Problem 64 - Algebra 1

Question

Find the difference. $$\frac{3}{4}-\frac{5}{12}$$

Step-by-Step Solution

Answer

The difference between \(\frac{3}{4}\) and \(\frac{5}{12}\) is \(\frac{1}{3}\).

Step 1: Find the Least Common Multiple (LCM)

The LCM is the smallest number that both 4 and 12 can divide into. Multiples of 4 are 4, 8, 12, 16, 20, ... and multiples of 12 are 12, 24, 36, .... The smallest multiple they share is 12.

Step 2: Convert Fractions to Equivalent Fractions

With the LCM, convert each fraction to equivalent fractions that have this denominator. The first fraction is already \(\frac{3}{4}\). To make 4 into 12, multiply by 3, therefore also multiply 3 by 3. The first fraction becomes \(\frac{9}{12}\). The second fraction is already \(\frac{5}{12}\), so no changes needed.

Step 3: Subtract Fractions

Now, subtract the equivalent fractions: \(\frac{9}{12} - \frac{5}{12} = \frac{9-5}{12} = \frac{4}{12}\)

Step 4: Simplify the Fraction

The fraction \(\frac{4}{12}\) can be simplified as both 4 and 12 have common factors. They can both be divided by 4, so the simplified fraction is \(\frac{1}{3}\).

Key Concepts

Least Common Multiple

In mathematics, finding the Least Common Multiple (LCM) is a fundamental skill that helps solve problems involving fractions, especially when adding or subtracting them. The LCM of two or more numbers is simply the smallest number into which all the original numbers can evenly divide. For instance, when you are working with fractions such as \( \frac{3}{4} \) and \( \frac{5}{12} \), identifying the LCM of their denominators, which are 4 and 12, is crucial.

To find the LCM:

• List the multiples of both numbers. For 4, these would be 4, 8, 12, 16, etc., and for 12, they would be 12, 24, 36, etc.
• The LCM is the first common number in both lists. In this example, the LCM of 4 and 12 is 12.

Recognizing the LCM simplifies combining or comparing fractions, because it provides a shared denominator.

Equivalent fractions

Equivalent fractions are different fractions that represent the same number or proportion. This concept is used when adjusting fractions to have a common denominator, such as when subtracting fractions with different denominators.

To create equivalent fractions, you multiply the numerator and the denominator of a fraction by the same number. For example, with \( \frac{3}{4} \), you want this fraction to have a denominator of 12 (the LCM). You can do so by multiplying both the numerator and the denominator by 3, resulting in \( \frac{9}{12} \).

Equivalent fractions are important because they allow you to add or subtract fractions by giving them the same denominator, changing only the form of the fraction but not its value.

Simplifying fractions

Simplifying fractions involves reducing them to their smallest possible form, where the numerator and the denominator have no common factors aside from 1. This makes fractions easier to understand and compare.

The process of simplifying involves:

• Identifying the greatest common factor (GCF) of the numerator and the denominator.
• Dividing both by this number. For example, with \( \frac{4}{12} \), the GCF is 4.
• Divide 4 and 12 by 4 to get the simplified \( \frac{1}{3} \).

Simplified fractions represent the same value but are more concise and easier to work with in calculations. Understanding how to simplify fractions is key to mastering operations involving fractions.