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Magazine Chapter 7: Problem 36
Solve each proportion. $$ \frac{27}{x}=\frac{9}{4} $$
Short Answer
Expert verified
The value of \( x \) is 12.
Step by step solution
01
Understand the Proportion
A proportion is an equation that states that two ratios are equal. We are given the proportion \( \frac{27}{x} = \frac{9}{4} \). Our goal is to find the value of \( x \).
02
Cross-Multiply
To solve for \( x \), use cross-multiplication. Multiply the numerator of the first fraction by the denominator of the second fraction, and the denominator of the first fraction by the numerator of the second fraction: \[27 \times 4 = 9 \times x\]
03
Simplify the Equation
After cross-multiplying, simplify the equation: \[108 = 9x\]This equation represents the relationship between the numbers after cross-multiplying.
04
Solve for X
To isolate \( x \), divide both sides of the equation by 9:\[\frac{108}{9} = x\]Simplify the expression on the left side to find the value of \( x \):\[ x = 12 \]
05
Verify the Solution
To ensure our solution is correct, substitute \( x = 12 \) back into the original proportion:\[\frac{27}{12} = \frac{9}{4}\]Simplify \( \frac{27}{12} \) to see if it equals \( \frac{9}{4} \):\[\frac{27 \div 3}{12 \div 3} = \frac{9}{4}\]This confirms our solution is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cross-Multiplication in Proportions
When dealing with proportions, a common technique to solve for a variable is cross-multiplication. This method involves multiplying across the equals sign in a crisscross or diagonal fashion. Consider the proportion given in the exercise: \( \frac{27}{x} = \frac{9}{4} \). Here鈥檚 how cross-multiplication works in this scenario:
- Step 1: Identify the numerators and denominators. In this case, 27 and 9 are the numerators, while \(x\) and 4 are the denominators.
- Step 2: Multiply the numerator of one side by the denominator of the other side. That is, multiply 27 by 4 and \(x\) by 9.
- Step 3: Set these two products equal to each other. You get: \(27 \times 4 = 9 \times x\).
Solving Equations from Proportions
Once you have used cross-multiplication to form an equation, the next step is to solve it. If we look at the equation \(27 \times 4 = 9 \times x\), it simplifies to \(108 = 9x\).
- First: Simplify both sides as much as possible. Here, our focus is to isolate \(x\).
- Second: Perform the necessary division. We divide both sides of the equation by 9 to solve for \(x\).
- Calculate: \( \frac{108}{9} = x \), thus \( x = 12 \).
Verifying Solutions to Proportions
It鈥檚 not enough to just find the solution; verifying your answer is crucial to ensure accuracy. In checking the solution for a proportion, you substitute the found value back into the original equation and see if the equality holds.
Here鈥檚 how the verification is performed for our original proportion \( \frac{27}{x} = \frac{9}{4} \), with \( x = 12 \):
Here鈥檚 how the verification is performed for our original proportion \( \frac{27}{x} = \frac{9}{4} \), with \( x = 12 \):
- Replace \(x\): Substitute 12 in place of \(x\) to get \(\frac{27}{12}\).
- Simplify: Simplify \(\frac{27}{12}\) by dividing both the numerator and the denominator by their greatest common factor, which is 3, resulting in \(\frac{9}{4}\).
- Compare: \(\frac{9}{4}\) equals \(\frac{9}{4}\), confirming the calculations are correct.
