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Chapter 7: Problem 8

Find the unknown number in each proportion. $$\frac{1}{6}=\frac{8}{x}$$

Short Answer

Expert verified
The unknown number \( x \) is 48.

Step by step solution

01

Understand the Proportion

The given proportion \( \frac{1}{6} = \frac{8}{x} \) means two ratios are equal. To solve for the unknown \( x \), cross-multiply the fractions.
02

Cross-Multiply

Cross-multiplying involves multiplying the numerator of one fraction with the denominator of the other and equating the products. For our proportion, we get: \( 1 \times x = 6 \times 8 \).
03

Simplify the Equation

Perform the multiplication on the right side of the equation: \( 1x = 48 \).
04

Solve for x

Since \( 1x = x \), this equation simplifies directly to \( x = 48 \).
05

Verify the Solution

Substitute \( x = 48 \) back into the original proportion to check: \( \frac{1}{6} = \frac{8}{48} \). Simplify \( \frac{8}{48} \) to get \( \frac{1}{6} \). Since both sides are equal, the solution is verified.

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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 powerful mathematical technique used to solve equations involving proportions. A proportion is an equation that states that two ratios are equivalent. When we encounter a mathematical statement such as \( \frac{1}{6} = \frac{8}{x} \), our goal is to find the unknown variable, in this case, \( x \). This process can be simplified using cross-multiplication.

To cross-multiply, follow these steps:
  • Multiply the numerator of the first ratio by the denominator of the second ratio.
  • Do the same with the numerator of the second ratio and the denominator of the first ratio.
  • Set these two products equal to each other.
In our example, this means multiplying \( 1 \times x \) and \( 6 \times 8 \). Thus, we have the equation \( 1x = 48 \). This step makes it straightforward to solve for the unknown.
Ratios and Proportions
Understanding ratios and proportions is essential in everyday life and mathematics. A ratio is a way to compare two quantities by division, while a proportion states that two ratios are equal. For instance, the problem \( \frac{1}{6} = \frac{8}{x} \) demonstrates how two ratios are being equated.

When working with ratios and proportions, consider the following:
  • Ratios express a relationship between two quantities, showing how many times one value contains another.
  • A proportion is an equation that equates two ratios, often used to find a missing value when other parts are known.
  • Solving proportions often involves understanding equivalent fractions and applying methods like cross-multiplication.
In our example, \( \frac{1}{6} \) is a ratio expressing one part out of six, and \( \frac{8}{x} \) is another ratio we adjust to maintain equivalency, ultimately finding the value of \( x \).
Verifying Solutions
Verifying a solution is a crucial step to ensure that the answer we obtained is indeed correct. After solving a proportion like \( \frac{1}{6} = \frac{8}{x} \) and finding \( x = 48 \), it's important to substitute back and check if our solution holds true.

Here's how to verify a solution:
  • Substitute the value of the unknown back into the original proportion.
  • Simplify each side of the equation to check if they are truly equal.
  • Trust this step to prevent mistakes that could arise from computational errors or oversight.
In this case, substituting \( x = 48 \) gives us the equation \( \frac{1}{6} = \frac{8}{48} \). Simplifying \( \frac{8}{48} \) results in \( \frac{1}{6} \), confirming both sides of the original proportion are equal. This verification ensures the reliability and correctness of our solution.

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