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Chapter 7: Problem 51

Find the value of \(x\) in each proportion. $$ \frac{x}{5}=\frac{8}{11} $$

Short Answer

Expert verified
x = \( \frac{40}{11} \)

Step by step solution

01

Understand the Proportion

We are given the proportion \( \frac{x}{5} = \frac{8}{11} \). Our task is to find the value of \( x \) that makes this equation true.
02

Set Up the Cross-Multiplication

In a proportion \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication means \( a \cdot d = b \cdot c \). Apply this to our proportion: \( x \cdot 11 = 5 \cdot 8 \).
03

Solve the Equation for x

First multiply the right-hand side: \( 5 \cdot 8 = 40 \). So the equation becomes \( 11x = 40 \). Now, solve for \( x \) by dividing both sides by 11: \( x = \frac{40}{11} \).
04

Simplify the Result (if necessary)

Check if \( \frac{40}{11} \) can be simplified. In this case, since 40 and 11 have no common divisors, \( \frac{40}{11} \) is already in its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cross-Multiplication
The concept of cross-multiplication is a handy method used in algebra to solve equations involving proportions. A proportion is an equation that states two ratios are equal. For example, in the exercise, the proportion given is \( \frac{x}{5} = \frac{8}{11} \).
To use cross-multiplication, you multiply each number in the numerator of one ratio by the denominator of the other, which allows you to eliminate the fractions and work with a simple equation. Here鈥檚 how you handle it:
  • Identify the cross-products in the proportion. For \( \frac{x}{5} = \frac{8}{11} \), the cross-products are \( x \cdot 11 \) and \( 5 \cdot 8 \).
  • You set these two cross-products equal to each other, forming the equation: \( x \cdot 11 = 5 \cdot 8 \).
When you do this, you create an equation without fractions, making the problem easier to solve in the next steps.
Solving Equations
Once you鈥檝e cross-multiplied, you're left with an equation that appears simpler to solve. In this example, you have: \( 11x = 40 \). Here, our only variable is \( x \), and our job is to isolate it to find its value.
Solving an equation generally involves:
  • Performing operations that help isolate the variable on one side of the equation.
  • Whatever operation you perform on one side, you must also perform on the other.
For \( 11x = 40 \), solving involves dividing both sides of the equation by 11 to solve for \( x \). This gives:
  • \( x = \frac{40}{11} \)
At this stage, you've found the numerical value for \( x \) in the proportion equation.
Simplifying Fractions
When you reach a solution that is a fraction, as was \( x = \frac{40}{11} \) in this example, it is important to check if the fraction can be simplified further.
A simplified fraction is one in which the numerator and denominator have no factors in common except 1. The steps are:
  • Find the greatest common divisor (GCD) of the numerator and denominator.
  • Divide the numerator and the denominator by the GCD.
In this case, 40 and 11 were checked for common factors, and none were found other than 1, which confirms \( \frac{40}{11} \) is already in its simplest form. Simplifying fractions is important for clarity and often required for final answers in mathematical problems.

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