Problem 21 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{42}{210}$$

Step-by-Step Solution

Answer

The reduced form is 1/5.

Step 1 - Identify the GCD

Determine the greatest common divisor (GCD) of the numerator and the denominator. For 42 and 210, the GCD is 42.

Step 2 - Divide Numerator by the GCD

Divide the numerator by the GCD. o: \[\frac{42}{42} = 1\]

Step 3 - Divide Denominator by the GCD

Divide the denominator by the GCD. o: \ \[\frac{210}{42} = 5\]

Step 4 - Write the Simplified Expression

Combine the results of Step 2 and Step 3 to write the simplified expression: 1/5.

Key Concepts

Greatest Common Divisor

Simplifying rational expressions often starts with finding the greatest common divisor (GCD). The GCD is the largest number that can exactly divide both the numerator (the top number of a fraction) and the denominator (the bottom number of a fraction) without leaving a remainder.
To find the GCD:

• List the factors of the numerator.
• List the factors of the denominator.
• Identify the largest factor that appears in both lists.

For example, in the fraction \(\frac{42}{210}\), the factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42, while the factors of 210 are 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, and 210. The largest common factor here is 42, so the GCD is 42.

Numerator

The numerator is the top part of a fraction. It represents how many parts of the whole are being considered. In our exercise example, the numerator is 42.
Once the GCD is found, we divide the numerator by this GCD. This helps to simplify the fraction. For instance, dividing 42 by its GCD, 42, results in:
\[\frac{42}{42} = 1\]
This process is important because it reduces the fraction step by step, making it easier to understand and use.

Denominator

The denominator is the bottom part of a fraction. It indicates the total number of equal parts the whole is divided into. In the given exercise, the denominator is 210.
After identifying the GCD, the next step is to divide the denominator by this GCD. For instance, dividing 210 by the GCD 42 results in:
\[\frac{210}{42} = 5\]
This division helps to simplify the fraction, making it more manageable and straightforward.

Fraction Reduction

Fraction reduction simplifies a complex fraction into its simplest form. It involves finding the GCD of the numerator and the denominator, and then dividing both by this GCD.
For example, in the exercise provided:\[\frac{42}{210}\]
We first found the GCD, which is 42. Then we divided the numerator and denominator by 42 into their simplest forms:
\( \frac{42}{42} = 1 \) (numerator) and \( \frac{210}{42} = 5 \) (denominator).
So, the simplified fraction is \( \frac{1}{5} \). This makes the fraction easier to understand and work with, and is especially useful for further mathematical operations.