Problem 21 - Prealgebra

Question

A new graphics card can increase the resolution of a computer's monitor. Suppose a monitor has a horizontal resolution of 800 pixels and a vertical resolution of 600 pixels. By adding a new graphics card, the resolutions remain in the same proportions, but the horizontal resolution increases to \(1,280\) pixels. What is the new vertical resolution?

Step-by-Step Solution

Answer

The new vertical resolution is 960 pixels.

Step 1: Understand the Given Problem

The problem states that a monitor has an initial horizontal resolution of 800 pixels and a vertical resolution of 600 pixels. After an upgrade, the horizontal resolution increases to 1,280 pixels, while the aspect ratio is maintained.

Step 2: Set Up the Proportionality Equation

The resolutions must maintain the same proportion. The original aspect ratio is \( \frac{800}{600} \). With the new horizontal resolution, let's represent the new vertical resolution as \( x \). Thus, \( \frac{1280}{x} = \frac{800}{600} \).

Step 3: Simplify Original Ratio

Simplify the original ratio \( \frac{800}{600} \) to its simplest form. The greatest common divisor of 800 and 600 is 200, so \( \frac{800}{600} = \frac{800 \div 200}{600 \div 200} = \frac{4}{3} \).

Step 4: Solve the Equation

We now have the equation \( \frac{1280}{x} = \frac{4}{3} \). To find \( x \), cross-multiply to get \( 1280 \times 3 = 4 \times x \), leading to \( 3840 = 4x \).

Step 5: Calculate the New Vertical Resolution

Divide both sides by 4 to solve for \( x \): \( x = \frac{3840}{4} = 960 \).

Step 6: Verify the Solution

Verify that the new ratio \( \frac{1280}{960} \) simplifies to \( \frac{4}{3} \). Simplifying gives \( \frac{4}{3} \), confirming the aspect ratio is the same.

Key Concepts

Aspect Ratio

Aspect ratio is a concept used to describe the proportional relationship between two dimensions, such as the width and height of a screen. In the context of the monitor resolution problem, the original aspect ratio is given as \( \frac{800}{600} \). This ratio tells us how the horizontal and vertical resolutions relate to each other.
When upgrading or modifying the resolution, it's essential to maintain the aspect ratio to ensure that images or videos are displayed correctly and are not distorted.

• A constant aspect ratio provides a consistent viewing experience.
• It helps prevent images from appearing stretched or squished.
• In this scenario, maintaining the aspect ratio ensures that the proportions of the original monitor are kept even after the resolution change.

Cross-Multiplication

Cross-multiplication is a method used to solve proportions, which are equations where two ratios are equal. The technique involves multiplying the numerator of one ratio by the denominator of the other and vice versa. This method is particularly useful when the equation consists of fractions.
In the given solution, we have the equation \( \frac{1280}{x} = \frac{4}{3} \). By cross-multiplying, we arrive at the equation \( 1280 \times 3 = 4 \times x \). This results in \( 3840 = 4x \), making it an easier expression to solve for \( x \).

• Cross-multiplication simplifies complex fractions into more accessible equations.
• It provides a straightforward path to solve for unknown variables.
• It's a quick and efficient strategy applied here to maintain the aspect ratio in the problem.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. This means finding a common factor for both the numerator and the denominator and dividing them by it. Simplifying makes fractions easier to work with and prevents unnecessary complexity.
Initially, the ratio \( \frac{800}{600} \) from the problem needs to be simplified for easier calculations. The common factor of 800 and 600 is 200, so when we divide both by 200, we get the simplified fraction \( \frac{4}{3} \).

• Simplifying helps to clarify and reduce fractions to their most basic form.
• It often provides a clearer picture of the mathematical relationship being described.
• For the aspect ratio problem, a simplified fraction offers a better understanding of how the resolutions relate to each other.