Problem 28 - Beginning and Intermediate Algebra

Question

Subtract. See Examples 1 through 5 $$ -\frac{4}{7}-\left(-\frac{1}{7}\right) $$

Step-by-Step Solution

Answer

The answer is \(-\frac{3}{7}\).

Step 1: Understand the Problem

The problem requires us to subtract a negative fraction from another negative fraction. The operation subtraction of a negative number can be simplified.

Step 2: Simplify the Expression

When you subtract a negative number, it is equivalent to adding the positive equivalent of that number. Therefore, \(-\frac{4}{7} - (-\frac{1}{7})\) becomes \(-\frac{4}{7} + \frac{1}{7}\).

Step 3: Find a Common Denominator

Both fractions, \(-\frac{4}{7}\) and \(\frac{1}{7}\), already have a common denominator of 7.

Step 4: Subtract the Numerators

Subtract the numerators while keeping the common denominator: \(-4 + 1 = -3\), thus the fraction becomes \(-\frac{3}{7}\).

Step 5: Write the Final Answer

The result of the subtraction \(-\frac{4}{7} - (-\frac{1}{7})\) is \(-\frac{3}{7}\).

Key Concepts

Negative Fractions

Negative fractions can seem confusing at first, but they're actually quite straightforward once you get the hang of them. A negative fraction is simply a fraction that has a negative sign either in the numerator, denominator, or in front of the fraction itself. All these representations are equivalent. For example,

• \(-\frac{4}{7}\)
• \(\frac{-4}{7}\)
• \(\frac{4}{-7}\)

all mean the same thing and indicate a value less than zero.

When performing operations with negative fractions, treat them like negative whole numbers. If you're subtracting a negative fraction from another fraction, it becomes an addition of the positive counterpart of that fraction. This might remind you of the rule of double negatives in grammar, where two negatives make a positive, just like in mathematics.

Common Denominator

To effectively subtract fractions, they must have a common denominator. The denominator is the bottom part of the fraction that tells you into how many parts the whole is divided.

When two fractions have the same denominator, it means both fractions are referring to the same sized parts. This makes subtraction (or addition) much simpler, as you only have to operate with the numerators and keep the common denominator as is. In our example problem, both fractions had a denominator of 7, which made the process straightforward.

If fractions do not have the same denominator, you will first need to change them to equivalent fractions that do. This often involves finding the least common multiple (LCM) of the original denominators. Once this is done, you can carry out your operation on the numerators with ease.

Numerators

The numerator is the top part of the fraction that indicates how many parts you are considering out of the whole. In the subtraction problem

• \(-\frac{4}{7}\)
• \(\frac{1}{7}\)

the numerators are -4 and 1, respectively.

When you perform operations like subtraction on fractions with a common denominator, you actually perform the operation on the numerators. So, \(-4 + 1 = -3\). Notice that the operation involved adding because we changed the subtraction of a negative to addition. This resulted in the new numerator -3.

Understanding the role of numerators helps break down what can seem like a complex operation into something much simpler and manageable.

Fraction Operations

Operating with fractions involves basic arithmetic but requires an understanding of how fractions represent parts of a whole. Key operations—addition, subtraction, multiplication, and division—often require the use of specific rules.

For subtraction in particular, the process requires the fractions to first have the same denominator. Once we confirm a common denominator, focus shifts to the numerators, where the actual arithmetic takes place. For the subtraction operation, as with \(-\frac{4}{7} - (-\frac{1}{7})\), remember that subtracting a negative turns into addition. Thus, we simplified it to \(-\frac{4}{7} + \frac{1}{7}\).

These operations might initially seem daunting, but when you break them down step by step and apply the rules consistently, they become much simpler. Understanding each part of a fraction and the operation helps to demystify the process and build confidence in solving these types of problems.