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Chapter 3: Problem 13

Solve each proportion. See Example 2 . $$\frac{120}{50}=\frac{x}{4}$$

Short Answer

Expert verified
We are given the proportion \(\frac{120}{50}=\frac{x}{4}\). Solve for x using cross-multiplication: \(120 * 4 = 50 * x\) \(480 = 50x\) Divide both sides by 50: \(\frac{480}{50} = x\) Simplify the fraction: \(x = \frac{480}{50} = \frac{48}{5}\)

Step by step solution

01

Write down the proportion

We are given the proportion \(\frac{120}{50}=\frac{x}{4}\). Keep in mind that our goal is to find the value of x.
02

Cross-multiply

Cross-multiplication means we multiply the numerator of the first fraction by the denominator of the second fraction and set the result equal to the product of the denominator of the first fraction and the numerator of the second fraction. So in our case, we do: \(120 * 4 = 50 * x\)
03

Simplify the equation

Multiply the numbers on both sides of the equation: \(480 = 50x\)
04

Solve for x

Divide both sides of the equation by 50 to isolate x: \(\frac{480}{50} = x\)
05

Simplify the fraction

Simplify the fraction \(\frac{480}{50}\) to its lowest terms: \(x = \frac{480}{50} = \frac{48}{5}\) So, \(x = \frac{48}{5}\).

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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 handy technique used to solve proportions, which are equations that express two ratios as equal. Imagine you have the proportion \(\frac{120}{50} = \frac{x}{4}\). To cross-multiply, you multiply across the equal sign diagonally.
  • Multiply the numerator of the left fraction (120) with the denominator of the right fraction (4).
  • Multiply the denominator of the left fraction (50) with the numerator of the right fraction (x).
After carrying out these steps, the products you obtain are set equal to each other, giving us the equation:\[120 \times 4 = 50 \times x\]Cross-multiplication helps us transform the proportion into a straightforward equation. This makes solving for the unknown variable much simpler.
Solving for x
Solving for an unknown variable like \(x\) in an equation involves isolating the variable to one side. Once you've cross-multiplied your proportion, you get an equation:\[480 = 50x\]To find the value of \(x\), you need to isolate it on one side of the equation. You can do this by dividing both sides by 50, which is the coefficient of \(x\):\[x = \frac{480}{50}\]This technique of rearranging an equation to make a variable the subject forms the basis of solving any algebraic equation. It essentially involves performing reverse operations such as addition, subtraction, multiplication, or division to isolate \(x\).
Simplifying Fractions
Simplifying fractions to their lowest terms is often the final step when solving problems involving ratios and proportions. Given the fraction \(\frac{480}{50}\), the idea is to express it in the simplest form possible by finding the greatest common divisor (GCD).
  • Calculate the GCD of the numerator and the denominator. Here, 10 is the GCD of 480 and 50.
  • Divide both the numerator and the denominator by their GCD.
When you simplify \(\frac{480}{50}\) by dividing both numbers by 10, you end up with \(\frac{48}{5}\). This is the fraction in its lowest terms. Simplifying fractions makes them easier to read and understand, removing common factors from the numerator and the denominator.

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