Fractions represent parts of a whole. For example, 18 over 36 means 18 parts out of 36 total parts. We can simplify fractions to make them easier to work with.
To simplify a fraction, we need to find common factors of the numerator and denominator. Let's list the factors of 18 and 36. The factors of 18 are 1 times 18, 2 times 9, and 3 times 6. The factors of 36 are 1 times 36, 2 times 18, 3 times 12, 4 times 9, and 6 times 6. The common factors are 1, 2, 3, 6, 9, and 18.
The Greatest Common Factor (GCF) is the largest number that divides both the numerator and denominator. Looking at our lists, the common factors are 1, 2, 3, 6, 9, and 18. The greatest of these is 18, so the GCF of 18 and 36 is 18.
We simplify the fraction by dividing both numerator and denominator by their GCF. So we have 18 over 36 divided by 18. This gives us 18 divided by 18 over 36 divided by 18, which equals 1 over 2.
We can also simplify in multiple steps by dividing by smaller common factors. First, divide both 18 and 36 by 2 to get 9 over 18. Then divide both 9 and 18 by 3 to get 3 over 6. Finally, divide both 3 and 6 by 3 to get 1 over 2. Both methods lead to the same result!
Let's verify that 18 over 36 and 1 over 2 represent the same value visually. In the first diagram, we've shaded 18 out of 36 parts. In the second diagram, we've shaded 1 out of 2 parts. As you can see, both shaded areas represent exactly half of the whole, confirming that our simplification is correct.
Let's work through a few more examples to reinforce the simplification process. For 12 over 24, the GCF is 12, so we get 1 over 2. For 15 over 45, the GCF is 15, so we get 1 over 3. For 20 over 50, the GCF is 10, so we get 2 over 5. All of these fractions have been successfully simplified.