Problem 67 - Prealgebra

Question

Write each fraction as an equivalent fraction with denominator 12. $$\frac{1}{3}$$

Step-by-Step Solution

Answer

The equivalent fraction is \(\frac{4}{12}\).

Step 1: Understand the Problem

We need to convert the fraction \(\frac{1}{3}\) so that it has a denominator of 12. This means we need to find an equivalent fraction with the same value but a different denominator.

Step 2: Find an Equivalent Denominator

Since we want the denominator to be 12, we must figure out what number to multiply 3 by to get 12. We can determine this by dividing the desired denominator, 12, by the original denominator, 3.

Step 3: Calculate the Multiplier

Divide 12 by 3 to find the multiplier: \(12 \div 3 = 4\). This tells us we need to multiply both the numerator and the denominator of \(\frac{1}{3}\) by 4 to make the denominator 12.

Step 4: Apply the Multiplier to the Numerator and Denominator

Multiply both the numerator and the denominator of \(\frac{1}{3}\) by 4: \(1 \times 4 = 4\) and \(3 \times 4 = 12\). This gives us the equivalent fraction \(\frac{4}{12}\).

Step 5: Verify the Solution

We obtained \(\frac{4}{12}\) using a valid multiplication process. Check that the simplified form of \(\frac{4}{12}\) is indeed \(\frac{1}{3}\) by dividing both the top and bottom of \(\frac{4}{12}\) by 4, which confirms \(\frac{1}{3}\).

Key Concepts

Finding Common Denominators

To find an equivalent fraction for a given fraction, one of the first steps is finding a common denominator. Simply put, a common denominator is a shared multiple of the denominators of two or more fractions. This becomes particularly useful when you need to add, subtract, or compare fractions. In our exercise, we want to rewrite \( \frac{1}{3} \) with a denominator of 12.
To do this, identify how many times the original denominator fits into the new desired denominator. For \( \frac{1}{3} \) to have a denominator of 12, we divide 12 by 3, resulting in a quotient of 4. This quotient is our multiplier, which we'll use in the next steps. Understanding how to find common denominators will make it much easier to work with fractions in various operations.

Fraction Multiplication

When converting one fraction to have a new denominator, multiplication is the tool you'll use. Multiplying both the numerator and denominator by the same number will generate an equivalent fraction. For the fraction \( \frac{1}{3} \), we found that the number 4 is essential to create the new denominator of 12. The next step is to multiply both parts of the fraction by this number.
Multiplying Fractions:

• Multiply the numerator by 4: \( 1 \times 4 = 4 \)
• Multiply the denominator by 4: \( 3 \times 4 = 12 \)

Thus, the fraction \( \frac{1}{3} \) becomes \( \frac{4}{12} \). This process keeps the value of the fraction unchanged but adjusts it to have a new denominator. Understanding fraction multiplication is crucial for many other mathematical operations, such as division and finding least common multiples (LCMs).

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form, where no common factor exists between the numerator and the denominator except 1. This process is often the final step to verify the equivalence of fractions after manipulation.
For instance, we made the fraction \( \frac{4}{12} \) after applying the multiplication strategy. To ensure it's equivalent to \( \frac{1}{3} \), we simplify \( \frac{4}{12} \) by dividing both the numerator and denominator by their greatest common divisor (GCD), which in this case is 4.

• Divide the numerator by 4: \( 4 \div 4 = 1 \)
• Divide the denominator by 4: \( 12 \div 4 = 3 \)

Upon simplification, we return to the fraction \( \frac{1}{3} \), confirming its equivalence with \( \frac{4}{12} \). Being adept at simplifying fractions is valuable not only for verifying solutions but also for making calculations simpler and results easier to interpret.