Problem 56 - Algebra 1

Question

Simplify the fraction. $$\frac{27}{108}$$

Step-by-Step Solution

Answer

Upon simplifying the given fraction \(\frac{27}{108}\), the result is \(\frac{1}{4}\).

Step 1: Finding the Greatest Common Divisor

The GCD of 27 and 108 is found by listing all the positive integers that can divide both numbers. For 27: 1, 3, 9, 27. For 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108. The GCD is the highest number that appears in both lists, which in this case is 27.

Step 2: Dividing by Greatest Common Divisor

Once we’ve determined that 27 is the GCD, we can proceed to divide the numerator and denominator of the fraction \(\frac{27}{108}\) by 27. As such; (27 ÷ 27)/(108 ÷ 27) = \(\frac{1}{4}\)

Key Concepts

Greatest Common Divisor

Understanding the Greatest Common Divisor (GCD) is essential in simplifying fractions. It is the largest number that can divide two or more numbers without leaving a remainder. Identifying the GCD involves listing out all the divisors of each number and finding the highest number common to each list.

For instance, with the numbers 27 and 108, the divisors for 27 are 1, 3, 9, and 27. Similarly, for 108 the divisors are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108. The GCD is the largest number that both lists share, which, in this case, is 27. This value is key in fraction reduction because it helps to scale down the fraction to its simplest form.

Fraction Reduction

The process of fraction reduction makes a fraction easier to understand and often easier to work with. After finding the GCD, we're ready to reduce the fraction. This is done by dividing both the numerator (top number) and the denominator (bottom number) by the GCD.

Reducing the fraction \(\frac{27}{108}\) involves dividing both 27 and 108 by their GCD, which is 27. This gives us \(\frac{1}{4}\), a much simpler representation of the original fraction. Fraction reduction is not just a neat arithmetic trick; it is also foundational in algebra, probability, and various real-world applications where simplified ratios are preferably used.

Numerical Division

Numerical division is a basic arithmetic operation where a number, the dividend, is divided by another number, the divisor, to find a quotient and possibly a remainder. In the context of fraction simplification, once the GCD is known, the numerical division is straightforward: divide both the numerator and the denominator by the GCD.

In our example, the division of 27 by 27 yields 1, and 108 divided by 27 equals 4. Numerical division, when applied correctly, renders a simplified fraction that represents the same ratio as the original fraction but in a more manageable form. Mastering numerical division, especially without a calculator, strengthens one's mental math skills and lays a foundation for more complex mathematical concepts.