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Use of Parentheses or Brackets (The Distributive Law)
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Multiplying Decimals
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Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
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Rise and Run
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Writing the Terms of a Polynomial in Descending Order
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Powers of ten (Scientific Notation)
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The Quadratic Formula
Writing a Quadratic with Given Solutions
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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.

 
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