Problem 67 - Elementary and Intermediate Algebra

Question

Perform the indicated operation and, if possible, simplify. If there are no variables, check using a calculator. $$ \frac{9}{7}-\frac{2}{7} $$

Step-by-Step Solution

Answer

1

Step 1 - Identify the Common Denominator

Both fractions \(\frac{9}{7}\) and \(\frac{2}{7}\) have the same denominator, which is 7.

Step 2 - Subtract the Numerators

Since the denominators are the same, subtract the numerators: \(9 - 2 = 7\).

Step 3 - Write the Result

Place the result from Step 2 over the common denominator: \(\frac{7}{7}\).

Step 4 - Simplify the Fraction

Simplify \(\frac{7}{7}\) to 1, because any number divided by itself equals 1.

Step 5 - Verify Using a Calculator

If you're unsure, use a calculator to verify your result: \(\frac{9}{7} - \frac{2}{7} = \frac{7}{7} = 1\).

Key Concepts

common denominator

When subtracting fractions, it is crucial to have a common denominator. A common denominator is a shared bottom number in the fractions you are working with. In fractions like \(\frac{9}{7}\) and \(\frac{2}{7}\), the number 7 is the common denominator. It means that both fractions are parts of the same whole, making it easy to subtract the numerators. This step ensures that we are comparing like parts when performing operations on fractions. Finding a common denominator is essential for correctly adding or subtracting fractions, preventing mistakes in your calculations.

simplify fractions

After performing arithmetic operations with fractions, simplifying them gives you the cleanest form of the result. Simplifying means reducing the fraction to its simplest form, where the numerator and denominator have no common factors other than 1. For instance, \(\frac{7}{7}\) simplifies to 1. This simplification is vital as it offers a clear and concise way to present your answer. To simplify, look for common factors in the numerator and the denominator or recognize basic identities, like any number over itself equals 1.

subtracting fractions

Subtracting fractions involves a few straightforward steps. First, ensure that the fractions have a common denominator. If they don't, find equivalent fractions that do. For fractions \(\frac{9}{7}\) and \(\frac{2}{7}\), they already share the same denominator, which streamlines our process. Next, subtract the numerators: \9 - 2 = 7\. After that, place the result over the common denominator, resulting in \(\frac{7}{7}\). Finally, simplify the fraction. In this case, \(\frac{7}{7}\) simplifies to 1. Use these steps to confidently handle subtracting fractions in any problem you encounter.