Problem 1 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{180}{420} $$

Step-by-Step Solution

Answer

\frac{3}{7}

Step 1: Identify the greatest common divisor (GCD)

To simplify the fraction \(\frac{180}{420}\), find the greatest common divisor (GCD) of 180 and 420. List out the factors of each number: \[ 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180 \] \[ 420: 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420 \] The greatest common divisor is 60.

Step 2: Divide the numerator and denominator by the GCD

Now, divide both the numerator and the denominator of \(\frac{180}{420}\) by the GCD, which is 60: \[ \frac{180 \div 60}{420 \div 60} = \frac{3}{7} \]

Step 3: Write the simplified fraction

After dividing both the numerator and the denominator by their greatest common divisor, the fraction simplifies to \(\frac{3}{7}\).

Key Concepts

greatest common divisor

The greatest common divisor (GCD) is a key concept in fraction simplification. It is the largest number that divides both the numerator and the denominator without leaving a remainder. To find the GCD of two numbers, list out all the factors of each number. For example, with 180 and 420, the factors are:

• 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
• 420: 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420

The greatest common factor here is 60. This number helps in reducing fractions to their simplest form.

numerator

In a fraction, the numerator is the number above the line. It represents the number of parts we have. For instance, in the fraction \(\frac{180}{420}\), 180 is the numerator. When simplifying, we divide the numerator by the GCD. So, 180 divided by 60 (GCD) gives us 3. This helps in reducing the numerator while keeping the fraction equivalent.

denominator

The denominator is the number below the line in a fraction. It shows the total number of equal parts the whole is divided into. For the fraction \(\frac{180}{420}\), 420 is the denominator. Similar to the numerator, we also divide the denominator by the GCD. Dividing 420 by 60 gives 7. This step is essential to accurately simplify the fraction.

fraction simplification

Fraction simplification is the process of making a fraction as simple as possible. This often involves dividing both the numerator and the denominator by their GCD. Doing this ensures the fraction remains equivalent to the original. Using our example: \(\frac{180}{420}\), the GCD is 60. Dividing both terms:
\(\frac{180 \div 60}{420 \div 60}\) simplifies to \(\frac{3}{7}\).
This resulting fraction, \(\frac{3}{7}\), is the simplest form of \(\frac{180}{420}\). It’s easier to understand and work with in calculations.