Problem 9 - Prealgebra

Question

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. $$\frac{12 c}{15 d}$$

Step-by-Step Solution

Answer

\( \frac{4c}{5d} \)

Step 1: Identify the Greatest Common Divisor (GCD)

First, find the greatest common divisor of the numerator and denominator, which are 12 and 15. List the factors of 12: 1, 2, 3, 4, 6, 12 and the factors of 15: 1, 3, 5, 15. The greatest common factor is 3.

Step 2: Simplify the Fraction

Divide both the numerator and the denominator by their GCD. Divide 12 by 3 to get 4, and divide 15 by 3 to get 5. The fraction simplifies to \( \frac{4c}{5d} \).

Key Concepts

Greatest Common Divisor

The greatest common divisor, often abbreviated as GCD, is a key concept when simplifying fractions. It refers to the largest number that can exactly divide both the numerator and the denominator without leaving a remainder. Identifying the GCD is the first crucial step in simplification. To find the GCD of two numbers:

• List all the factors of each number. Factors are numbers that can be multiplied together to give the original number.
• Compare the lists to find the largest factor that appears in both.

In our example, for the numbers 12 and 15, the greatest common divisor is 3. Recognizing the GCD helps in breaking down the fraction into smaller, exactly divisible parts.

Numerator and Denominator

Fractions consist of two main parts: the numerator and the denominator. The numerator is the top part of the fraction and indicates how many parts of the whole are being considered. The denominator is the bottom part, showing the total number of equal parts in the whole.Understanding these terms is essential for simplifying fractions:

• The numerator (12 in the original fraction \( \frac{12c}{15d} \)) tells us how many parts we have.
• The denominator (15 in the original fraction) tells us how many parts make up a whole.

By dividing both the numerator and denominator by their GCD, we can simplify the fraction without changing its overall value.

Simplest Form of a Fraction

Reaching the simplest form of a fraction means reducing it so that the numerator and denominator are as small as possible, yet still representative of the same value. After identifying the GCD, each part of the fraction is divided by this number.In the example of \( \frac{12c}{15d} \):

• Divide the numerator by the GCD: \( \frac{12}{3} = 4 \).
• Divide the denominator by the GCD: \( \frac{15}{3} = 5 \).

After division, the fraction \( \frac{12c}{15d} \) simplifies to \( \frac{4c}{5d} \). This form is considered simplest because no whole number other than 1 can further divide both the numerator and the denominator.