Here are three other ways to calculate a product of two decimals, such as .

• First, we can multiply each decimal by the same power of 10 to obtain whole-number factors.

  Because we multiplied both 0.04 and 0.07 by 100 to get 4 and 7, the product 28 is times the original product, so we need to divide 28 by 10,000.

• Second, we can write each decimal as a fraction and multiply them.

• Third, we can use an area diagram. The product can be thought of as the area of a rectangle with side lengths of 0.04 unit and 0.07 unit.

  In this diagram, each small square is 0.01 unit by 0.01 unit. The area of each square, in square units, is therefore , which is .

  

  Because the rectangle is composed of 28 small squares, the area of the rectangle, in square units, must be:

All three calculations show that .

To find the product of two two-digit numbers, such as , we can think of finding the area of a rectangle with those numbers, 3.4 units and 1.2 units, as side lengths.

Then, we decompose the rectangle into four smaller sub-rectangles and find their areas.

Area diagram. A rectangle partitioned into 4 rectangles, A, B, C, D. D, vertical side, 1, horizontal side, 3. C, vertical side, 1, horizontal side, 0 point 4. B, vertical side, 0 point 2, horizontal side, 3. A, vertical side, 0 point 2, horizontal side, 0 point 4.

A:

B:

C:

D:

Each multiplication gives a partial product that represents the area of a sub-rectangle. The sum of the four partial products gives the area of the entire rectangle, 4.08 square units.

We can show the same partial-product calculations vertically. Here are two ways:

Two vertical calculations of 3 point 4 times 1 point 2. First calculation, 7 rows. First row: 3 point 4. Second row: multiplication symbol, 1 point 2. Horizontal line. Third row: 0 point 0 8, A. Fourth row: 0 point 6, B. Fifth row: 0 point 4, C. Sixth row: plus 3, D. Horizontal line. Seventh row: 4 point 0 8. Second calculation, 5 rows. First row: 3 point 4. Second row: multiplication symbol, 1 point 2. Horizontal line. Third row: 0 point 6 8, A plus B. Fourth row: plus 3 point 4, C plus D. Horizontal line. Fifth row: 4 point 0 8.

The calculation on the left shows four partial products, one for the area of each sub-rectangle.

The calculation on the right shows two partial products:

• 0.68 is the value of , or the combined area of A and B.
• 3.4 is the value of , or the combined area of C and D.

In both calculations, adding the partial products gives a total of 4.08, which is the product of and the area (in square units) of the entire rectangle.