Problem 23 - Prealgebra

Question

Find sum or difference. Write in simplest form. \(\frac{10}{11}-\frac{8}{11}\)

Step-by-Step Solution

Answer

The difference is \( \frac{2}{11} \).

Step 1: Identify the Operation

The given expression involves subtraction of fractions with the same denominator: \( \frac{10}{11} - \frac{8}{11} \). This means we need to subtract the numerators while keeping the denominator the same.

Step 2: Subtract the Numerators

Subtract the numerator of the second fraction from the numerator of the first fraction. So, we have:\[ 10 - 8 = 2 \].

Step 3: Write the Resulting Fraction

Since the denominators are the same, write the result as a new fraction: \( \frac{2}{11} \).

Step 4: Simplify the Fraction

Check if \( \frac{2}{11} \) can be simplified further. Since 2 and 11 have no common factors (other than 1), \( \frac{2}{11} \) is already in its simplest form.

Key Concepts

Subtraction of Fractions

When you are dealing with the subtraction of fractions, the main focus is on the numerators, provided that your denominators are already the same. This simplifies your work greatly because you don't need to modify the denominator or find a common one, which is usually the main hurdle in fraction calculations. Instead, you can go straight to subtracting the numerators.

Here's how you manage this operation:

• Make sure the denominators are the same. This allows us to directly subtract the numerators.
• Subtract the second fraction's numerator from the first fraction's numerator.
• Place this result over the original denominator, as it remains unchanged.

For instance, solving \( \frac{10}{11} - \frac{8}{11} \), you just look at the numerators: 10 and 8. Subtract as 10 - 8, resulting in the numerator 2. Therefore, the resulting fraction is \( \frac{2}{11} \). Using common denominators like this makes subtraction clear and straightforward.

Simplifying Fractions

After you perform operations like addition or subtraction, it's crucial to check if your result can be simplified. Simplifying fractions means reducing them to their most basic form, where the numerator and denominator share no common factors other than 1.

To simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

In our example, we end up with \( \frac{2}{11} \). Here:

• The numerator is 2, and the denominator is 11.
• The GCD of 2 and 11 is 1, which means they don't have common factors other than 1.

Thus, \( \frac{2}{11} \) is already in its simplest form, needing no further simplification. The value remains accurately represented when expressed in this simplest form.

Common Denominators

Understanding common denominators is key in fraction operations like addition and subtraction. A common denominator is a shared multiple of two or more denominators and is essential because you need uniformity in denominators to perform these operations without altering their values.

When you see fractions with the same denominator:

• You can directly perform the operation (addition or subtraction) on the numerators.
• The denominator remains unchanged, maintaining its role as a uniform reference point.

In our subtraction exercise with \( \frac{10}{11} - \frac{8}{11} \):

• Both fractions share the denominator of 11.
• This common factor allows us to focus on a simple subtraction of the numerators to obtain \( \frac{2}{11} \).

Equal denominators make the entire process more straight-forward, letting us concentrate on solving rather than transforming fractions. Ensuring that denominators match is critical to correct and efficient fraction arithmetic.