Multiplying Decimals
Objective Learn to apply the standard
algorithm for multiplication of whole numbers to multiplying
decimals.
When multiplying decimals, the algorithm for multiplying whole
numbers can be used. One then only needs to determine where the
decimal point should be placed in the result.
The Algorithm
Take a look at the algorithm for multiplying two decimals.
Multiplication Algorithm for Decimals
1. First multiply the decimals as if they were whole numbers,
without regard to the decimal points.
2. Determine the number of digits to the right of the decimal
point in each of the decimals, and add these two numbers
together.
3. The sum in Step 2 will be the number of digits to the right
of the decimal point in the answer. Place the decimal point in
the answer accordingly.
Example 1
Multiply 2.3 and 1.11.
Solution
First multiply the numbers as if they were whole numbers,
without considering the decimal points.
| 1.11 |
| × 2.3 |
| 3 3 3 |
| 2 2 2 |
| 2 5 5 3 |
Look at the decimals 1.11 and 2.3. The decimal 1.11 has two
digits to the right of its decimal point, and 2.3 has one digit
to the right of its decimal point. Together, there are 2 + 1 or 3
digits to the right of the decimal points. So, in the answer,
there should be 3 digits to the right of the decimal point. This
means the decimal point should be placed between the 2 and the
first 5, making the answer 2.553.
Example 2
Multiply 3.21 × 0.02.
Solution
Ignoring the decimal points leads to multiplying 321 × 2,
which gives 642. Since 3.21 and 0.02 each have two digits to the
right of the decimal point, their product will have four digits
to the right of the decimal point.
3.21 × 0.02 = 0.0642
Complete several decimal multiplication problems using this
algorithm. Pay special attention to the placement of the decimal
point in the product, since this is the new skill.
Estimation
Just as when multiplying whole numbers, estimation should be
used to check the reasonableness of the product of two decimals.
This check can help students catch any possible mistake in the
placement of the decimal point in their answer.
Example 3
Multiply 3.7 and 4.2. Use estimation to check the
reasonableness of your answer.
Solution
First, carry out the multiplication.
| 4.2 |
| × 3.7 |
| 2 9 4 |
| 1 2 6 |
| 1 5 5 4 |
The decimals 3.7 and 4.2 each has one digit to the right of
its decimal point. So the decimal point in the answer should be
placed between the two 5’s, making the answer 15.54. To
check the reasonableness of this result, round 4.2 to 4 and round
3.7 to 4. The answer should be approximately 4 × 4 or 16. Since
15.54 is close to 16, this answer is reasonable. In addition to
checking the reasonableness of a product, estimation can be used
to actually determine the correct placement of the decimal point
in the product of two decimals.
Example 4
Multiply 4.03 × 3.04.
Solution
Multiply the numbers without regard to the decimal points.
| 4.03 |
| × 3.04 |
| 1 6 1 2 |
| |
0 0 0 |
| 1 2 0 9 |
| 1 2 2 5 1 2 |
To help determine where the decimal point should be in the
answer, notice that 4.03 is about 4 and 3.04 is about 3. So the
answer should be close to 4 × 3 or 12. In order for the answer
to be close to 12, the decimal point needs to be placed between
the second and third digits from the left, so the answer is
12.2512. Notice that if the decimal point had been written one
place further to the right, the answer would have been 122.512,
which is much too large; and if the decimal point had been
written one place further to the left, the answer would have been
1.22512, which is too small.
|
