By the end of this section, you will be able to:
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 $20 bill for a $14.65 purchase? We will show the steps to calculate this in the next example.
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 Table 5.3. Look for a pattern.
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.
In many fields, especially in the sciences, it is common to multiply decimals by powers of 10. Let’s see what happens when we multiply 1.9436 by some powers of 10.
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. Table 5.4 summarizes the results.
|Multiply by||Number of zeros||Number of places decimal point moves|
|10||1||1 place to the right|
|100||2||2 places to the right|
|1,000||3||3 places to the right|
|10,000||4||4 places to the right|
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 10 and then move the decimal point that same of places to the right.
So, for example, to multiply 45.86 by 100, move the decimal point 2 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 2.4 by 100. We need to move the decimal point 2 places to the right. Since there is only one digit to the right of the decimal point, we must write a 00 in the hundredths place.
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?” Figure 5.5 shows that there are four groups of two-tenths in eight-tenths. So 0.8÷4=0.2.
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 24 water bottles cost $3.99. To find the price per water bottle, we would divide $3.99 by 24, and round the answer to the nearest cent (hundredth).
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 0.2 go into 0.8?” Because (0.2)(4)=0.8, we can say that 0.2 goes into 0.8 four times. This means that 0.8 divided by 0.2 is 4.
We would get the same answer, 4, if we divide 8 by 2, both whole numbers. Why is this so? Let’s think about the division problem as a fraction.
We multiplied the numerator and denominator by 10 and ended up just dividing 8 by 2. To divide decimals, we multiply both the numerator and denominator by the same power of 10 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:
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.
Add and Subtract Decimals
In the following exercises, add or subtract.
In the following exercises, multiply.
In the following exercises, divide.
In the following exercises, simplify.
In the following exercises, use the strategy for applications to solve.
ⓑ If she had worked all 23 hours as a tutor instead of working both jobs, how much more would she have earned?
ⓑ If he had worked all 17 hours at the art gallery instead of working both jobs, how much more would he have earned?
ⓐ 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?