Problem 2 - Elementary Algebra

Question

For exercises \(1-4\), evaluate. $$ \frac{9}{50}+\frac{3}{50} $$

Step-by-Step Solution

Answer

The result is \( \frac{6}{25} \).

Step 1: Understand the Problem

The task is to add two fractions with the same denominator: \ \( \frac{9}{50}+\frac{3}{50} \ \)

Step 2: Add the Numerators

Since the denominators are the same, add the numerators: \ \( 9 + 3 = 12 \ \)

Step 3: Write the Result

Place the sum of the numerators over the common denominator: \ \( \frac{12}{50} \ \)

Step 4: Simplify the Fraction

Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD). The GCD of 12 and 50 is 2. \ \ \( \frac{12 \div 2}{50 \div 2} = \frac{6}{25} \ \)

Key Concepts

common denominator

When adding fractions, it's crucial that they share the same denominator. The denominator is the bottom number of a fraction, representing the total number of equal parts. Having a common denominator means you can directly add or subtract the fractions, because the parts you're working with are the same size. In the given problem, the fractions already have the same denominator of 50. This simplifies the process significantly, as you can directly proceed to adding the numerators.

simplifying fractions

After adding the numerators and forming a new fraction, the next step is to simplify it. Simplifying a fraction means making it as simple as possible, usually by dividing both the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD). For example, with the fraction \( \frac{12}{50} \), you look for the largest number that evenly divides both 12 and 50. In this case, that number is 2. So, we divide both the numerator and the denominator by 2, resulting in \( \frac{6}{25} \). The process of simplifying helps make the fraction easier to understand and work with.

greatest common divisor (GCD)

The greatest common divisor (GCD) is the largest number that can exactly divide both the numerator and the denominator of a fraction without leaving a remainder. Finding the GCD is a key step in simplifying fractions. To find the GCD of two numbers, you can list the factors of both numbers and then identify the largest factor they have in common. For the numbers 12 and 50, their GCD is 2. Once you know the GCD, you can divide both the numerator and the denominator by this number to simplify the fraction.