# Operations with Decimals

## Objectives

After completing this unit you should be able to:

• Subtract numbers containing decimals.
• Multiply numbers containing decimals.
• Divide numbers containing decimals.

## Working with Decimals

In this unit we will give a review of some basic operations with decimals. Performing these operations with decimals is like doing them with whole numbers. The difference is the way that you need to keep track of the decimal point. In each section of this unit there is an example that should help remind you of this.

Addition and subtraction of decimals is like adding and subtracting whole numbers. The only thing we must remember is to line up the place values correctly. The easiest way to do that is to line up the decimal points.

In this section we will provide a few examples to remind you of the procedure for adding and subtracting decimals. Look at the examples below, and then read through the detailed examples.

### Example

 Here is an example of adding 12.35 and 5.287. Notice how the decimal points are lined up. Here is an example of subtracting 2.28 from 12.993. Notice how the decimal points are lined up.

Add the following numbers 1.19 and .16

The answer to this is: 1.19 + .16 = 1.35.

 First line the numbers up in a column, lining up the decimal points. Add down the columns, starting at the right. Notice that 9 + 6 = 15, so we need to carry a 1 to the tenths column. Continue to add down the columns, moving from right to left.

### Detailed Example of Subtraction

Subtract 1.387 from 12.17.
The answer to this is: 12.17 - 1.387 = 10.783
 First line the numbers up in a column, lining up the decimal points. Since the number 1.387 has three digits after the decimal point, we add a zero on the end of 12.17 so it also has three digits showing past the decimal point. Subtract down the columns, starting at the right. Notice that in the thousandths column 7 > 0. We must "borrow" from the hundredths column. When we move to the hundredths column, we notice that 8 > 6. We must "borrow" from the tenths column. Continue to subtract down the columns, moving from right to left. Again, we need to borrow from the ones place to be able to subtract the tenths.

## Multiplication of Decimals

When multiplying numbers with decimals, we first multiply them as if they were whole numbers. Then, the placement of the number of decimal places in the result is equal to the sum of the number of decimal places of the numbers being multiplied.

 For example, if we multiply 2.3 times 4.5, each number has one digit to the right of the decimal, so each has one decimal place. When they are multiplied, the result will have two digits to the right of the decimal, or two decimal places. Now let's look at a detailed example of multiplying two number with decimals.

### Detailed Example of Multiplication

Multiply 12.3 by 3.54.

 When multiplying multi-digit numbers, we start with the digit at the furthest right of the bottom number and multiply that by each of the digits of the top number. Multiply the second digit from the right of the bottom number by the top. Notice how this second number is offset below the first. Do this for each digit of the bottom number, moving from right to left. Continue doing this for each digit of the bottom number, moving from right to left. Remember to offset each number as you multiply. Now add the numbers up going down the columns. To determine how many digits are to the right of the decimal point in the result, we count the decimal places in the two numbers being multiplied and add these together.

## Division of Decimals

Division with decimals is easier to understand if the divisor (the dividend is divided by the divisor) is a whole number.

 If the divisor has a decimal in it, we can make it a whole number by moving the decimal point the appropriate number of places to the right. However, if you shift the decimal point to the right in the divisor, you must also do this for the dividend. Once you have moved the decimal point so the divisor is a whole number, you can do the division.

Now let's look at a detailed example of dividing with decimals.

### Detailed Example of Division

Divide 0.2334 by .004