Problem 95 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{36}{100}$$

Step-by-Step Solution

Answer

The fraction \( \frac{36}{100} \) reduces to \( \frac{9}{25} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify the fraction \( \frac{36}{100} \), we need to find the greatest common divisor (GCD) of the numerator (36) and the denominator (100). The GCD of 36 and 100 is 4.

Step 2: Divide by the GCD

Divide both the numerator and the denominator of the fraction by the GCD (4) to reduce it to the lowest terms. \[\text{Numerator: } \frac{36}{4} = 9\]\[\text{Denominator: } \frac{100}{4} = 25\]So, \( \frac{36}{100} \) simplifies to \( \frac{9}{25} \).

Step 3: Verify the Result

Let’s confirm the fraction cannot be reduced further. The GCD of the new numerator (9) and the new denominator (25) is 1, which means \( \frac{9}{25} \) is already in its simplest form.

Key Concepts

Greatest Common Divisor (GCD)

When simplifying a fraction, finding the greatest common divisor (GCD) is a crucial step. The GCD is the largest positive integer that can divide both numbers without leaving a remainder. For fractions, these numbers are usually the numerator and the denominator.
To find the GCD of two numbers, you can use different methods such as:

• Listing Factors: Write down the factors of each number. The highest common factor is the GCD.
• Prime Factorization: Break down both numbers into their prime factors and multiply the common factors.
• Euclidean Algorithm: An efficient way to find the GCD using division and remainders.

For instance, to simplify \( \frac{36}{100} \), the GCD is 4, as it is the largest number that divides both 36 and 100 evenly.

Reducing Fractions

Once the greatest common divisor (GCD) is identified, reducing the fraction becomes straightforward. Reducing a fraction means making it smaller without changing its value, by dividing both the numerator and denominator by their GCD.
This process is important because it makes the fraction easier to understand and compare with others. A reduced fraction is in its simplest form, representing the proportion with the smallest possible whole numbers.
In our example, to reduce \( \frac{36}{100} \), we divide both the numerator (36) and the denominator (100) by the GCD, which is 4. As a result, the fraction \( \frac{36}{100} \) simplifies to \( \frac{9}{25} \). This is the equivalent but much simpler representation of the same value.

Fraction Simplification

Simplification of fractions is all about expressing them in their simplest form. It's like cleaning up the math to make it look neater and more understandable. **Why simplify fractions?**

• Clarity: A simplified fraction is easier to read and interpret.
• Comparison: It's easier to compare when fractions are in their simplest form.
• Calculation Efficiency: Simplified fractions make calculations quicker and less error-prone.

The fraction \( \frac{9}{25} \) is already in its simplest form because the GCD of 9 and 25 is 1. This means no further reduction is possible. Simplifying helps ensure that the fraction is as straightforward as it can be, assisting in better math understanding and communication.