# Decimals

## Decimal Operations


### Add and Subtract Decimals

Let’s take one more look at the lunch order from the start of Decimals, this time noticing how the numbers were added together.
All three items (sandwich, water, tax) were priced in dollars and cents, so we lined up the dollars under the dollars and the cents under the cents, with the decimal points lined up between them. Then we just added each column, as if we were adding whole numbers. By lining up decimals this way, we can add or subtract the corresponding place values just as we did with whole numbers.
How much change would you get if you handed the cashier a  bill for a  purchase? We will show the steps to calculate this in the next example.

### Multiply Decimals

Multiplying decimals is very much like multiplying whole numbers—we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first review multiplying fractions.
Do you remember how to multiply fractions? To multiply fractions, you multiply the numerators and then multiply the denominators.
So let’s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side in . Look for a pattern.
There is a pattern that we can use. In A, we multiplied two numbers that each had one decimal place, and the product had two decimal places. In B, we multiplied a number with one decimal place by a number with two decimal places, and the product had three decimal places.
How many decimal places would you expect for the product of  If you said “five”, you recognized the pattern. When we multiply two numbers with decimals, we count all the decimal places in the factors—in this case two plus three—to get the number of decimal places in the product—in this case five.
Once we know how to determine the number of digits after the decimal point, we can multiply decimal numbers without converting them to fractions first. The number of decimal places in the product is the sum of the number of decimal places in the factors.
The rules for multiplying positive and negative numbers apply to decimals, too, of course.
When you multiply signed decimals, first determine the sign of the product and then multiply as if the numbers were both positive. Finally, write the product with the appropriate sign.
In the next example, we’ll need to add several placeholder zeros to properly place the decimal point.

### Multiply by Powers of 

In many fields, especially in the sciences, it is common to multiply decimals by powers of  Let’s see what happens when we multiply  by some powers of 
Look at the results without the final zeros. Do you notice a pattern?



The number of places that the decimal point moved is the same as the number of zeros in the power of ten.  summarizes the results.
We can use this pattern as a shortcut to multiply by powers of ten instead of multiplying using the vertical format. We can count the zeros in the power of  and then move the decimal point that same of places to the right.
So, for example, to multiply  by  move the decimal point  places to the right.
Sometimes when we need to move the decimal point, there are not enough decimal places. In that case, we use zeros as placeholders. For example, let’s multiply  by  We need to move the decimal point  places to the right. Since there is only one digit to the right of the decimal point, we must write a  in the hundredths place.

### Divide Decimals

Just as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed.
To understand decimal division, let’s consider the multiplication problem
Remember, a multiplication problem can be rephrased as a division problem. So we can write
We can think of this as “If we divide 8 tenths into four groups, how many are in each group?”  shows that there are four groups of two-tenths in eight-tenths. So 
Using long division notation, we would write
Notice that the decimal point in the quotient is directly above the decimal point in the dividend.
To divide a decimal by a whole number, we place the decimal point in the quotient above the decimal point in the dividend and then divide as usual. Sometimes we need to use extra zeros at the end of the dividend to keep dividing until there is no remainder.
In everyday life, we divide whole numbers into decimals—money—to find the price of one item. For example, suppose a case of  water bottles cost  To find the price per water bottle, we would divide  by  and round the answer to the nearest cent (hundredth).

### Divide a Decimal by Another Decimal

So far, we have divided a decimal by a whole number. What happens when we divide a decimal by another decimal? Let’s look at the same multiplication problem we looked at earlier, but in a different way.
Remember, again, that a multiplication problem can be rephrased as a division problem. This time we ask, “How many times does  go into  Because  we can say that  goes into  four times. This means that  divided by  is 
We would get the same answer,  if we divide  by  both whole numbers. Why is this so? Let’s think about the division problem as a fraction.
We multiplied the numerator and denominator by  and ended up just dividing  by  To divide decimals, we multiply both the numerator and denominator by the same power of  to make the denominator a whole number. Because of the Equivalent Fractions Property, we haven’t changed the value of the fraction. The effect is to move the decimal points in the numerator and denominator the same number of places to the right. 
We use the rules for dividing positive and negative numbers with decimals, too. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.
It may help to review the vocabulary for division:
Now we will divide a whole number by a decimal number.

### Use Decimals in Money Applications

We often apply decimals in real life, and most of the applications involving money. The Strategy for Applications we used in The Language of Algebra gives us a plan to follow to help find the answer. Take a moment to review that strategy now.
Be careful to follow the order of operations in the next example. Remember to multiply before you add.

### Key Concepts

1. Add or subtract decimals.

2. Multiply decimal numbers.

3. Multiply a decimal by a power of 10.

4. Divide a decimal by a whole number.

5. Divide decimal numbers.

6. Strategy for Applications


### Practice Makes Perfect

Add and Subtract Decimals 
In the following exercises, add or subtract.
Multiply Decimals 
In the following exercises, multiply.
Divide Decimals 
In the following exercises, divide.
Mixed Practice 
In the following exercises, simplify.
Use Decimals in Money Applications 
In the following exercises, use the strategy for applications to solve.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?
