Problem 104 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ \frac{19}{35}-\frac{12}{35} $$

Step-by-Step Solution

Answer

The result is \( \frac{1}{5} \).

Step 1: Ensuring Like Denominators

Both fractions have the same denominator of 35, so the subtraction can be performed directly.

Step 2: Subtract the Numerators

Subtract the numerators while keeping the denominator the same: \( 19 - 12 = 7 \). Thus, the operation becomes \( \frac{19}{35} - \frac{12}{35} = \frac{7}{35} \).

Step 3: Simplify the Fraction

Determine if \( \frac{7}{35} \) can be simplified. The greatest common divisor of 7 and 35 is 7. Divide both the numerator and the denominator by 7: \( \frac{7}{35} = \frac{1}{5} \).

Key Concepts

Simplifying Fractions

Understanding how to simplify fractions is essential for dealing with fractions effectively. Simplifying a fraction means reducing it to its simplest form, where the numerator and the denominator have no common divisors other than 1. This helps make calculations easier and results clearer.

When simplifying a fraction, follow these steps:

• Identify the greatest common divisor (GCD) of the numerator and the denominator. This is the largest number that can divide both numbers without leaving a remainder.
• Divide both the numerator and the denominator by their GCD.

For instance, in the exercise, we have the fraction \( \frac{7}{35} \). The GCD of 7 and 35 is 7, so dividing both by 7 simplifies the fraction to \( \frac{1}{5} \).

Practicing this skill will improve your ability to handle complex fraction operations with ease.

Subtraction of Fractions

Subtracting fractions can seem tricky at first, but it's all about ensuring the denominators are the same. Fractions with like denominators can be subtracted directly by working with the numerators.

Here's the step-by-step process:

• Verify that the fractions have the same denominator. If they don't, find a common denominator.
• Once the denominators match, subtract the numerators while keeping the denominator unchanged.
• Consider simplifying the resulting fraction, if possible.

In our case, with \( \frac{19}{35} - \frac{12}{35} \), both fractions already have the same denominator, so we can proceed to subtract the numerators: \( 19 - 12 = 7 \). This gives us \( \frac{7}{35} \).

Understanding these steps prepares you for dealing with more complicated addition and subtraction problems in the future.

Greatest Common Divisor

The greatest common divisor (GCD) is an essential concept when simplifying fractions. It is the largest number that divides two or more integers without leaving a remainder.

Here's how you can find the GCD:

• List the factors of each number.
• Identify the largest factor that appears in both lists.

In the exercise, the numbers are 7 and 35:

• Factors of 7: 1, 7
• Factors of 35: 1, 5, 7, 35

The largest common factor is 7, which makes it the GCD. Using this, we reduce \( \frac{7}{35} \) to \( \frac{1}{5} \). Knowing how to find the GCD allows you to simplify fractions quickly, making your math work cleaner and more efficient.

Keep practicing finding the GCD and soon it will become second nature!