Problem 47 - Prealgebra

Question

A hamburger is formed into the shape of a circle with a radius of \(1 \frac{3}{4}\) inches. If a grill is 28 inches wide, how many hamburgers can fit across the grill?

Step-by-Step Solution

Answer

8 hamburgers can fit across the grill.

Step 1: Understand the problem

We need to find out how many hamburgers can be placed in a straight line across a grill that is 28 inches wide. Each hamburger is circular with a diameter that we have to calculate first.

Step 2: Convert the radius to a mixed number

The radius of the hamburger is given as \(1 \frac{3}{4}\) inches. We can convert this into an improper fraction for easier calculation. \(1 \frac{3}{4} = \frac{7}{4}\).

Step 3: Calculate the diameter of one hamburger

The diameter of a circle is twice the radius. Therefore, the diameter of one hamburger is \[ 2 \times \frac{7}{4} = \frac{14}{4} = 3.5 \text{ inches.} \]

Step 4: Calculate how many hamburgers fit across the grill

To find out how many hamburgers fit across the grill, divide the width of the grill by the diameter of one hamburger:\[ \frac{28}{3.5} = 8. \] This means 8 hamburgers can fit across the grill.

Key Concepts

Understanding Geometry

When we approach problems involving shapes, such as hamburgers shaped like circles, we are diving into the world of geometry. Geometry helps us understand the properties and relations of points, lines, surfaces, and solids. In this exercise, we specifically looked at a circle, which is essential since each hamburger is formed into a circular shape. To solve problems involving circles, remember a few key terms:

• Radius: This is the distance from the center of the circle to any point on its edge. For our hamburger, the radius is given as \(1 \frac{3}{4}\) inches.
• Diameter: This is the distance across the circle, passing through the center. It is simply twice the radius.

Using these properties, you can calculate how many circular hamburgers fit across a linear space like a grill, as we did by moving from radius to diameter.

Dealing with Fractions

Fractions are everywhere, and understanding them is crucial for solving many mathematical problems, especially when measurements aren't whole numbers. In this exercise, the radius is given as a mixed number \(1 \frac{3}{4}\), which combines a whole number and a fraction. To make calculations easier, it helps to convert mixed numbers to improper fractions:

• In \(1 \frac{3}{4}\), multiply the whole number 1 by the denominator 4, giving 4.
• Add the numerator 3, resulting in 7.
• The improper fraction becomes \(\frac{7}{4}\).

Improper fractions allow for straightforward arithmetic operations like multiplication, which is seen when calculating the diameter. Remember, fractions help to simplify and manage numbers that aren't whole, making them essential in precise measurements.

Effective Problem Solving

Approaching a problem strategically can make it much easier to solve, as seen in our hamburger exercise. Effective problem-solving involves:

• Understanding the Problem: Begin by identifying what you need to find out, such as the number of hamburgers that can fit on a grill.
• Breaking Down Information: Analyze the given information step by step, converting measurements if necessary, like changing the radius to a diameter.
• Performing Calculations: Use basic arithmetic to handle measurements. For example, after determining the diameter, dividing the grill width by the hamburger diameter gives the number desired.
• Evaluating the Results: After calculations, confirm that results make sense in the context.

Problem-solving is about organizing the approach, applying the math, and verifying the outcomes to ensure they answer the question correctly.

Using Measurement

Measurement is a core concept in math, especially in geometry and real-world applications. Measurements include units like inches, crucial in determining how things fit together. For our problem, understanding the measurement process involved:

• Recognizing the radius and converting it to a useful number (diameter) for calculations.
• Applying the diameter to different units of length across a given distance, such as the grill's width of 28 inches.
• Using straightforward units consistently helps avoid mistakes and makes estimating and calculating much easier.

Remember, accurate measurements allow for efficiency and precision when solving problems, and this skill is vital in everything from science to engineering and beyond.