Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(\frac{x}{48}=\frac{7}{8}\)

Short answer

x = 42

Step-by-step solution

Identify the Cross Multiplication Method

When dealing with proportions, we use cross multiplication for finding the unknown. Given the proportion \( \frac{x}{48} = \frac{7}{8} \), the principle of cross-multiplication states you can find \( x \) by computing \( x \times 8 = 7 \times 48 \).

Perform the Cross Multiplication

Multiply across the equality: \( x \times 8 = 7 \times 48 \). Simplify the right side of the equation by calculating \( 7 \times 48 = 336 \). This gives us the equation \( 8x = 336 \).

Isolate the Variable

To solve for \( x \), you need to isolate it by dividing both sides of the equation by 8: \( x = \frac{336}{8} \).

Calculate the Division

Perform the division to find the value of \( x \). Calculate \( \frac{336}{8} = 42 \). Thus, \( x = 42 \).

Key concepts

Cross Multiplication

Cross multiplication is a fundamental technique in algebra, particularly when working with proportions. It involves multiplying the numerator of one fraction by the denominator of the other fraction. This creates an equation where these two products are equal. This method helps us find unknown variables within a proportion.
For example, in the given proportion \( \frac{x}{48} = \frac{7}{8} \), cross multiplication implies we multiply \( x \) by \( 8 \) and \( 7 \) by \( 48 \). This results in the equation:

• \( x \times 8 = 7 \times 48 \)

This technique effectively eliminates the fractions, allowing us to work with a simple linear equation. It's an essential skill for solving proportion problems because it provides a straightforward pathway to find unknown variables.

Proportion Problems

Proportion problems involve equations where two ratios are set equal to each other. It's like comparing two fractions. When dealing with proportions, the key is to understand that the two ratios are equivalent. This balance allows us to solve for an unknown variable.
To tackle a proportion problem, you should follow these steps:

• Identify the fractions within the equation.
• Use cross multiplication to create an equation without the fractions.
• Solve the resulting equation to find the unknown variable.

In the example \( \frac{x}{48} = \frac{7}{8} \), we identify \( x \) as the unknown and use cross multiplication to set up the equation \( x \times 8 = 7 \times 48 \). Solving this equation reveals the value of \( x \). Proportion problems are common in algebra and are useful in many real-world applications, such as calculating distances, scaling recipes, and more.

Solve for x

Solving for \( x \) means finding the value of this variable that makes the equation true. After using cross multiplication and setting up our equation, the next step is to isolate \( x \) on one side of the equation.
In our specific problem, we've arrived at the equation \( 8x = 336 \). To isolate \( x \), we divide both sides of the equation by \( 8 \):

• \( x = \frac{336}{8} \)

Performing this division gives \( x = 42 \). It's crucial to understand each step in this process because solving for \( x \) is a common task in algebra. It requires balance and manipulation of equations to find the desired value, helping us make sense of numerous mathematical problems.