MULTIPLYING and DIVIDING DECIMALS

Multiplying Decimal Numbers

If we know our multiplication tables, this is the easiest math lesson ever. To multiply decimal numbers together, we ignore the decimal points until we've found the product of the numbers. Then we insert the decimal in the right place in the product. Here are some examples:

Notice how we just find the product of the numbers, then we count how many decimal places there are in the question and put that many places in the answer. In the first example, we needed one decimal place from the 2.6. In the 2nd, we used 2 decimal places from the 0.26. In the 3rd example, we used 2 decimal places one from each number (1.2 and 2.6).

 To multiply decimal numbers, multiply the numbers ignoring the decimals, count the number of decimal places in all parts of the question put the decimal in the product in the proper place.

Why Does It Work?

Let's look at these numbers as fractions. The first example was 12 × 2.6.

Written as a product of fractions it is

The 2nd example was 12 × 0.26.

Written as a product of fractions it is

The 3rd example was 1.2 × 2.6.

Written as a product of fractions it is

Sometimes, there aren't enough digits in the product to put in the right number of decimal places. In such a case, we add zeros (0's) at the left end of the product until we can insert the decimal point where it should be.

Example

There are only 3 digits in the product, but we need 4 numbers after the decimal, so we insert zeros to fill the ones and tenths columns. Now we can put 4 decimal places in the product and we have a zero to hold the 1's place.

Written as a product of fractions it is

.

Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.

Practice Exercise 1: Multiplying Decimals

1) Find these products. Put the correct number of decimal places in the answer.

 a) b) c) d)

.

Dividing Decimal Numbers

Division questions with decimals come in 2 categories:

1) Whole Number Divisor -- Decimals in the Dividend

Example:

Reminder: sometimes we need to put zeros (0's) in the quotient to hold a place value column.
Here's an example:

2) Decimals in the Divisor

To do these, we use our knowledge of equivalent fractions. Since fractions are just another way to express division, let's write a division question as a fraction and investigate the method.

We know that means exactly the same thing as 6.219 ÷ 0.09.

We also know that we can multiply any fraction by 1 without changing its value. Our goal is to make the denominator a whole number so that we can use the approach we learned before.

In this example, we want to turn 0.09 into 9, so we multiply by 100 to move the decimal 2 places. Since 100 / 100 = 1, we multiply BOTH terms by 100 to get 621.9 ÷ 9. This quotient will be exactly the same as 6.219 ÷ 0.09.

 To divide by a decimal in the divisor, multiply both divisor and dividend by the power of 10 that turns the divisor into a WHOLE NUMBER. Then divide as usual.

To find 2.65 ÷ 0.053, we will multiply both numbers by 1000 to move the divisor's decimal point 3 places to the right. When we do, we get 2650 ÷ 53. Let's do this one as a last example.

Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.

Practice Exercise 2: Dividing Decimals

1) Rewrite each question as an equivalent fraction with a whole number divisor,
then use short division to find the quotient.

 1) 2) 3) 4)

2) Estimate then find the quotient:

 a) 97.2 ÷ 5.4 = b) 12.22 ÷ 2.6 = c) 22.26 ÷ 5.3 = d) 10.8 ÷ 0.72 =

Solutions

Practice Exercise 1: Multiplying Decimals

1) Find these products. Put the correct number of decimal places in the answer.

 a) b) c) d)

.

Practice Exercise 2: Dividing Decimals

1) Rewrite each question as an equivalent fraction with a whole number divisor,
then use short division to find the quotient.

 1) 2) 3) 4)

2) Estimate then find the quotient:

 a) 97.2 ÷ 5.4 =est: 100 ÷ 5 = 20972 ÷ 54 = 18 b) 12.22 ÷ 2.6 =est: 12 ÷ 3 = 4122.2 ÷ 26 = 4.7 c) 22.26 ÷ 5.3 =est: 20 ÷ 5 = 4222.6 ÷ 53 = 4.2 d) 10.8 ÷ 0.72 =est: 10 ÷ 1 = 101080 ÷ 72 = 15