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Magazine Chapter 9: Problem 37
Solve each proportion. $$\frac{84}{52}=\frac{m}{13}$$
Short Answer
Expert verified
The solution to the proportion is \( m = 21 \).
Step by step solution
01
Understand the Proportion
The given proportion is \( \frac{84}{52} = \frac{m}{13} \). A proportion means two fractions are equal. We are required to find the value of \( m \).
02
Use Cross-Multiplication
Cross-multiplication helps solve proportions. Multiply the numerator of one fraction by the denominator of the other fraction and set them equal. Thus, \( 84 \times 13 = 52 \times m \).
03
Perform Multiplication
Calculate \( 84 \times 13 \). This equals \( 1092 \). Now, rewrite the equation as \( 1092 = 52 \times m \).
04
Isolate the Variable
To solve for \( m \), divide both sides of the equation by 52. This gives \( m = \frac{1092}{52} \).
05
Simplify the Division
Perform the division \( \frac{1092}{52} \), which simplifies to \( 21 \).
06
Verify the Solution
Recheck by substituting \( m = 21 \) back into the original proportion: \( \frac{84}{52} = \frac{21}{13} \). Simplifying \( \frac{84}{52} \) gives \( \frac{21}{13} \), verifying the 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 Explained
Cross-multiplication is an essential technique for solving proportions. It's used to find an unknown variable in a proportion, which is an equation stating that two ratios or fractions are equal. When employing cross-multiplication, follow these simple steps:
- Identify the two fractions in the proportion. In our example, these fractions are \( \frac{84}{52} \) and \( \frac{m}{13} \).
- Multiply the numerator of one fraction by the denominator of the second fraction, then do the reverse. You multiply \( 84 \times 13 \) and \( 52 \times m \).
- Set the two results equal to each other: \( 84 \times 13 = 52 \times m \). This creates a simple equation with one unknown.
Solving Equations through Cross-Multiplication
Once you have utilized cross-multiplication to create an equation, it's crucial to know how to solve it correctly. In our example, cross-multiplication gave us the equation: \( 1092 = 52 \times m \).
Here's how you solve it:
Here's how you solve it:
- First, perform any necessary arithmetic on the side with numbers only, like multiplying \( 84 \times 13 \) to simplify the equation.
- Once simplified, isolate the variable by using inverse operations. In this case, divide both sides by 52 to find \( m \).
- This results in \( m = \frac{1092}{52} \), which upon division equals 21.
Understanding Fractions in Proportions
Fractions are a central part of proportions, representing parts of a whole. In proportions like \( \frac{84}{52} = \frac{m}{13} \), they essentially compare two ratios. Here are essential points to remember about fractions:
- A fraction consists of a numerator (top number) and a denominator (bottom number).
- Proportions indicate that two fractions are equal, meaning they express the same part-to-whole relationship.
- It's crucial to simplify fractions when possible. For example, both \( \frac{84}{52} \) and \( \frac{21}{13} \) simplify to the same ratio, confirming the proportion's equality.
