Problem 33 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each fraction by reducing it to its lowest terms. $$\frac{35}{50}$$

Step-by-Step Solution

Answer

The fraction \(\frac{35}{50}\) simplified to its lowest terms is \(\frac{7}{10}\).

Step 1: Identify the Greatest Common Divisor

Firstly, the greatest common divisor of 35 and 50 needs to be identified. Both numbers can be divided by 5 without leaving a remainder, so the greatest common divisor is 5.

Step 2: Divide the Numerator and Denominator by their GCD

Next, divide both the numerator (35) and the denominator (50) by the greatest common divisor, which is 5. \(\frac{35}{5} = 7\) and \(\frac{50}{5} = 10\). So the simplified form of the fraction is \(\frac{7}{10}\).

Key Concepts

Greatest Common Divisor

The greatest common divisor, often abbreviated as GCD, is a critical concept when simplifying fractions. To reduce a fraction to its simplest form, you must find a common number that divides both the numerator and the denominator evenly. This shared number is the GCD.

Finding the GCD involves examining all divisors of the two numbers and choosing the largest one they share. For example, to simplify \( \frac{35}{50} \), consider the numbers 35 and 50. Both can be divided by 1, 5, and themselves, but their greatest shared divisor is 5. This means 5 is the GCD, and using it efficiently reduces your fraction.

• List factors of each number.
• Identify the largest common factor.
• Use this GCD to simplify the fraction.

This step uncovers the simplest form of the fraction, enabling easier arithmetic operations.

Numerator

The numerator is the top part of a fraction, representing the number of parts we are considering. In the fraction \( \frac{35}{50} \), 35 is the numerator. When reducing fractions, we focus on simplifying both the numerator and the denominator by dividing them by their GCD.

In the example of \( \frac{35}{50} \), dividing the numerator 35 by the GCD 5 gives us 7. This step changes the numerator to fit the new form of the fraction, resulting in \( \frac{7}{10} \). It's key to:

• Understand its role as the 'portion' of the fraction.
• See it reduced by dividing by the GCD.
• Maintain its relationship with the denominator.

Hence, the fraction maintains its value but is expressed more simply.

Denominator

The denominator is the bottom number in a fraction, indicating the total number of equal parts. For instance, in the fraction \( \frac{35}{50} \), the denominator is 50. It tells us that 35 parts out of 50 equal parts are being considered.

To simplify a fraction, the denominator must also be divided by the GCD. In our ongoing example, the number 50 divided by the GCD 5 equals 10. This changes the entire denominator, reconfiguring the fraction into \( \frac{7}{10} \), thus:

• Indicating a simpler division of the whole.
• Maintaining proportional consistency with the numerator.
• Aiding in easier comprehension and calculation.

Every part of the operation contributes to a streamlined fraction, making calculations straightforward for learners.