Problem 68 - Elementary and Intermediate Algebra

Question

Perform the indicated operation and, if possible, simplify. If there are no variables, check using a calculator. $$ \frac{12}{5}-\frac{2}{5} $$

Step-by-Step Solution

Answer

The simplified result is 2.

Step 1: Identify the Common Denominator

Observe that both fractions, \(\frac{12}{5}\) and \(\frac{2}{5}\), have the same denominator, which is 5.

Step 2: Subtract the Numerators

Since the denominators are the same, subtract the numerators: \(12 - 2\). Keep the common denominator: \(\frac{12 - 2}{5}\).

Step 3: Simplify the Expression

Perform the subtraction in the numerator: \(12 - 2 = 10\). So, the fraction simplifies to \(\frac{10}{5}\).

Step 4: Simplify the Fraction

Divide the numerator by the denominator: \( \frac{10}{5} = 2 \).

Step 5: Verify Using Calculator

To ensure the solution is correct, use a calculator to find \( \frac{12}{5} - \frac{2}{5} \), which should also be 2.

Key Concepts

Common Denominator

When dealing with the subtraction of fractions, a necessary first step is to identify a common denominator. The common denominator is essentially the same denominator for both fractions, making it possible to subtract them directly. In this exercise, the fractions \(\frac{12}{5}\) and \(\frac{2}{5}\) each have a denominator of 5.

Identifying the common denominator is crucial because it ensures the fractions are referring to the same parts of a whole. Without a common denominator, we can't directly add or subtract fractions. For example, if you're given \(\frac{3}{4}\) and \(\frac{2}{3}\) to subtract, you'd first need to convert them to have a common denominator, in this case, 12, before you could proceed.

Remember, for fractions where the denominators are already the same, this step is straightforward. However, when they are different, you must find the least common multiple (LCM) of the denominators to proceed.

Subtracting Fractions

After confirming that the fractions have a common denominator, the next step is to subtract the numerators while keeping the common denominator. In our given exercise: \(\frac{12}{5} - \frac{2}{5}\)

We subtract the numerators 12 and 2, making it:

\[ \frac{12 - 2}{5} \]

So we get \(\frac{10}{5}\).

This step is about combining the fractions. As long as the denominators match, we don't change them; they remain the same, while our attention is purely on the numerators.

Subtracting the numerators of fractions enables us to work with parts of the same whole, without changing the base or the scale of the fractions themselves. It’s much like subtracting slices of a pizza; as long as the pizza size doesn't change, you're accurately keeping track of what's been taken away.

Simplifying Fractions

Finally, it's time to simplify the resulting fraction. Simplifying a fraction involves reducing it to its most basic form. This often means dividing the numerator and the denominator by their greatest common divisor (GCD).

From our earlier steps, we ended up with \(\frac{10}{5}\). To simplify, divide both the numerator (10) and the denominator (5) by 5, the GCD here.

Perform the division:

\[ \frac{10}{5} = 2 \]

Thus, the fraction simplifies to 2.

Simplification makes the fraction easier to understand and work with. Always ensure that the fraction is in its simplest form unless the problem specifies otherwise. Simplified fractions are simpler and more precise, which is always helpful when communicating mathematical concepts or preparing for further calculations.