Owen likes to help his mom cut out coupons from the paper on Sunday mornings. This week he found 5 coupons that will save them $4.10 on their purchases at the grocery store. Once Owen and his mom get to the grocery store, they fill up their cart with everything on their list. At the cash register, their total is $85.91 before Owen hands over the coupons. How could Owen determine what the new total is going to be after the coupons are applied?

In this concept, you will learn how to subtract decimals with and without rounding.

### Subtracting Decimals

Recall that in a decimal number, the **decimal point** divides the whole part of the number from the fractional part of the number.

To subtract decimal numbers:

- Arrange the numbers vertically so that the decimal points line up.
- Add zeros for any missing spaces so that each number has the same number of digits to the right of the decimal point.
- Subtract from right to left, borrowing as necessary just as if you were subtracting whole numbers.
- Insert a decimal point into your answer so that it lines up with the decimal points of the original numbers.

Here is an example.

Subtract 4.5 from 18.98.

First, notice that you are going to find 18.98 minus 4.5.

Now, arrange the numbers vertically so that the decimal points line up.

\begin{align*}& \ \ 18.98\\ & \underline{- \ 4.5}\end{align*}

Next, add a zero after the 5 in the second number so that both numbers have two digits to the right of the decimal point.

\begin{align*}& \ \ 18.98\\ & \underline{- \ 4.50}\end{align*}

Now, subtract from right to left. Start by subtracting 8 minus 0 to make 8. Then 9 minus 5 to make 4. Then 18 minus 4 to make 14. Insert a decimal point into your answer directly under the decimal points in the original numbers.

The answer is

.Sometimes you will want to round numbers before subtracting them. Rounding is useful when you only need an approximate answer instead of an exact answer. Read the problem carefully to determine if you should round before or after you subtract.

Here is an example.

Round each number to the nearest tenth and then find the difference.

First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point. Look at each digit in the tenths place and then the digit to the right of that to decide if you will need to round up the digit in the tenths place or keep it the same.

- 72.953 rounds to 73.0
- 52.418 rounds to 52.4

Now, subtract the rounded numbers. Start by arranging the numbers vertically so that the decimal points line up.

Next, subtract from right to left. Borrow from the 3 and then subtract 10 minus 4 to make 6. Then subtract 2 minus 2 to make 0. Finally subtract 7 minus 5 to make 2. Insert a decimal point into your answer directly under the decimal points in the original numbers.

\begin{align*}& \ \ \ \ 73.0\\ & \underline{- \ 52.4} \\ & \ \ \ \ 20.6\end{align*}

The answer is

.### Examples

#### Example 1

Earlier, you were given a problem about Owen, who likes clipping coupons with his mom.

This week Owen found $4.10 worth of coupons. At the store, their items cost $85.91 before the coupons are applied. Owen wants to determine what their total will be after he gives the coupons to the cashier.

First, Owen needs to realize that he will need to subtract 85.91 minus 4.10.

Then, he should arrange the numbers vertically so that the decimal points line up.

Now, he should subtract from right to left. First subtract 1 minus 0 to make 1. Then 9 minus 1 to make 8. Then 85 minus 4 to make 81. He needs to insert a decimal point into his answer directly under the decimal points in the original numbers.

\begin{align*}& \ \ 85.91\\ & \underline{- \ 4.10} \\ & \ \ 81.81\end{align*}

The answer is \begin{align*}85.91-4.10=81.81\end{align*}. The total will be $81.81 after the coupons are applied.

#### Example 2

Subtract

.First, arrange the numbers vertically so that the decimal points line up.

\begin{align*}& \ \ \ \ 4.56\\ & \underline{- \ 2.37} \end{align*}

Now, subtract from right to left. Start by borrowing from the 5 and then subtracting 16 minus 7 to make 9. Then subtract 4 minus 3 to make 1. Finally subtract 4 minus 2 to make 2. Insert a decimal point into your answer directly under the decimal points in the original numbers.

The answer is

.#### Example 3

Subtract \begin{align*}5.674-2.5\end{align*}.

First, arrange the numbers vertically so that the decimal points line up.

\begin{align*}& \ \ \ \ 5.674 \\ & \underline{- \ 2.5 \;\;\;\;\;}\end{align*}

Next, add two zeros after the 5 in the second number so that both numbers have three digits to the right of the decimal point.

\begin{align*}& \ \ \ \ 5.674 \\ & \underline{- \ 2.500}\end{align*}

Now, subtract from right to left. Start by subtracting 4 minus 0 to make 4. Then 7 minus 0 to make 7. Then 6 minus 5 to make 1. Finally 5 minus 2 to make 3. Insert a decimal point into your answer directly under the decimal points in the original numbers.

\begin{align*}& \ \ \ \ 5.674 \\ & \underline{- \ 2.500}\\ & \ \ \ \ 3.174\end{align*}

The answer is

.#### Example 4

Take 5.67 from 12.378.

First, notice that you are going to find 12.378 minus 5.67.

Now, arrange the numbers vertically so that the decimal points line up.

\begin{align*}& \ \ 12.378\\ & \underline{- \ 5.67 \;\;}\end{align*}

Next, add a zero after the 7 in the second number so that both numbers have three digits to the right of the decimal point.

\begin{align*}& \ \ 12.378\\ & \underline{- \ 5.670}\end{align*}

Now, subtract from right to left. Start by subtracting 8 minus 0 to make 8. Then 7 minus 7 to make 0. Then borrow from the 2 and subtract 13 minus 6 to make 7. Finally subtract 11 minus 5 to make 6. Insert a decimal point into your answer directly under the decimal points in the original numbers.

\begin{align*}& \ \ 12.378\\ & \underline{- \ 5.670}\\ & \ \ \ \ 6.708\end{align*}

The answer is

.#### Example 5

Round to the nearest tenth and then subtract:

.First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point. Look at each digit in the tenths place and then the digit to the right of that to decide if you will need to round up the digit in the tenths place or keep it the same.

- 83.56 rounds to 83.6
- 1.258 rounds to 1.3

Now, subtract the rounded numbers. Start by arranging the numbers vertically so that the decimal points line up.

\begin{align*}& \ \ 83.6\\ & \underline{- \ 1.3}\end{align*}

Next, subtract from right to left. Subtract 6 minus 3 to make 3. Then 83 minus 1 to make 82. Insert a decimal point into your answer directly under the decimal points in the original numbers.

\begin{align*}& \ \ 83.6\\ & \underline{- \ 1.3} \\ & \ \ 82.3\end{align*}

The answer is

.### Review

Find each difference.

- \begin{align*}6.57-5.75\end{align*}
- \begin{align*}.0826-.044\end{align*}
- \begin{align*}19.315-6.8116\end{align*}
- \begin{align*}2056.04-2044.1\end{align*}
- \begin{align*}303.45-112.05\end{align*}
- \begin{align*}16.576-8.43\end{align*}
- \begin{align*}199.2-123.45\end{align*}
- \begin{align*}1.0009-.234\end{align*}
- \begin{align*}789.12-.876\end{align*}
- \begin{align*}102.03-.27\end{align*}

Find the difference after rounding each decimal to the nearest hundredth.

- \begin{align*}63.385-50.508\end{align*}
- \begin{align*}.535-.361\end{align*}
- \begin{align*}747.005-47.035\end{align*}
- \begin{align*}.882-.546\end{align*}
- \begin{align*}.9887-.0245\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.7.