Problem 38 - Prealgebra

Question

Find each sum or difference. Write in simplest form. \(\frac{3 r}{10}+\frac{3 r}{10}\)

Step-by-Step Solution

Answer

Sum is \(\frac{3r}{5}\).

Step 1: Identify the operation

We need to find the sum or difference of the given fractions. In this exercise, we are adding the two fractions: \(\frac{3r}{10}+\frac{3r}{10}\).

Step 2: Ensure the denominators are the same

The denominators of both fractions are the same, which is 10. This allows us to simply add the numerators while keeping the same denominator.

Step 3: Add the numerators

Add the numerators, keeping the denominator the same: \(3r + 3r = 6r\). So, the resulting fraction is \(\frac{6r}{10}\).

Step 4: Simplify the fraction

To simplify \(\frac{6r}{10}\), find the greatest common factor of 6 and 10, which is 2. Divide both the numerator and denominator by 2: \(\frac{6r \div 2}{10 \div 2} = \frac{3r}{5}\).

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means that the numerator (the number on top) and the denominator (the number on the bottom) should be divided by their greatest common factor (GCF). The GCF is the largest number that divides both the numerator and the denominator evenly.
For example, when we have the fraction \(\frac{6r}{10}\), we want to determine the GCF of 6 and 10, which is 2. By dividing both the numerator and the denominator by 2, we effectively simplify the fraction to \(\frac{3r}{5}\). The fraction becomes easier to understand and work with.
Key points to remember:

• Always check for the largest number that can divide both elements without a remainder.
• Simplifying doesn’t change the value of the fraction, just its appearance.
• Simplifying fractions makes further calculations more straightforward.

Common Denominators

When adding or subtracting fractions, having a common denominator is essential. A common denominator is a shared multiple of the denominators in the fractions you are working with.
In this exercise, both fractions are \(\frac{3r}{10}\), which means the denominators are already the same, simplifying our task. This uniform denominator indicates that the fractions represent parts of the same whole, allowing us to add the numerators directly.
Understanding common denominators:

• Without common denominators, you cannot directly add or subtract the numerators.
• Find a common denominator by identifying the least common multiple (LCM) when they differ.
• Converting fractions to have common denominators keeps the setup consistent across different operations.

Numerator Addition

Adding the numerators is the step in fraction addition where you sum up the top parts while keeping the denominator unchanged. This is possible only when the denominators of the fractions are the same.
For instance, in the given exercise: \(\frac{3r}{10} + \frac{3r}{10}\), we add the numerators while leaving the denominator as 10. This gives us \(3r + 3r = 6r\), resulting in the fraction \(\frac{6r}{10}\).
Why numerator addition is straightforward:

• It allows for simple arithmetic operations without altering the base (denominator).
• Direct addition helps maintain the relationships that fractions represent.
• Remember: Only add numerators when denominators are equivalent, ensuring the fractions align properly.