Jeremy and his family are driving to visit his grandparents. On the first day they drove 234.8 miles and on the second day they drove 251.6 miles. How many more miles did they drive on the second day?

### Subtracting Decimals

To subtract decimals, first write the decimals using the vertical alignment method. The decimal points must be kept directly under each other and the digits in the same place value must be kept in line with each other. If the decimal numbers are signed numbers, the rules for adding integers are applied to the problem. The number of greater magnitude should be placed above the number of smaller magnitude. Magnitude is simply the size of the number without respect to its sign. The number -42.8 has a magnitude of 42.8.

#### Let's practice subtracting positive decimals:

- \begin{align*}57.62 - 6.18\end{align*}
57.62−6.18

Subtracting decimals is similar to subtracting whole numbers. Line up the decimal points so that you can subtract corresponding place value digits (e.g. tenths from tenths, hundredths from hundredths, and so on). As with whole numbers, start from the right and work toward the left remembering to borrow when it is necessary.

\begin{align*}& \quad 57. \cancel{\overset{5}{6}} \, ^1 2\\
& \underline{ \; \; - \ 6. \ \, 1 \; \; \ 8}\\
& \quad 51. \ 4 \ \ \ \, 4\end{align*}

- \begin{align*}(98.04)-(32.801)\end{align*}
(98.04)−(32.801)

Begin by writing the question using the vertical alignment method. To ensure that the digits are aligned correctly, add zero to 98.04.

\begin{align*}& {\color{white}-} 98.04 {\color{blue}0}\\
& \underline{-32.801}\\
& \end{align*}

Subtract the numbers.

\begin{align*}& {\color{white}-} 9 \overset{7}{\cancel{8}}. \, \! ^1 0 \overset{3}{\cancel{4}} ~ ^1 {\color{blue}0}\\
& \underline{-3 \ 2 \ . {\color{white} ^1} 8 \ \, 0 {\color{white}~ ^1} 1}\\
& {\color{white}-} 6 \ 5 \ . {\color{white} ^1} 2 \ \, 3 {\color{white}~ ^1} 9\end{align*}

- \begin{align*}(137.4)-(+259.687)\end{align*}
(137.4)−(+259.687)

The first step is to write the problem as an addition problem and to change the sign of the original number being subtracted. In other words, add the opposite.

\begin{align*}(137.4)+(-259.687)\end{align*}

Now write the problem using the vertical alignment method. Remember to put 259.687 above 137.4 because 259.687 is the number of greater magnitude. The two numbers that are being added have opposite signs. Apply the same rule that you used when adding integers that had opposite signs – subtract the numbers and use the sign of the larger number in the answer.

\begin{align*} -259.687 & \\
\underline{ +137.4 {\color{white} 00}} & \end{align*}

To ensure that the digits are aligned correctly, add zeros to 137.4.

\begin{align*} -259.687 & \\
\underline{+137.4 {\color{blue}00}} & \end{align*}

Subtract the numbers.

\begin{align*} -259.687 & \\
\underline{ +137.4 {\color{blue}00}} & \\
-122.287 & \end{align*}

The numbers being added have opposite signs. This means that the sign of the answer will be the same sign as that of the number of greater magnitude. In this problem the answer has a negative sign.

#### Now, let's subtract positive and negative decimals:

\begin{align*}(67.65)-(-25.43)\end{align*}

The first step is to write the problem as an addition problem and to change the sign of the original number being subtracted. In other words, add the opposite.

\begin{align*}(67.65)+(+25.43)\end{align*}

Now, write and solve the problem using the vertical alignment method.

\begin{align*} 67.65 & \\
\underline{ +25.43} & \\
+93.08 & \end{align*}

### Examples

#### Example 1

Earlier, you were told that Jeremy and his family drove 234.8 miles on the first day and 251.6 miles on the second day. How many more miles did Jeremy and his family drive on the second day of their trip than on the first day.

The decimal number 251.6 is of greater magnitude than 234.8. The numbers must be vertically aligned with the larger one above the smaller one. Now the numbers can be subtracted.

They drove 16.8 miles more on the second day.

#### Example 2

Subtract the decimals: \begin{align*}(243.67)-(196.3579)\end{align*}

\begin{align*}(243.67)-(196.3579)=47.3121\end{align*}

#### Example 3

Subtract the decimals: \begin{align*}(32.47)-(-28.8)-(19.645)\end{align*}

Write the question as an addition problem and change the sign of the original number being subtracted.

\begin{align*}(32.47)+(+28.8)+(-19.645)\end{align*}

Follow the rules for adding integers.

\begin{align*}(32.47)-(-28.8)-(19.645)=41.625\end{align*}

#### Example 4

Josie has $59.27 in her bank account. She went to the grocery store and wrote a check for $62.18 to pay for the groceries. Describe Josie’s balance in her bank account now.

$59.27-$62.18=$-2.91. The account will have a negative value. This means that her account is overdrawn.

### Review

Subtract the following numbers:

- \begin{align*}42.37-15.32\end{align*}
42.37−15.32 - \begin{align*}37.891-7.2827\end{align*}
37.891−7.2827 - \begin{align*}579.237-45.68\end{align*}
579.237−45.68 - \begin{align*}4.2935-0.327\end{align*}
4.2935−0.327 - \begin{align*}16.074-7.58\end{align*}
16.074−7.58 - \begin{align*}(-17.39)-(-49.68)\end{align*}
(−17.39)−(−49.68) - \begin{align*}(92.75)+(-106.682)\end{align*}
(92.75)+(−106.682) - \begin{align*}(-72.5)-(-77.57)-(31.724)\end{align*}
(−72.5)−(−77.57)−(31.724) - \begin{align*}(-82.456)-(279.83)+(-567.3)\end{align*}
(−82.456)−(279.83)+(−567.3) - \begin{align*}(-57.76)-(-85.9)-(33.84)\end{align*}
(−57.76)−(−85.9)−(33.84)

Determine the answer to the following problems.

- The diameter of No. 12 bare copper wire is 0.08081 in., and the diameter of No. 15 bare copper wire is 0.05707 in. How much larger is the diameter of the No.12 wire compared to the diameter of the No. 15 wire?
- The resistance of an armature while it is cold is 0.208 ohm. After running for several minutes, the resistance increases to 1.340 ohms. Find the increase in resistance of the armature.
- The highest temperature recorded in Canada this year was \begin{align*}114.8^\circ F\end{align*}
114.8∘F . The lowest temperature of \begin{align*}-62.9^\circ F\end{align*}−62.9∘F was recorded in February this year. Find the difference between the highest and lowest temperatures recorded in Canada this year. - The temperature in Alaska was recorded as \begin{align*}-78.64^\circ F\end{align*}
−78.64∘F in January of 2010 and as \begin{align*}-59.8^\circ F\end{align*}−59.8∘F on the same date in 2011. What is the difference between the two recorded temperatures? - Laurie has a balance of -$32.16 in her bank account. Write a problem that could represent this balance.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.6.