Problem 76 - Algebra 1: Concepts and Skills

Question

Subtract. Write the answer in simplest form. \begin{equation} \frac{3}{4}-\frac{1}{4} \end{equation}

Step-by-Step Solution

Answer

The answer in simplest form is \(\frac{1}{2}\).

Step 1: Identify the Problem

We are asked to subtract the fraction \(\frac{1}{4}\) from \(\frac{3}{4}\). An important point here is that both fractions share the same denominator.

Step 2: Perform the Subtraction

In subtraction of fractions with common denominators, keep the denominator and subtract the numerators. So, we will subtract the numerator of the second fraction from the numerator of the first fraction. This gives us: \(\frac{3}{4}-\frac{1}{4}=\frac{3-1}{4}=\frac{2}{4}\).

Step 3: Simplify the Result

The result, \(\frac{2}{4}\), can be simplified by dividing both the numerator and the denominator by 2. This gives us: \(\frac{2}{4}=\frac{2\div2}{4\div2}=\frac{1}{2}\).

Key Concepts

Common Denominator

When it comes to subtracting fractions, having a common denominator makes the process much easier. A common denominator is simply when two fractions share the same bottom number, known as the denominator. For example, in our subtraction problem \(\frac{3}{4} - \frac{1}{4}\), both fractions have the denominator 4. This means we can directly subtract the numerators: 3 and 1.

• Common denominators allow you to focus only on the numerators during subtraction.
• It simplifies the operation since the denominator remains unchanged.

To find a common denominator, especially when the fractions are different, you might need to find the least common multiple (LCM) of the denominators. In this case, as the denominators were already the same, we could skip this step.

Simplifying Fractions

After subtracting your fractions, it’s often necessary to simplify the result. Simplifying a fraction means reducing it to its simplest form. For example, after subtracting \(\frac{3}{4} - \frac{1}{4}\), we ended up with \(\frac{2}{4}\).

• Simplification involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
• In our example, the GCD of 2 and 4 is 2.

By dividing both the numerator and the denominator by 2, we can express \(\frac{2}{4}\) as \(\frac{1}{2}\). This involves reducing the top and bottom as much as possible without changing the value of the fraction.

Numerators and Denominators

It’s crucial to understand the role of numerators and denominators in fractions. The numerator is the top number, indicating how many parts we have, while the denominator is the bottom number, indicating into how many parts the whole is divided.

• In \(\frac{3}{4}\), the numerator is 3, and the denominator is 4.
• Similarly, in \(\frac{1}{4}\), the numerator is 1, and the denominator is still 4.

When subtracting fractions with the same denominator, the operation only occurs between the numerators. In our example, this would be 3 minus 1, resulting in a new numerator of 2. The denominator remains 4. This is why it’s essential to pay attention to both parts of a fraction when performing operations like addition and subtraction.