Problem 30 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. $$\frac{1}{7}-\left(-\frac{3}{7}\right)$$

Step-by-Step Solution

Answer

The result of the operation \(\frac{1}{7}-\left(-\frac{3}{7}\right)\) equals \(\frac{4}{7}\).

Step 1: Understand the Problem

The problem asked to subtract \(\frac{3}{7}\) from \(\frac{1}{7}\). It is noted here that a negative sign precedes the fraction to be subtracted, which transforms the subtraction into an addition.

Step 2: Change the Operation

Remember the rule 'minus a negative number becomes a plus', thus the operation changes from subtraction to addition. The expression now becomes: \(\frac{1}{7} + \frac{3}{7}\)

Step 3: Perform the Operation

Since the fractions have the same denominator, simply add the numerators. This will give: \(\frac{1 + 3}{7} = \frac{4}{7}\)

Key Concepts

Negative Numbers

Understanding negative numbers is crucial in arithmetic and algebra because they frequently pop up in mathematical operations. Negative numbers are numbers less than zero, represented with a minus sign (-). For example, -3 is a negative number.

When dealing with negative numbers, especially in operations like subtraction, a common rule is used. This rule is: "Subtracting a negative number is the same as adding its positive counterpart."

For instance, in the expression \( \frac{1}{7} - (-\frac{3}{7}) \), the subtraction of a negative fraction \( -\frac{3}{7} \) is equivalent to the addition of \( \frac{3}{7} \). That's why you change the operation from subtraction to addition. This step is often simplified in the expression as: \( \frac{1}{7} + \frac{3}{7} \).

Keeping this rule in mind helps simplify and solve problems without running into common mistakes.

Adding Fractions

Adding fractions involves combining two fractions into one. When fractions have the same denominator, the process is straightforward.

You simply need to add their numerators. This step does not change the denominator. It remains the same. For example, with \( \frac{1}{7} \) and \( \frac{3}{7} \):

• The numerators are 1 and 3. So, add them together: \( 1 + 3 = 4 \).
• The denominator stays as 7.
• This results in the fraction: \( \frac{4}{7} \).

This kind of addition is very manageable, as you focus only on the top numbers (numerators) and ensure you do not alter the bottom number (denominator).

It's like stacking fractions that fit perfectly due to their matching bases.

Common Denominators

Common denominators simplify many fraction operations, such as addition and subtraction. A common denominator means that fractions share the same bottom number.

Having a common denominator makes it easier to perform calculations, as shown in our example where both fractions are \( \frac{1}{7} \) and \( -\frac{3}{7} \). They both have the denominator 7.

For operations needing common denominators:

• Check the denominator of each fraction.
• If they're the same, proceed with the operation, such as adding numerators.
• If not, you need to find the least common denominator (LCD) to match the fractions.

In our example, the task was simpler due to the already common denominator, allowing quick calculations without finding an LCD.

Understanding this concept ensures you handle fractional computations efficiently.