Examine the problem 2/3 x 3/4. These are common fractions (fractions that have integers for both numerator and denominator). The problem asks, "What is 2/3 of 3/4 of a group?". Using the simple rules for multiplying fractions, you know the answer is 6/12, or 1/2.
Draw a circle on the paper. Draw a cross-hair over the circle, segmenting it into four quadrants. These are the four parts of the whole unit. Think of it as a pizza cut into four slices.
Shade in one of the four quadrants with a very light color. This represents one part of the whole unit, which ultimately has four parts. Think of it as removing one of the slices from the pizza, which leaves you with three-fourths of a pizza.
Draw two lines in one of the quadrants, dividing the fourth into thirds. Use a different-colored ink for these lines so that you can see your work.
Repeat the process, dividing each of the quadrants into thirds. By the time you are done, you should have nine mini-slices of pizza.
Examine one of the quadrants. In it, you should have three mini-slices before you. Color in two of those mini slices with a different-colored ink from the one used before. You now have a visual reference for the two-thirds part of the problem.
Repeat the process for the other quadrants. You should have six mini-slices shaded and three mini-slices left blank. The shaded parts show 2/3 of 3/4. However, they do not represent the whole pizza. To do this, you must divide all four quadrants into thirds.
Draw two more lines over the lightly shaded quadrant. You should have three mini-slices in this quadrant.
Count all of the mini-slices in the pizza. There should be 12, which represents the whole pizza. Write the 12 slices in the denominator and the number of shaded parts as the numerator: 6/12. Therefore, 2/3 of 3/4 = 6/12.
Reduce the solution "6/12" to its simplest form. One way to do this is to understand that both parts of the fraction are multiples of 6 and therefore can be reduced by dividing each part by 6 and writing the remainders as a fraction: 6/12 = 1/2. There is another way, however.
Factor the fraction 6/12 to prime numbers. Keep them in fraction form, however, to see how this process works. You should have 2 x 3 over 2 x 2 x 3.
Cancel out the factors that are the same. A simple way to do this is to cross out the factors that are the same in both parts. Simplify the fraction by multiplying the integers in the numerator: 1 x 1 = 1 and by multiplying the integers in the denominator: 1 x 2 x 1 = 2. Therefore, 6/12 reduces to 1/2.