Problem 68 - Beginning and Intermediate Algebra

Question

Find each difference. $$ \frac{1}{3}-\left(-\frac{1}{12}\right) $$

Step-by-Step Solution

Answer

The difference is \frac{5}{12}\.

Step 1: Rewrite the Subtraction as Addition

Rewrite the subtraction of \(-\frac{1}{12}\) as addition. Subtracting a negative is the same as adding a positive:\[ \frac{1}{3} - \left(-\frac{1}{12}\right) = \frac{1}{3} + \frac{1}{12} \]

Step 2: Find the Common Denominator

To add fractions, first find a common denominator. The least common multiple of 3 and 12 is 12, so convert all fractions to have this common denominator:

Step 3: Convert \frac{1}{3}\ to an Equivalent Fraction

Convert \frac{1}{3}\ to an equivalent fraction with a denominator of 12:\[ \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \]

Step 4: Add the Fractions

Now add the fractions with the common denominator:\[ \frac{4}{12} +\frac{1}{12} = \frac{4 + 1}{12} = \frac{5}{12} \]

Step 5: Simplify the Fraction

The fraction \frac{5}{12}\ is already in its simplest form. So, the final answer is:\[ \frac{5}{12} \]

Key Concepts

common denominator

To successfully add or subtract fractions, it's essential to have a common denominator. A common denominator is a shared multiple of the denominators of the fractions involved. For instance, in our exercise, we needed to add \(\frac{1}{3}\) and \(\frac{1}{12}\). To do so, we found a common denominator.

Here are a few steps to find a common denominator:

• Identify the denominators of the fractions.
• Determine the least common multiple (LCM) of the denominators. For 3 and 12, the LCM is 12.
• Rewrite each fraction as an equivalent fraction with the common denominator.

Finding a common denominator ensures that we can easily add or subtract the fractions without mistakes. Once you have the same denominator, you can simply add or subtract the numerators while keeping the denominator the same.

equivalent fractions

Equivalent fractions are fractions that represent the same value, even though they may look different. For example, in our problem, we converted \(\frac{1}{3}\) into \(\frac{4}{12}\) to match the common denominator.

To convert fractions to equivalent forms, follow these steps:

• Multiply the numerator and the denominator of the fraction by the same number.
• Ensure that the multiplication brings the denominators to the common denominator.

Here's a small example: \(\frac{1}{3}\) can be converted to \(\frac{4}{12}\) by multiplying both the numerator and the denominator by 4:

\[ \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \]
Using equivalent fractions makes the process of addition or subtraction straightforward because it aligns the denominators, simplifying the computation.

addition of fractions

Adding fractions might seem tricky at first, but it's quite simple if you follow the right steps. In the exercise, we had to add \(\frac{4}{12}\) and \(\frac{1}{12}\). Since they have a common denominator, adding them is straightforward.

Steps to add fractions:

• If necessary, convert the fractions to have a common denominator.
• Add the numerators while keeping the common denominator unchanged.
• Simplify the result, if possible.

For example, in our exercise:
\(\frac{4}{12} + \frac{1}{12} = \frac{4 + 1}{12} = \frac{5}{12}\)

In this problem, the final fraction, \(\frac{5}{12}\), was already in its simplest form. Understanding how to properly add fractions is a crucial skill for solving many mathematical problems. Always ensure to check if your final fraction can be simplified.