Problem 75 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{16}{7}-\frac{2}{7}$$

Step-by-Step Solution

Answer

The result of \( \frac{16}{7} - \frac{2}{7} \) is \(2\)

Step 1: Identify the fractions

The provided fractions to subtract are \( \frac{16}{7} \) and \( \frac{2}{7} \). Since both fractions have the same denominator, they can directly be subtracted.

Step 2: Subtract the Numerators

Subtract the numerators and keep the denominator same. \(\frac{16}{7} - \frac{2}{7} = \frac{(16-2)}{7}\)

Step 3: Calculate the Numerator

The subtaction \(16-2\) is 14. So, the resulting fraction is \( \frac{14}{7} \).

Step 4: Simplify the Resulting Fraction

The fraction \( \frac{14}{7} \) can be simplified to \(2\) as both numerator and denominator are divisible by 7.

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing a fraction to its simplest form in which the numerator and the denominator have no common factors other than 1. This process is also known as 'reducing fractions.' When simplifying, you're searching for the greatest common divisor (GCD) that both the numerator and the denominator can be divided by.

For example, in the fraction \( \frac{14}{7} \), the GCD of 14 and 7 is 7. You'd divide both the numerator (14) and the denominator (7) by this largest common factor to simplify the fraction. This results in \( \frac{14 \div 7}{7 \div 7} = \frac{2}{1} \), which is simply the whole number 2. Simplifying does not change the value of the fraction, it just makes it easier to understand and work with.

Subtracting Numerators

Subtracting numerators is a straightforward process when dealing with fractions that have the same denominator. When the denominators are the same, you keep the denominator constant and simply subtract the numerator of the second fraction from the numerator of the first fraction.

Let's use the fractions \( \frac{16}{7} \) and \( \frac{2}{7} \) as an example. Since the denominators are both 7, you subtract the numerators as follows: \( 16 - 2 = 14 \). The resulting fraction is \( \frac{14}{7} \). It's important to only subtract the numerators—changing the denominator would result in a different, incorrect fraction value.

Lowest Terms

A fraction is considered to be in its lowest terms when the numerator and the denominator are as small as possible and cannot be divided by any common number except for 1. Simplifying a fraction until it is in its lowest terms ensures that the fraction is presented in its most basic form, which can be desirable for clarity and can make it easier to perform further mathematical operations.

Returning to the example of the fraction \( \frac{14}{7} \) — once we subtract the numerators, we are left with a fraction that can still be simplified because 14 and 7 share a common factor, 7. Dividing both by 7 brings the fraction to its lowest terms. In this case, the fraction simplifies completely to the whole number 2, which is the fraction in its absolute lowest term.