5.2 Decimal Operations

The topics covered in this section are:

1. Add and subtract decimals
2. Multiply decimals
3. Divide decimals
4. Use decimals in money applications

5.2.1 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 TO: Add or subtract decimals.

1. Write the numbers vertically so the decimal points line up.
2. Use zeros as place holders, as needed.
3. Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.

Example 1

Add: $3.7+12.4$.

Example 2

Add: $23.5+41.38$.

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.

Example 3

Subtract: $20-14.65$.

Example 4

Subtract: $2.51-7.4$.

5.2.2 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 the table below. Look for a pattern.

	A	B
	$(0.3)(0.7)$	$(0.2)(0.46)$
Convert to fractions.	$( \frac{3}{10} ) ( \frac{7}{10} )$	$( \frac{2}{10} ) ( \frac{46}{100} )$
Multiply.	$\frac{21}{100}$	$\frac{92}{100}$
Convert back to decimals.	$0.21$	$0.092$

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 $(0.001)(0.0004)$? 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.

MULTIPLYING TWO NUMBERS

When multiplying two numbers,

• if their signs are the same, the product is positive.
• if their signs are different, the product is negative.

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.

HOW TO: Multiply decimal numbers.

1. Determine the sign of the product.
2. Write the numbers in vertical format, lining up the numbers on the right.
3. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
4. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders.
5. Write the product with the appropriate sign.

Example 5

Multiply: $(3.9)(4.075)$.

Example 6

Multiply: $(-8.2)(5.19)$.

In the next example, we’ll need to add several placeholder zeros to properly place the decimal point.

Example 7

Multiply: $(0.03)(0.045)$.

Multiply by Powers of $10$

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?

$1.9436(10) = 19.436$

$1.9436(100) = 194.36$

$1.9436(1000) = 1943.6$

The number of places that the decimal point moved is the same as the number of zeros in the power of ten. The table below 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 $0$ in the hundredths place.

HOW TO: Multiply a decimal by a power of $10$.

1. Move the decimal point to the right the same number of places as the number of zeros in the power of $10$.
2. Write zeros at the end of the number as placeholders if needed.

Example 8

Multiply $5.63$ by factors of…

1. $10$
2. $100$
3. $1000$

5.2.3 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

$(0.2)(4)=0.8$

Remember, a multiplication problem can be rephrased as a division problem. So we can write

$0.8 \div 4=0.2$

We can think of this as “If we divide 8 tenths into four groups, how many are in each group?” The figure below shows that there are four groups of two-tenths in eight-tenths. So $0.8 \div 4 =0.2$.

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.

HOW TO: Divide a decimal by a whole number.

1. Write as long division, placing the decimal point in the quotient above the decimal point in the dividend.
2. Divide as usual.

Example 9

Divide: $0.12 \div 3$.

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).

Example 10

Divide: $ \$ 3.99 \div 24$.

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.

$(0.2)(4)=0.8$

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.

$\large \frac{0.8}{0.2}$

$\large \frac{(0.8)10}{(0.2)10}$

$\large \frac{8}{2}$

$\large 4$

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:

HOW TO: Divide decimal numbers.

1. Determine the sign of the quotient.
2. Make the divisor a whole number by moving the decimal point all the way to the right. Move the decimal point in the dividend the same number of places to the right, writing zeros as needed.
3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
4. Write the quotient with the appropriate sign.

Example 11

Divide: $-2.89 \div (3.4)$

Example 12

Divide: $-25.65 \div (-0.06)$

Now we will divide a whole number by a decimal number.

Example 13

Divide: $4 \div 0.05$

5.2.4 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.

STRATEGY FOR APPLICATIONS

1. Identify what you are asked to find.
2. Write a phrase that gives the information to find it.
3. Translate the phrase to an expression.
4. Simplify the expression.
5. Answer the question with a complete sentence.

Example 14

Paul recieved $ \$50$ for his birthday. He spent $ \$ 31.64$ on a video game. How much of Paul’s birthday money was left?

Example 15

Jessie put 88 gallons of gas in her car. One gallon of gas costs $ \$ 3.529$. How much does Jessie owe for the gas? (Round the answer to the nearest cent.)

Example 16

Four friends went out for dinner. They shared a large pizza and a pitcher of soda. The total cost of their dinner was $ \$ 31.76$. If they divide the cost equally, how much should each friend pay?

Be careful to follow the order of operations in the next example. Remember to multiply before you add.

Example 17

Marla buys $6$ bananas that cost $ \$ 0.22$ each and $4$ oranges that cost $ \$ 0.49$ each. How much is the total cost of the fruit?