Problem 47 - Elementary and Intermediate Algebra

Question

Simplify. $$ \frac{42}{50} $$

Step-by-Step Solution

Answer

\( \frac{21}{25} \)

Step 1: Find the Greatest Common Divisor (GCD)

Find the greatest common divisor (GCD) of the numerator and the denominator. For the fraction \( \frac{42}{50} \), determine the GCD of 42 and 50.

Step 2: List the Factors

List all the factors of 42 and 50. Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42.Factors of 50: 1, 2, 5, 10, 25, 50.

Step 3: Identify the Highest Common Factor

From the lists, identify the highest number that appears in both lists. The highest common factor (HCF) of both numbers is 2.

Step 4: Simplify the Fraction

Divide both the numerator and the denominator by their GCD, which is 2.\[ \frac{42 \div 2}{50 \div 2} = \frac{21}{25} \]

Key Concepts

Greatest Common Divisor

When simplifying fractions, we must first understand the **Greatest Common Divisor (GCD)**. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.
To find the GCD of two numbers, we list their factors and identify the largest one they have in common. For example, with the fraction \(\frac{42}{50}\), we need to find the GCD of 42 and 50.

Factors

Factors are the integers that exactly divide a given number. Knowing how to list all factors is essential in finding the GCD.
For instance, the factors of 42 are:

• 1
• 2
• 3
• 6
• 7
• 14
• 21
• 42

And the factors of 50 are:

• 1
• 2
• 5
• 10
• 25
• 50

Make sure to list all factors carefully to avoid missing any common factors.

Highest Common Factor

Once we have the lists of factors for both numbers, we look for the largest number that appears in both lists. This number is the **Highest Common Factor (HCF)**, and it is the key to simplifying the fraction.
In our example, the common factors of 42 and 50 are 1 and 2. Therefore, the HCF is 2.
Always double-check your common factors to ensure accuracy!

Fraction Simplification

After identifying the GCD (or HCF), we use it to simplify the fraction.
To do this, divide both the numerator and the denominator by the GCD. For \(\frac{42}{50}\), the GCD is 2, so we get:
\[ \frac{42 \div 2}{50 \div 2} = \frac{21}{25} \]
This means that \(\frac{42}{50}\) simplified is \(\frac{21}{25}\).
Remember, simplifying a fraction makes it easier to work with in further calculations.