Problem 43 - Prealgebra

Question

The problems below will allow you to review subtraction of fractions and mixed numbers. $$\frac{1}{3}-\frac{1}{4}$$

Step-by-Step Solution

Answer

The answer is \(\frac{1}{12}\).

Step 1: Find Common Denominator

To subtract fractions, we need a common denominator. The denominators here are 3 and 4. The least common multiple of 3 and 4 is 12, so we use 12 as the common denominator.

Step 2: Convert Fractions

Convert \(\frac{1}{3}\) and \(\frac{1}{4}\) to have the denominator of 12. \(\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}\) and \(\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}\).

Step 3: Subtract the Fractions

Now, with the common denominator, subtract the numerators: \(\frac{4}{12} - \frac{3}{12} = \frac{4 - 3}{12} = \frac{1}{12}\).

Step 4: Simplify the Fraction

The fraction \(\frac{1}{12}\) is already in its simplest form.

Key Concepts

Common Denominator

When subtracting fractions, it's crucial that the denominators of both fractions are the same. This is what we call a "common denominator." It allows you to perform operations like addition or subtraction straightforwardly. Think of denominators as parts of a whole. If each fraction represents a different number of parts of different sizes, they can't be subtracted directly.

To find a common denominator, you may need to adjust the fractions by multiplying both the numerator and the denominator to get them to have the same denominator. For example, if you have fractions like \( \frac{1}{3} \) and \( \frac{1}{4} \), you look for a number that both denominators (3 and 4 in this case) can divide into evenly. By doing so, you can ensure that the fractions are referring to the same sized parts of a whole.

• This adjustment ensures that you're comparing "apples to apples," or in this case, fractions to fractions with a uniform denominator.

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. For example, if you have 1 and \( \frac{2}{3} \), this can be written as \( 1\frac{2}{3} \). Mixed numbers are often used in everyday life, like in recipes or measurements.

When you need to subtract mixed numbers, it's first important to convert them to improper fractions. This makes arithmetic operations more straightforward. Here's how you can do it:

• Multiply the whole number by the denominator of the fraction part.
• Add this result to the numerator of the fraction part.
• Place the sum over the original denominator.

By converting mixed numbers to improper fractions, you align them under a unified format that is easier to manipulate mathematically for processes like subtraction.

Least Common Multiple

The "least common multiple" (LCM) is the smallest number that is a multiple of two or more numbers. Finding the LCM is a key step when seeking a common denominator for subtraction of fractions because it identifies a shared foundation to build upon.

For the denominators 3 and 4, the multiples of 3 are 3, 6, 9, 12... and the multiples of 4 are 4, 8, 12, 16... The smallest number they both share is 12. Therefore, 12 is the LCM for 3 and 4.

• To find the LCM, list the multiples of each number, and then identify the smallest multiple they share.
• This common denominator facilitates the conversion of the original fractions into a format ready for addition or subtraction by aligning them to the same base.

Using the LCM to find the common denominator streamlines the process of subtracting fractions, ensuring accurate and simple calculations.