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Chapter 5: Problem 21

Solve. $$ \frac{6}{11}=\frac{12}{x} $$

Short Answer

Expert verified
x = 22

Step by step solution

01

Understand the equation

The given equation is \ \( \frac{6}{11}=\frac{12}{x} \). This is a proportion, which states that two ratios are equal.
02

Cross-multiply to solve for x

Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction. Multiply 6 by x and 12 by 11 to get: \ \( 6x = 12 \times 11 \)
03

Simplify the multiplication

Next, simplify the right side of the equation: \ \( 6x = 132 \)
04

Solve for x

Now, solve for x by dividing both sides of the equation by 6: \ \( x = \frac{132}{6} \). Simplify the fraction to get: \ \( x = 22 \)

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Key Concepts

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

cross-multiplication
Cross-multiplication is a straightforward way to solve proportions. It involves multiplying the numerator of one fraction by the denominator of the other, and vice versa, to eliminate the fractions and get an equation with whole numbers. For example, given \(\frac{6}{11}=\frac{12}{x}\), cross-multiplying gives us \(6 \cdot x = 11 \cdot 12\). This transformation simplifies our problem, making it easier to solve for the unknown variable.
solving equations
Solving equations is a key skill in algebra. Once you've set up your equation using cross-multiplication, the next step is to isolate the variable you鈥檙e solving for. From our example, we have \(6x = 132\). To isolate \(x\), divide both sides of the equation by 6: \(x = \frac{132}{6}\). Simplifying, we find \(x = 22\). The isolated variable tells us its value, helping us solve the proportion.
ratios
Ratios are a way to compare two quantities. They can be written in several forms, such as \(\frac{A}{B}\), \(A:B\), or \(A \, \text{to} \, B\). In our exercise, the ratios \(\frac{6}{11}\) and \(\frac{12}{x}\) compare quantities in proportion to each other. By understanding ratios, one can see that the relationship between 6 and 11 is the same as the one between 12 and 22, as confirmed by solving the equation. Recognizing and working with ratios helps us solve practical problems and understand relationships between numbers. Additionally, always check your final answer to ensure that the original proportions or relationships are maintained.

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