Problem 24 - Prealgebra

Question

A new graphics card can increase the resolution of a computer's monitor. Suppose a monitor has a horizontal resolution of 640 pixels and a vertical resolution of 480 pixels. By adding a new graphics card, the resolutions remain in the same proportions, but the vertical resolution increases to 786 pixels. What is the new horizontal resolution?

Step-by-Step Solution

Answer

The new horizontal resolution is 1048 pixels.

Step 1: Understand the Proportion

The aspect ratio (proportion) of the monitor must remain the same. The original aspect ratio is the horizontal resolution divided by the vertical resolution: \( \frac{640}{480} \). This proportion must be maintained with the new vertical resolution.

Step 2: Set Up the Proportion Equation

To find the new horizontal resolution (let's call it \( x \)), we can set up a proportion equation based on the original and new resolutions: \( \frac{640}{480} = \frac{x}{786} \).

Step 3: Solve for the New Horizontal Resolution

Cross-multiply to solve for \( x \): \( 640 \times 786 = 480 \times x \). Simplifying gives \( x = \frac{640 \times 786}{480} \).

Step 4: Calculate the Value

Calculate the value of \( x \) using the equation from Step 3: \( x = \frac{640 \times 786}{480} = 1048 \).

Key Concepts

Solving Equations

In mathematics, solving equations is all about finding an unknown value that satisfies a given mathematical expression. These equations often come in the form of algebraic expressions where one side of the equation is set equal to the other. Solving them involves determining the value that can replace the variables (often denoted by letters like \( x \)) to make the equation true.
To solve an equation, we use various mathematical techniques such as addition, subtraction, multiplication, and division to isolate the variable on one side of the equation. This often involves rearranging the terms in a logical sequence.
Let’s consider our example. We have a proportional equation: \( \frac{640}{480} = \frac{x}{786} \). Here, \( x \) represents the unknown horizontal resolution we want to find. By employing techniques such as cross-multiplication (which we will explore further below), we can solve for \( x \) with ease.

Aspect Ratio

An aspect ratio is the relationship between two quantities displayed in a ratio format. In the context of monitor resolutions, it is expressed as the proportion of its width to its height, usually in pixel count. Common aspect ratios include 4:3, 16:9, etc., which are expressions of the format \( \frac{width}{height} \).
For the monitor in the given exercise, the original resolution is 640 by 480 pixels. When divided, this gives us an aspect ratio of \( \frac{640}{480} = \frac{4}{3} \). This ratio indicates a 4:3 aspect ratio, a traditional format for many screens.
Keeping the aspect ratio consistent is crucial when changing resolutions, as it ensures the displayed image retains its intended shape and does not appear stretched or squished. In our exercise, after the new graphics card is added, the vertical resolution increased to 786 pixels, to maintain the same image proportions, we use the aspect ratio to compute the new horizontal resolution.

Cross-Multiplication

Cross-multiplication is a technique often used to solve proportions or equations involving fractions. It is particularly useful when dealing with equations where we have two ratios set equal to each other.
Consider, for example, our proportion \( \frac{640}{480} = \frac{x}{786} \). In this, cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa. This technique eliminates the fractions, giving us an equation that is easier to solve.
In our scenario, cross-multiplying gives us: \( 640 \times 786 = 480 \times x \). Solving this equation by isolating \( x \) ( \( x = \frac{640 \times 786}{480} \)) allows us to determine the new horizontal resolution of 1048 pixels efficiently.
By using cross-multiplication, we can very effectively bridge from proportions in fractions to more easily managed multiplication and division problems that yield quick answers.