Problem 34 - Beginning and Intermediate Algebra

Question

Add or subcract as indicared. Write the answer in lower terms. See Erample 5 $$ \frac{6}{7}-\frac{1}{7} $$

Step-by-Step Solution

Answer

\(\frac{5}{7}\) is the simplified result.

Step 1: Identify the Operation

Given the problem involves two fractions, we need to subtract the second fraction from the first. The operation to perform is subtraction.

Step 2: Check Denominators

Notice that both fractions have the same denominator: 7. Since the denominators are identical, we can subtract the numerators directly.

Step 3: Subtract the Numerators

Subtract the numerators of the fractions: \(6 - 1 = 5\). Hence, the result of the subtraction is \(\frac{5}{7}\).

Step 4: Simplify the Fraction

Check if \(\frac{5}{7}\) can be simplified. Since 5 and 7 have no common factors other than 1, \(\frac{5}{7}\) is already in its simplest form.

Key Concepts

Common Denominators

When adding or subtracting fractions, it's crucial to have *common denominators*. This means that the bottom number of the fractions, known as the denominator, should be the same for both fractions. Having a common denominator allows us to directly add or subtract the top numbers, or numerators. If the denominators differ, you will need to find a common denominator by finding the least common multiple (LCM). In this exercise, since both fractions have the same denominator of 7, you can proceed with the subtraction without needing to adjust the fractions further.

Simplifying Fractions

After performing an operation like addition or subtraction on fractions, the next step is often simplifying the result. Simplifying fractions involves reducing the fraction to its simplest form. You do this by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by this number.
For instance, if you end up with a fraction like \(\frac{8}{12}\), you can simplify it by dividing both the numerator and denominator by 4 (the GCD of 8 and 12) to get \(\frac{2}{3}\).
In the exercise, the result \(\frac{5}{7}\) cannot be simplified further, as 5 and 7 share no common factors other than 1.

Numerators and Denominators

Understanding the role of numerators and denominators in fractions can greatly help in operations like addition and subtraction. The numerator is the top number of a fraction, representing how many parts we have, whereas the denominator is the bottom number, indicating how many parts make up a whole.
During addition or subtraction, only the numerators are combined, assuming the fractions have the same denominator. This is because the denominator sets the scale, or size, of the fraction's parts.

• If denominators are the same, like 7 in the example \(\frac{6}{7} - \frac{1}{7}\), you simply subtract the numerators: \(6 - 1 = 5\).
• The denominator remains unchanged, maintaining the structure and scale of the fraction.