Question

A large-screen television has an aspect ratio of 16:9. How tall is the screen if it is 96 centimeters wide? A. 48 centimeters B. 54 centimeters C. 56 centimeters D. 60 centimeters

Short answer

The height of the screen is 54 centimeters. Option B.

Step-by-step solution

- Understand the Aspect Ratio

The aspect ratio of a screen describes the proportional relationship between its width and height. Here, the aspect ratio is given as 16:9, which means for every 16 units of width, there are 9 units of height.

- Set up the Proportion

To find out the height when the width is 96 centimeters, set up the proportion equation: \[\frac{width}{height} = \frac{16}{9}\]Since the width is 96 cm, substitute it into the equation to get: \[\frac{96}{height} = \frac{16}{9}\]

- Solve for Height

To find the height, cross-multiply to get: \[96 \times 9 = 16 \times height\]Calculate: \[864 = 16 \times height\]Now, divide both sides by 16: \[height = \frac{864}{16} = 54 \text{ cm}\]

- Verify the Answer

Check the answer choice that matches the calculated height of 54 cm, which is option B.

Key concepts

proportional relationships

The concept of proportional relationships is fundamental in solving aspect ratio problems. Proportional relationships mean that two quantities maintain a constant ratio. In the given problem, the aspect ratio of 16:9 means that for every 16 units of width, there are 9 units of height. This is a direct example of proportional relationships.
To understand it better, imagine stretching or shrinking the screen while keeping the same aspect ratio. Both the width and height must change proportionally so that the ratio remains 16:9. This proportionality can be represented mathematically as:
\[\frac{width}{height} = \frac{16}{9}\]
When we know one dimension (in this case, the width of 96 cm), we can use this ratio to calculate the other dimension. This highlights how proportional relationships simplify the problem-solving process by providing a clear mathematical pathway to find unknown values.

geometry

Geometry plays a key role in understanding aspect ratios, as it involves measuring shapes and their properties. Here, the screen's shape is a rectangle described by its width and height.
In the problem, we are given the width of the screen and need to find its height based on the aspect ratio. The formula shown earlier \[\frac{width}{height} = \frac{16}{9}\], allows us to visualize the relationship between the two dimensions geometrically. By setting up the proportion, we can see how changing one dimension (width) affects the other (height), while maintaining the same overall shape of the rectangle.
This method can be used in various geometric problems where dimensions need to be scaled while preserving the shape. Understanding the geometric implications of aspect ratios helps build intuition about how objects transform when their sizes change.

problem solving

Problem solving in mathematics often involves breaking down a complex problem into simpler steps. In this aspect ratio problem, we followed a structured approach to find the height of the television screen.
Let's recap the steps: We first understood the aspect ratio, then set up the proportion using the given width, and finally solved for the height using cross-multiplication followed by division.
This methodical approach ensures that we don't overlook any crucial steps and arrive at the correct solution. Key problem-solving strategies here include:

• Understanding the given information
• Setting up a proportion based on the aspect ratio
• Using algebraic manipulation to solve for the unknown value
• Verifying the answer by checking it against the possible choices

By practicing these strategies, students can enhance their problem-solving skills and handle various mathematical challenges with confidence.