Problem 37 - Prealgebra

Question

Solve each proportion. $$\frac{84}{52}=\frac{m}{13}$$

Step-by-Step Solution

Answer

The solution to the proportion is \( m = 21 \).

Step 1: 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 \( m \).

Step 2: 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, \( 84 \times 13 = 52 \times m \).

Step 3: Perform Multiplication

Calculate \( 84 \times 13 \). This equals \( 1092 \). Now, rewrite the equation as \( 1092 = 52 \times m \).

Step 4: Isolate the Variable

To solve for \( m \), divide both sides of the equation by 52. This gives \( m = \frac{1092}{52} \).

Step 5: Simplify the Division

Perform the division \( \frac{1092}{52} \), which simplifies to \( 21 \).

Step 6: 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.

Key Concepts

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.

Using cross-multiplication transforms a complex proportion into a straightforward equation. This method reduces potential confusion when working with fractions, making it more manageable to find the solution.

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:

• 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.

By following these steps, you methodically simplify and solve the equation created by cross-multiplication. This ensures you clearly and correctly find the value of the unknown in the original proportion.

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.

Proficiently handling fractions involves understanding their role within proportions and knowing how to manipulate them using methods like simplification and cross-multiplication. With practice, fractions become less daunting and more intuitive to work with.