Problem 9 - Fundamentals of Mathematics

Question

Reduce each fraction to lowest terms. $$\frac{21}{48}$$

Step-by-Step Solution

Answer

The reduced fraction is \(\frac{7}{16}\).

Step 1: Find the Greatest Common Divisor (GCD)

To reduce the fraction \(\frac{21}{48}\) to its lowest terms, we first need to find the greatest common divisor (GCD) of the numerator (21) and the denominator (48). Start by finding the prime factorization of each number: - For 21, the prime factors are 3 and 7: \(21 = 3 \times 7\).- For 48, the prime factors are 2 and 3: \(48 = 2^4 \times 3\).The only common prime factor is 3, hence the GCD is 3.

Step 2: Divide Both the Numerator and the Denominator by the GCD

Now that we know the GCD is 3, divide both the numerator and the denominator of the fraction by the GCD to reduce it:\[ \text{Numerator: } \frac{21}{3} = 7 \]\[ \text{Denominator: } \frac{48}{3} = 16 \]Thus, the fraction \(\frac{21}{48}\) simplifies to \(\frac{7}{16}\).

Step 3: Verify the Resulting Fraction

Ensure that the fraction \(\frac{7}{16}\) is in its simplest form. Check this by confirming that the numerator (7) and denominator (16) have no common factors other than 1:- 7 is a prime number (1 and 7 are its only divisors).- 16 is \(2^4\) and does not share any prime factor with 7.Since no common factors exist between 7 and 16, the fraction is simplified completely.

Key Concepts

Greatest Common Divisor

The concept of the Greatest Common Divisor (GCD) is very useful in reducing fractions to their simplest form. The GCD of two numbers is the largest integer that divides both numbers without leaving any remainder. To find the GCD, we often use the method of prime factorization.
You start by breaking down the numbers into their prime factors. For instance, with the fraction \( \frac{21}{48} \), you would find that:

• 21 is made up of the prime factors 3 and 7, so \( 21 = 3 \times 7 \).
• 48 is made of the prime factors 2 and 3, hence \( 48 = 2^4 \times 3 \).

The next step is to identify the common prime factors between the two numbers. In this case, the only common factor is 3. Therefore, the GCD of 21 and 48 is 3. This GCD is crucial because it tells us by what number we can evenly reduce both the numerator and denominator of the fraction.

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For any fraction, finding the prime factors of the numerator and denominator helps in identifying the greatest common divisor (GCD).

Here's how you can perform prime factorization:

• Start with the smallest prime number, which is 2. Check if the number is divisible by 2. If it is, divide it and repeat this step for the quotient.
• If the number is not divisible by 2, try the next prime number, which is 3, and continue the process.
• Continue this division process with successive prime numbers until the quotient is 1.

Using this method, you can find, for example, that:

• 21 = 3 × 7
• 48 = 2^4 × 3

This makes it easier to figure out the common factors necessary for simplifying a fraction.

Simplest Form

Reducing a fraction to its simplest form means expressing it in such a way that the numerator and the denominator are relatively prime. Numbers are relatively prime when their greatest common divisor (GCD) is 1. This process involves dividing both the numerator and the denominator by their GCD.

For the fraction \( \frac{21}{48} \), after determining that the GCD is 3 from the prime factorizations, you simplify the fraction:

• Divide 21 by 3 to get 7.
• Divide 48 by 3 to get 16.

Thus, the fraction simplifies to \( \frac{7}{16} \). To ensure that \( \frac{7}{16} \) is indeed in its simplest form, check that 7 and 16 share no common divisors other than 1, which they do not since 7 is a prime number and 16 is \( 2^4 \). This confirms that the fraction is fully simplified.