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Chapter 1: Problem 35

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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, \( \frac{3}{4} \).

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Most popular questions from this chapter

In Exercises \(147-149,\) perform the indicated operation. \(-6-(-3)\) (Section \(1.6,\) Example 1 )

In Exercises \(120-123\), use a calculator to find a decimal approximation for each irrational number, correct to three decimal places. Between which two integers should you graph each of these numbers on the number line? \(1-\sqrt{2}\)

Use a calculator to find a decimal approximation for each irrational number, correct to three decimal places. Between which two integers should you graph each of these numbers on the number line? $$-\sqrt{12}$$

Explain how to add two numbers with a number line. Provide an example with your explanation.

Give an example of a real number that is not an irrational number. (Section \(1.3,\) Example 5 ).
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