Problem 3 - Elementary Algebra

Question

For exercises \(1-4\), evaluate. $$ \frac{12}{35}-\frac{2}{35} $$

Step-by-Step Solution

Answer

The answer is \(\frac{2}{7}\).

Step 1: Identify the Common Denominator

Observe that both fractions \(\frac{12}{35}\) and \(\frac{2}{35}\) share the same denominator, 35.

Step 2: Subtract the Numerators

Since the denominators are the same, subtract the numerators directly: \(\frac{12-2}{35}\).

Step 3: Simplify the Fraction

Subtract the numerators: \(\frac{10}{35}\).

Step 4: Reduce the Fraction to the Lowest Terms

Simplify \(\frac{10}{35}\) by finding the greatest common divisor (GCD) of 10 and 35, which is 5. So, divide both the numerator and the denominator by 5: \(\frac{10 \div 5}{35 \div 5} = \frac{2}{7}\).

Key Concepts

Common Denominator

When subtracting fractions, it is easiest if the fractions already have the same denominator. The denominator is the bottom part of the fraction. In the exercise, both fractions \(\frac{12}{35}\) and \(\frac{2}{35}\) have the same denominator, which is 35. This makes the subtraction simpler because you don’t have to find a common denominator. If the denominators were different, you would need to find a common denominator by determining the least common multiple (LCM) of the two denominators. Usually, multiplying the two denominators together gives a common denominator, then adjusting the numerators accordingly.

Numerator Subtraction

Once you have a common denominator, you can focus on subtracting the numerators. The numerator is the top part of the fraction. In our exercise, after confirming the common denominator, we subtract the numerators 12 and 2: \[\frac{12 - 2}{35} = \frac{10}{35}\]. The denominator stays the same during this step. Always ensure to keep the denominator unchanged when performing numerator subtraction.

Simplifying Fractions

After subtracting the numerators, the next step is to simplify the fraction. Simplifying fractions helps to reduce them to their lowest terms. A fraction is simplified when the greatest common divisor (GCD) of both the numerator and the denominator is 1. From our exercise, we get the fraction \[\frac{10}{35}\]. This fraction can be simplified because both 10 and 35 share common divisors greater than 1. By simplifying, we make the fraction easier to understand and use in further calculations.

Greatest Common Divisor (GCD)

To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. In our example, the numbers 10 and 35 have a GCD of 5. We divide both the numerator and the denominator by their GCD: \[\frac{10 \div 5}{35 \div 5} = \frac{2}{7}\]. Now, the fraction \[\frac{2}{7}\] is in its simplest form. Practicing finding the GCD will help you become efficient in reducing fractions quickly and accurately.