in 3 crates and 72 glasses in an unknown number of crates. By creating a fraction for each scenario, we have

• \( \frac{12}{3} \)
• \( \frac{72}{m} \)

These fractions represent the relationship between glasses and crates in each scenario. By equating these ratios, which we call a proportion, it becomes possible to solve for the unknown quantity \( m \), ensuring both scenarios represent the same relationship.

The Technique of Cross-Multiplication

Cross-multiplication is a powerful technique used to solve proportions. It involves multiplying diagonally across the equal sign in a proportion to simplify the equation. For instance, given the proportion \( \frac{12}{3} = \frac{72}{m} \), you would multiply across as follows:

• 12 multiplied by \( m \) gives \( 12m \)
• 3 multiplied by 72 yields 216

These two products form the equation \( 12m = 216 \), making it much easier to solve for the missing variable. Cross-multiplication works because it maintains the balance of the equation, ensuring that both sides stay equal.

Solving for Variables

When you've set up your equation through cross-multiplication, the next step is solving for the unknown variable. In our case, the equation is \( 12m = 216 \). To isolate \( m \), perform a simple division:

• Divide both sides by 12
• \( m = \frac{216}{12} \)
• Simplify to \( m = 18 \)

This shows that 72 glasses fit into 18 crates. Solving for variables using proportions and cross-multiplication is a fundamental skill. It helps in problem-solving across various scientific and mathematical fields, where understanding how to isolate and solve for unknowns is essential to uncover relationships and outcomes.