Problem 63 - Prealgebra

Question

Add or subtract the following fractions, as indicated. $$\frac{9}{10}-\frac{3}{10}$$

Step-by-Step Solution

Answer

The result of \( \frac{9}{10} - \frac{3}{10} \) is \( \frac{3}{5} \).

Step 1: Identify the Operation

The problem requires you to subtract the second fraction from the first. The operation to be performed is subtraction: \( \frac{9}{10} - \frac{3}{10} \).

Step 2: Verify Like Denominators

Both fractions \( \frac{9}{10} \) and \( \frac{3}{10} \) have the same denominator, which is 10. Therefore, you can directly subtract the numerators.

Step 3: Subtract the Numerators

Since the denominators are the same, subtract the numerators: \( 9 - 3 = 6 \). So, \( \frac{9}{10} - \frac{3}{10} = \frac{6}{10} \).

Step 4: Simplify the Fraction

Check if the resulting fraction \( \frac{6}{10} \) can be simplified. Both the numerator and the denominator are divisible by 2. Simplify by dividing the numerator and the denominator by 2: \( \frac{6 \div 2}{10 \div 2} = \frac{3}{5} \).

Key Concepts

Subtraction of Fractions

When you are faced with subtracting fractions, the fundamental principle is to focus on the numerators (the top numbers) first, after ensuring the denominators (the bottom numbers) are the same.
In this case, you have two fractions: \(\frac{9}{10}\) and \(\frac{3}{10}\). Since subtraction is the operation you will perform, illustrated by the problem \(\frac{9}{10} - \frac{3}{10}\), you should proceed by directly subtracting the numerators.
This results in subtracting 3 from 9, which is 6. Now, you are left with the fraction \(\frac{6}{10}\).
At this stage, subtraction of fractions with the same denominators is complete. It's a straightforward chain of subtracting numerators once you validate that the denominators are identical.

Like Denominators

The term 'like denominators' refers to fractions that have the same denominator. It is crucial for fractions involved in addition or subtraction to have like denominators.
This concept is significant because, only when the denominators are the same, you can directly add or subtract the numerators. If they aren't, you would first need to convert them to like denominators by finding a common denominator before proceeding.
In our example, both \(\frac{9}{10}\) and \(\frac{3}{10}\) have the denominator 10. This means they are perfectly set up for the subtraction task, allowing you to proceed without the need for additional conversion steps.
Recognizing and confirming like denominators at the start ensures a smooth and correct fraction operation.

Simplifying Fractions

Simplifying a fraction means reducing it to its smallest possible numerator and denominator while maintaining the same value.
Once you've subtracted the fractions and obtained \(\frac{6}{10}\), simplifying it involves finding the greatest common factor (GCF) of 6 and 10.
The GCF in this instance is 2. Dividing both the numerator and the denominator by the GCF simplifies \(\frac{6}{10}\) to \(\frac{3}{5}\).
Simplification not only makes fractions easier to read and work with but is often required for the final answer. Whenever possible, always simplify your fractions to their smallest equivalent.