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Chapter 5: Problem 20

Solve. $$ \frac{8}{12}=\frac{20}{x} $$

Short Answer

Expert verified
x = 30

Step by step solution

01

Cross-Multiply

To solve the equation \( \frac{8}{12} = \frac{20}{x} \), start by cross-multiplying the terms. Multiplying across the equal sign gives: \[ 8x = 12 \times 20 \]
02

Simplify the equation

Next, perform the multiplication on the right side of the equation: \[ 8x = 240 \]
03

Solve for x

Divide both sides of the equation by 8 to isolate \( x \): \[ x = \frac{240}{8} \] Simplify the fraction: \[ x = 30 \]

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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 an essential technique when solving proportions. When you have two ratios set equal to each other, which looks like \(\frac{a}{b} = \frac{c}{d}\), you can solve this by cross-multiplying. What this means is that you multiply the numerator (top number) of the first fraction by the denominator (bottom number) of the second fraction, and the numerator of the second fraction by the denominator of the first fraction. This process looks like this: a * d = b * c. In our example problem \(\frac{8}{12} = \frac{20}{x}\), we cross-multiply to get: 8 * x = 12 * 20. This gives us the equation 8x = 240.
Simplifying Equations
After cross-multiplication, the next step is simplifying the equation. Simplifying involves performing operations that make the equation easier to solve. In the equation from our example, we have: 8x = 240. Here, all we need to do is carry out the multiplication on the right-hand side of the equation. By multiplying 12 and 20, we get 240. So, our simplified equation is still 8x = 240.
Division
Once your equation is simplified, the final step is division. We use division to isolate the variable (the unknown value we're solving for). In our example, we want to solve for x in the equation 8x = 240. To isolate x, we divide both sides of the equation by 8: \(\frac{8x}{8} = \frac{240}{8}\). This simplifies to x = 30. Hence, the solution to our proportion \(\frac{8}{12} = \frac{20}{x}\) is x = 30.

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