Problem 97 - Elementary Algebra

Question

For exercises 95-97, evaluate. $$ \frac{16}{21}-\frac{2}{21} $$

Step-by-Step Solution

Answer

The answer is \(\frac{2}{3}\).

Step 1: Identify the denominators

Notice that both fractions have the same denominator, which is 21.

Step 2: Subtract the numerators

Since the denominators are the same, subtract the numerators: 16 - 2.

Step 3: Simplify the fraction

After subtracting the numerators, you get \(\frac{14}{21}\). Simplify this fraction by finding the greatest common divisor (GCD) of 14 and 21, which is 7. Divide both the numerator and the denominator by 7.

Step 4: Write the simplified fraction

\(\frac{14 \, \div \, 7}{21 \, \div \, 7} = \frac{2}{3}\).

Key Concepts

common denominators

Understanding common denominators is crucial in fraction subtraction. When subtracting fractions, both fractions must have the same denominator. The denominator is the bottom number of a fraction and signifies the total number of equal parts the whole is divided into. In our exercise, both fractions \( \frac{16}{21} \) and \( \frac{2}{21} \) have 21 as their denominator. When the denominators are the same, you can move on to the next step: subtracting the numerators. This makes the process simpler as you don't need to perform any additional calculation to make the denominators the same.

numerator subtraction

Once you have common denominators, the next step is to subtract the numerators. The numerator is the top number in a fraction and represents the number of parts you have. In our exercise, the numerators are 16 and 2. To subtract these numerators, you perform a simple subtraction: 16 - 2.
After subtracting, you'll get \( \frac{14}{21} \). The denominator remains the same while you only subtract the numerators. This ensures the fractional parts are consistent and correctly represent the calculated portion of the whole.

fraction simplification

The final step in fraction subtraction is to simplify the fraction. Simplifying a fraction means reducing it to its lowest terms. This involves finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the highest number that divides both the numerator and the denominator without leaving a remainder.
In our example, \( \frac{14}{21} \) can be simplified by finding the GCD of 14 and 21, which is 7. Divide both the numerator and the denominator by 7:
\[ \frac{14 \div \7}{21 \div \7} = \frac{2}{3} \]
Thus, \( \frac{14}{21} \) simplifies to \( \frac{2}{3} \). Simplifying fractions makes them easier to understand and work with, and it’s a common final step in many fraction problems.