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Chapter 6: Problem 32

Solve the given proportion. \(\frac{3}{22}=\frac{21}{x}\)

Short Answer

Expert verified
\( x = 154 \)

Step by step solution

01

Identify the Cross Products

In a proportion, the cross products are equal. So, we can set up the equation by multiplying the numerator of the first fraction by the denominator of the second and vice versa: \( 3 \times x = 22 \times 21 \).
02

Calculate the Right-Hand Side Product

Now, calculate the product on the right-hand side of the equation: \( 22 \times 21 = 462 \). So, the equation becomes \( 3x = 462 \).
03

Solve for x

To find the value of \( x \), divide both sides of the equation by 3: \( x = \frac{462}{3} \).
04

Perform the Division

Calculate the division to isolate \( x \): \( x = 154 \).

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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 technique used to solve proportions. A proportion is an equation that states two ratios are equal. In the exercise \( \frac{3}{22} = \frac{21}{x} \), cross multiplication helps us find the unknown value of \( x \). We start by multiplying the numerator of one fraction by the denominator of the other fraction, and then set them equal. This means we calculate \( 3 \times x \) and \( 22 \times 21 \).
  • Step-by-step: Write down your two products: \( 3 \times x \) and \( 22 \times 21 \).
  • Set the two products equal: \( 3x = 462 \).
Cross multiplication simplifies the comparison of fractions and helps find unknown values quickly, especially when working with proportions in algebra.
Equivalent Fractions
Equivalent fractions are fractions that represent the same value, even though they have different numerators and denominators. The concept surfaces naturally when solving proportion problems. In the exercise, the fractions \( \frac{3}{22} \) and \( \frac{21}{x} \) are equated, indicating they hold the same value.
To understand equivalent fractions, consider:
  • Both fractions are parts of the same whole.
  • The cross-multiplied equation \( 3x = 462 \) confirms their equivalence once solved.
Recognizing fractions as equivalent can simplify solving equations since we can adjust and scale them while preserving their value.
Solving Equations
Solving equations is a fundamental skill in algebra, allowing one to find the unknown value that makes an equation true. In our exercise, we solve for \( x \) by manipulating an equation established through cross multiplication.
Here's how it's done:
  • We start with the equation \( 3x = 462 \).
  • To isolate \( x \), divide both sides by 3, which gives \( x = \frac{462}{3} \).
  • Perform the calculation to find \( x = 154 \).
Solving equations often involves similar steps: rearranging the equation, isolating the variable, and performing basic arithmetic operations. Mastery of this process is crucial in solving not just simple proportions but more complex equations later on.

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