Problem 26 - Prealgebra

Question

Divide the numerator and the denominator of each of the following fractions by 2 . $$\frac{106}{142}$$

Step-by-Step Solution

Answer

The fraction \( \frac{106}{142} \) simplifies to \( \frac{53}{71} \).

Step 1: Understanding the Problem

We need to simplify the fraction \( \frac{106}{142} \) by dividing both the numerator and the denominator by their greatest common divisor.

Step 2: Finding the GCD

To simplify \( \frac{106}{142} \), we first find the greatest common divisor (GCD) of 106 and 142. We need to check the common divisors, specifically starting from smaller prime numbers until divisibility is found.

Step 3: Dividing by GCD

The GCD of 106 and 142 is 2. To simplify, divide both the numerator and denominator by 2: \( \frac{106 \div 2}{142 \div 2} = \frac{53}{71} \).

Step 4: Checking Simplicity

Now verify that \( \frac{53}{71} \) is fully simplified. Since 53 and 71 are prime numbers with no common divisors besides 1, the fraction is indeed in its simplest form.

Key Concepts

Greatest Common Divisor (GCD)

When simplifying fractions, identifying the greatest common divisor (GCD) is a key first step. The GCD is the largest number that can divide both the numerator and the denominator of a fraction without leaving a remainder. It allows us to simplify fractions by reducing them to their simplest form.

For example, if we have a fraction \( \frac{106}{142} \), our task is to simplify it. To do this, we must find the GCD of 106 and 142. We check the divisibility of both numbers by smaller prime numbers such as 2, 3, or 5, to identify if they share any common factors.

• Step 1: Start by checking the smallest prime number, which is 2. Since both 106 and 142 are even numbers, they are divisible by 2.
• Step 2: Compute \(106 \div 2 \) and \(142 \div 2 \), resulting in \(\frac{53}{71}\).

Notice here, 2 is found to be the GCD itself. Dividing by the GCD ensures that the fraction is simplified correctly.

Numerator and Denominator

In any fraction, we have two main components: the numerator and the denominator. The numerator is the upper part of the fraction and indicates how many parts are being considered, while the denominator is the lower part and tells us the total number of equal parts in the whole. Understanding their roles is essential in simplifying fractions.

Consider the fraction \( \frac{106}{142} \). Here, **106** is the numerator, representing the part of interest, and **142** is the denominator, representing the total. To simplify this fraction, we divide both parts by their GCD, because:

• It maintains the proportion of the fraction.
• It helps to reveal the simplest form of the fraction.
• It makes it easier to understand or compute further operations.

Dividing each by 2 (which is our GCD found), results in \( \frac{53}{71} \). Always remember, the simplified fraction has the same value as the original one!

Prime Numbers

Prime numbers play a pivotal role in simplifying fractions and finding the greatest common divisor. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In other words, it has only two divisors: 1 and itself.

Relying on prime numbers is crucial when checking if we've reached a fraction's simplest form. For instance, once we simplified \( \frac{106}{142} \) to \( \frac{53}{71} \), the next task was to ensure it's fully simplified. This is confirmed by recognizing that both 53 and 71 are prime numbers.

• Prime numbers cannot be factored further into any smaller natural numbers besides 1 and themselves.
• Since 53 and 71 are primes, they share no common divisors other than 1.

Thus, knowing the concept of prime numbers helps to verify that the fraction \( \frac{53}{71} \) cannot be simplified further, confirming it's in its most reduced form.