Problem 59 - Elementary Algebra Within Reach

Question

In Exercises 57-60, evaluate the expression. $$ \frac{2}{7}-\frac{1}{3} $$

Step-by-Step Solution

Answer

The result of the subtraction \(\frac{2}{7} - \frac{1}{3}\) is \(-\frac{1}{21}\).

Step 1: Identify the Least Common Denominator (LCD)

The first step is to identify the least common denominator (LCD) between the two fractions. The denominators are 7 and 3. To find a common denominator, simply multiply the two denominators. So, in this case, the LCD is \(7 \times 3 = 21\).

Step 2: Convert the fractions

Next, convert each fraction to have the obtained least common denominator. Hence, \[\frac{2}{7} becomes \frac{2 \times 3}{21} = \frac{6}{21}\]and \[\frac{1}{3} becomes \frac{1 \times 7}{21} = \frac{7}{21}\]in order to make both fractions have the same denominator.

Step 3: Perform the subtraction

Now, subtract the two fractions, keeping the common denominator constant. Hence,\[\frac{6}{21} - \frac{7}{21} = \frac{-1}{21}\]

Key Concepts

Least Common Denominator

When subtracting fractions, it's important they share the same denominator. This shared denominator is known as the Least Common Denominator (LCD). The LCD is the smallest number that both denominators can divide into evenly. In our example, we had the fractions \( \frac{2}{7} \) and \( \frac{1}{3} \).

To find the LCD, you can choose a simple trick if you have small denominators: multiply them together. In this case, \( 7 \times 3 = 21 \). Thus, 21 becomes the Least Common Denominator.

Finding the LCD ensures that both fractions are converted to the same base so that they can be easily added or subtracted. Without the same denominator, the fractions are talking about different "sized" parts, which makes it impossible to directly subtract them.

Converting Fractions

After identifying the Least Common Denominator, the next step is converting each fraction to have that denominator. This involves adjusting the numerators to maintain the same value as the original fractions.

Let's see how it works:

• Start with \( \frac{2}{7} \). To convert it, multiply both the numerator and denominator by 3 to match the denominator of 21. This gives \( \frac{2 \times 3}{21} = \frac{6}{21} \).
• Now, take \( \frac{1}{3} \). Multiply both its numerator and denominator by 7. You'll get \( \frac{1 \times 7}{21} = \frac{7}{21} \).

Ensure that the value of the fractions remains unchanged even though they look different now. Both fractions have been effectively re-expressed with a common denominator.

Subtraction of Fractions

After ensuring fractions have a common denominator, subtraction becomes straightforward. Keep the denominator the same and perform the subtraction on the numerators.

In our exercise, the fractions \( \frac{6}{21} \) and \( \frac{7}{21} \) are ready for subtraction. Since they already share the denominator 21, you subtract the numerators: \( 6 - 7 = -1 \).

So, \( \frac{6}{21} - \frac{7}{21} = \frac{-1}{21} \).

The result shows that the second fraction was larger than the first, hence the subtraction gives a negative fraction. Remember, the denominator remains unchanged in the subtraction process, allowing you to focus only on the numerators.