1Simplifying means dividing the numerator and denominator by the same number
Divide a fraction's numerator and denominator by the same number, and the numbers get smaller but the size the fraction represents never changes at all. Turning a fraction into a simpler one this way is called simplifying.
Dividing both 8 and 12 by 4 gives 2/3. The shaded proportion in the bar diagram stays exactly the same.
2Dividing by the GCD gets you to simplest form in one step
You can divide the numerator and denominator down bit by bit, but dividing both by their greatest common divisor at once gets you straight to the simplest form that can't be divided any further. If the GCD is 1, the fraction is already in simplest form.
3Finding a common denominator means matching the denominators
Fractions with different denominators are hard to compare directly. So we multiply the numerator and denominator by the same number to make two fractions' denominators match. This process is called finding a common denominator.
Using 12, the LCM of 3 and 4, as the common denominator lets you compare the two fractions on the same footing.
4Matching to the LCM keeps the numbers smallest
A common denominator can actually be any common multiple of the two denominators. But matching to the LCM keeps the numbers as small as possible, making the calculation much easier.
Simplifying makes the denominator smaller, and finding a common denominator makes it bigger as needed. But both do the same thing at heart: multiplying or dividing the numerator and denominator by "the same number."
5Try this
- Keep changing the denominator with the slider. Check how the on-screen explanation differs when the GCD is 1 versus when it isn't.
- Compare the bars before and after finding a common denominator. See how two fractions that were hard to compare with different denominators become easy to compare at a glance once they share one.
- Try solving the quiz in your head first. Simplify or find a common denominator yourself before looking at the screen, then compare with the correct answer.