Problem 35 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. $$\frac{1}{2}-\left(-\frac{1}{4}\right)$$

Step-by-Step Solution

Answer

The short answer is \( \frac{3}{4} \).

Step 1: Understand the problem

The problem is to find the solution of \( \frac{1}{2}-\left(-\frac{1}{4}\right) \). Note that subtraction of a negative number is the same as addition.

Step 2: Change the operation

The given operation is \( \frac{1}{2}-\left(-\frac{1}{4}\right) \). This can be rewritten as \( \frac{1}{2} + \frac{1}{4} \) as per the rules of subtraction and negation.

Step 3: Perform the addition

Now that we have two positive fractions, we can directly add them. The addition of \( \frac{1}{2} \) and \( \frac{1}{4} \) is \( \frac{3}{4} \) because the denominator is the same (2 and 4 are both divisible by 2), and we just add the numerators 1 and 1 to get 2.

Key Concepts

Understanding Addition of Fractions

When you add fractions, converting them to a common denominator is crucial.
This means you need the bottom numbers (denominators) to be the same.
Only then, can you add the numerators, which are the top numbers.

• Let's say you need to add \( \frac{1}{2} \) and \( \frac{1}{4} \).
  First, find the least common denominator.
• Both 2 and 4 can be multiplied by 2 to equal 4.
• Therefore, the least common denominator is 4.

Once you know the least common denominator, adjust the fractions:
\( \frac{1}{2} \) becomes \( \frac{2}{4} \) since \( 1 \times 2 = 2 \) and \( 2 \times 2 = 4 \).
Now it’s easy! Just add the numerators: \( \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \).

Handling Negative Numbers in Subtraction

Negatives flip the operation around in subtraction problems.
Think of subtracting as adding a negative number. For example,
\( a - (-b) \) is the same as \( a + b \).

• This is because subtracting a negative is essentially giving a positive effect.
• So, the problem \( \frac{1}{2} - (-\frac{1}{4}) \) becomes \( \frac{1}{2} + \frac{1}{4} \).

When you see two minus signs next to each other, they become a plus.
Always check for double negatives as an opportunity for an easier operation.

Importance of Step-by-Step Problem Solving

Approaching math with a step-by-step process helps you break down complex problems into manageable parts.
It keeps your work organized and makes it easier to find and fix mistakes.

• First, read and understand the problem: Know what is being asked.
• Next, translate the problem into a mathematical expression.
• Then, perform operations attentively, one at a time.
• Lastly, verify the solution by checking the work for any errors.

For the exercise \( \frac{1}{2} - (-\frac{1}{4}) \), following these steps reveals that subtraction can turn into addition.
It also illustrates how finding a common denominator and adjusting fractions can lead to the correct solution, \( \frac{3}{4} \).