Fraction Multiplication and Division

We will use a “zoomed-in” view of the grid to study fractional multiplication and division.

The fractional grid

Notice that the grid to the left now has dashed lines in addition to solid lines. The dashed lines mark fractional amounts. For example, if every third horizontal line is solid, every dashed horizontal line counts the fraction $$1/3$$.

How tall are each of these rectangles?
How wide are each of these rectangles?

Multiplying fractions

The green rectangle you see to the left covers exactly one unit square, so it represents the number $1$.

How many pieces do the dashed lines divide the green rectangle into?
What fraction does each of these pieces represent?

Multiplying two fractions on the grid is exactly like multiplying two whole numbers: it means finding an area. The only difference is that the area may involve fractional pieces of grid squares. You can count those fractional pieces because they’re set apart by dashed lines.

For example, the rectangle currently shown on the grid has width $$5/2$$ and height $$4/3$$, so it gives a picture for the multiplication $$5/2(4/3)$$.

What is the area of each colored rectangle? (Remember that the area will be positive if the rectangle is green or negative if it is pink.)
What is the area of each colored rectangle? (Remember that the product of two negative numbers is positive.)

So, if you want to multiply two fractions:

  • The denominator of the product can be found by multiplying the denominators of the original fractions.
  • The numerator of the product can be found by multiplying the numerators of the original fractions.

Dividing fractions

When dividing two fractions, it’s useful to think of division problems as asking you to find the height of a rectangle, when you already know its area and the width of its base. Unlike in the rest of the course, we’ll use the division sign (÷) in this lesson.

So $${3/4} ÷ {1/2}$$ should be read as “$$3/4$$ divided by $$1/2$$.” As before, you can picture this division problem as a rectangle with area $$3/4$$, and base width $$1/2$$.

For each row of the table to the right, type in a division problem that makes the bottom grid match the top grid. (Remember that a negative number divided by a positive number is negative.)
Use the heights of the colored rectangles on the grid to the left to solve these division problems. (Remember that a negative number divided by a negative number is positive, and a positive number divided by a negative number is negative.)

Division and equivalent fractions

In order to divide fractions in the way you did in the last two problems, there need to be enough fractional (dashed) grid lines on the grid. Sometimes the fractions you start with won’t line up perfectly with the dashed lines, and then you’ll need to find equivalent fractions that do.

In the next question, we’ll be looking at the division problem $${2/1} ÷ {3/2}$$. Notice that the top line of the rectangle currently shown for that problem does not match up with the dashed lines on the grid.

Each row of the table to the right corresponds to a different number of fractional (dashed) grid lines on the grid. In each row, find an equivalent fraction to $$2/1$$ that has the requested denominator. Look at the grid to the left and figure out if that fraction splits the grid into pieces that let you find the height of the rectangle by counting.
Division problem Does it split the grid up correctly? (yes or no)
Which of these denominators splits the grid up in the right way?
Click to look at the picture for that denominator again. What is the height of the green rectangle?

You probably learned to solve division problems with fractions by cross-multiplying. For example, you could divide two fractions like this: $${2/3} ÷ {5/4}={2(4)} / {3(5)}=8/15$$. You can think of cross-multiplying as a way of making sure the grid is always split up into small enough pieces.

Solve each of these problems by cross-multiplying. (Write your final answer in lowest terms.)