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To simplify a fraction like \(\frac{18r}{27s}\), we first identify the **greatest common divisor (GCD)**. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. For the numbers 18 and 27, their GCD is 9.

We find the GCD by listing out the factors of each number and choosing the largest one they have in common.

• Factors of 18: 1, 2, 3, 6, 9, 18
• Factors of 27: 1, 3, 9, 27

Among these, 9 is the greatest common factor, so we use 9 to simplify our fraction.

Once we have the GCD, the next step is **simplifying the expression**. This involves breaking down the fraction to its simplest form by dividing both the numerator and the denominator by the GCD.

So, for \(\frac{18r}{27s}\), we divide both the numerator (18) and the denominator (27) by 9:

• 18 divided by 9 = 2
• 27 divided by 9 = 3

This turns our fraction into \(\frac{2r}{3s}\). Important note: The variables r and s stay the same as they are, we only divide the coefficients.

The concept of **algebraic fractions** extends the idea of fractions to include variables. In the given example, \(\frac{18r}{27s}\), r and s are variables, and 18 and 27 are coefficients. When simplifying algebraic fractions, follow these steps:

1. Identify the greatest common divisor (GCD) for the coefficients.
2. Divide both the numerator and denominator by the GCD.
3. Re-write the fraction using the simplified coefficients.

In our example, by finding the GCD (9) and dividing both the 18r and 27s by this number, we achieve the simplified expression \(\frac{2r}{3s}\). Remember, this process makes the fraction easier to understand and work with in subsequent algebraic operations.