Problem 39 - Prealgebra

Question

The classrooms at Lincoln Middle School are painted every summer. If it takes 7 gallons of paint to paint 4 classrooms, how many gallons of paint does it take to paint 16 classrooms? A 9 B 28 C 64 D 112

Step-by-Step Solution

Answer

It takes 28 gallons to paint 16 classrooms.

Step 1: Understand the Problem

We need to calculate how many gallons of paint are needed to paint 16 classrooms, given that 7 gallons can paint 4 classrooms. This involves setting up a proportion based on the given information.

Step 2: Set Up the Proportion

We know that 7 gallons are needed for 4 classrooms. This can be expressed as the ratio \( \frac{7}{4} \). We want to find out how many gallons, say \( x \), are needed for 16 classrooms. Set up the equation: \( \frac{7}{4} = \frac{x}{16} \).

Step 3: Cross-Multiply to Solve for x

Cross-multiplying gives us: \( 7 \times 16 = 4 \times x \). Simplify to find the equation: \( 112 = 4x \).

Step 4: Solve for x

Divide both sides of the equation by 4 to isolate \( x \): \( x = \frac{112}{4} \). Simplifying this division gives us \( x = 28 \).

Step 5: Confirm the Solution

Ensure the solution makes sense by plugging it back into the context of the problem: 28 gallons of paint are needed for 16 classrooms, which matches the initial ratio set for the smaller number of classrooms.

Key Concepts

Ratios

A ratio is a comparison of two quantities expressed as a fraction. In word problems like the one we are discussing, ratios help to compare how much paint is needed for different numbers of classrooms. Consider our example: 7 gallons of paint cover 4 classrooms. We express this ratio as \( \frac{7}{4} \). This means for every 4 classrooms, 7 gallons are used.

• The numerator (top number) represents the paint needed (7 gallons).
• The denominator (bottom number) represents the number of classrooms (4 classrooms).

Think of ratios as a way to scale things up or down. They allow us to maintain the same relationship between two quantities as we deal with different sizes.

Cross-Multiplication

Cross-multiplication is a method for solving equations that involve two ratios. It's useful for determining an unknown value within a proportional relationship. In our classroom painting problem, we have two ratios:

• \( \frac{7}{4} \) — representing 7 gallons for 4 classrooms.
• \( \frac{x}{16} \) — representing an unknown number of gallons \(x\) for 16 classrooms.

By setting them equal to each other as fractions, we can solve for \(x\) using cross-multiplication: multiply the numerator of one fraction by the denominator of the other and set them equal. So, we get \(7 \cdot 16 = 4 \cdot x\). Solving this helps us find the number of gallons needed.

Word Problems

Word problems translate real-world situations into mathematical equations. They involve picking out the essential information and ignoring the fluff. In our paint problem, the key details are:

• 7 gallons cover 4 classrooms.
• We need paint for 16 classrooms.

Recognizing the need to set up a proportion is a critical skill. Word problems like these challenge us to form and solve equations based on the scenarios described, using concepts like ratios and cross-multiplication to find solutions.

Basic Arithmetic Operations

Basic arithmetic operations such as addition, subtraction, multiplication, and division, are fundamental to solving ratio problems. They allow us to handle and manipulate the numbers to find solutions. In our example, we used multiplication and division:

• First, we multiplied: \(7 \times 16\) using cross-multiplication.
• Then, we performed division: \(\frac{112}{4}\) to solve for \(x\).

These operations are the building blocks needed to calculate and verify proportions, ensuring the solution makes sense. Being comfortable with these operations makes problem-solving much easier and more efficient.