1.6: Subtraction of Decimals
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?
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Khan Academy Subtracting Decimals
Guidance
To subtract decimals, the decimal numbers should be written using the vertical alignment method. The decimal 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 or direction. The number 42.8 has a direction to the left and a magnitude of 42.8. The decimal points must be kept directly under each other as well as the digits must be kept in the same place value in line with each other. This means that digits in the ones place must be directly below digits in the ones place, digits in the tenths place must be in the tenths column, digits in the hundredths place must be in the hundredths column and so on. Once the numbers have been correctly aligned, the subtraction process is the same as subtracting whole numbers.
Example A
Subtract: \begin{align*}57.62  6.18\end{align*}
Subtracting decimals is similar to subtracting whole numbers. We line up the decimal points so that we can subtract corresponding place value digits (e.g. tenths from tenths, hundredths from hundredths, and so on). As with whole numbers, we start from the right and work toward the left remembering to borrow when it is necessary.
To answer this question, the decimal numbers should be written using the vertical alignment method. Don’t forget to put the number of greater magnitude above the smaller number.
\begin{align*}& \quad 57. \cancel{\overset{5}{6}} \ ^1 2\\
& \underline{ \; \; 6.1 \; \; \; 8}\\
& \quad 51.4 \ \ 4\end{align*}
Example B
\begin{align*}(98.04)(32.801)\end{align*}
Begin by writing the question using the vertical alignment method.
\begin{align*} 98.04\phantom{0} & \\
\underline{32.801} \end{align*}
The decimal points must be kept directly under each other as well as the digits must be kept in the same place value in line with each other. This means that digits in the ones place must be directly below digits in the ones place, digits in the tenths place must be in the tenths column, digits in the hundredths place must be in the hundredths column and so on. 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{32. {\color{white} ^1} 80 {\color{white}~ ^1} 1}\\
& {\color{white}} 65. {\color{white} ^1} 23 {\color{white}~ ^1} 9\end{align*}
Example C
\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 the problem using the vertical alignment method. The two decimal numbers that are being added have positive signs. Apply the same rule that you used when adding integers that had same signs – add the numbers and use the sign of numbers in the answer.
\begin{align*} 67.65 & \\
\underline{ +25.43} & \\
+93.08 & \end{align*}
Example D
\begin{align*}(137.4)(+259.687)\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*}(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 decimal 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 decimal 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 will have a negative sign.
Concept Problem Revisited
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.
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.
Vocabulary
 Decimal Number
 A decimal number is a fraction whose denominator is 10 or some multiple of 10.
 Decimal Point
 A decimal point is the place marker in a decimal number that separates the whole number and the fraction part. The decimal number 326.45 has the decimal point between the six and the four.
 Magnitude
 A magnitude is the size of a number without respect to its direction. The number 35.6 has a direction to the left and a magnitude of 35.6.
Guided Practice
1. Subtract these decimal numbers: \begin{align*}(243.67)(196.3579)\end{align*}
2. \begin{align*}(32.47)(28.8)(19.645)\end{align*}
3. Josie has $59.27 in her bank account. She went to the grocery store and wrote a cheque for $62.18 to pay for the groceries. Describe Josie’s balance in her bank account now.
Answers:
1. \begin{align*}(243.67)(196.3579)\end{align*}
Write the decimal numbers using the vertical alignment method.
\begin{align*} 243.67\phantom{00} & \\
\underline{196.3579} & \end{align*}
To ensure that the digits are properly aligned, add zeros to 243.67
\begin{align*} 243.67 {\color{blue}00} & \\
\underline{196.3579} & \end{align*}
Subtract the numbers. Work from right to left and borrow when it is necessary.
\begin{align*}& \ \ \overset{1}{\cancel{2}} \ \overset{13}{\cancel{4}} \ {^1} 3.6 \overset{6}{\cancel{7}} \ \overset{9}{^1 {\color{blue}\bcancel{0}}} \ {^1} {\color{blue}0}\\
& \underline{ 1 \; 9 \; \;\; 6.35 \;\; 7 \;\;\; 9\;\;}\\
& \quad \ \ 4 \ \ 7.31 \ \ 2 \ \ 1\end{align*}
2. \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. Like signs – add and use the sign of the numbers Being added.
\begin{align*} 32.47 & \text{~~~~ Add a zero to} ~ 28.8 & \\
\underline{ +28.8 {\color{blue}0}} & & \end{align*}
Add the numbers
\begin{align*}& \ \ 32.47\\
& \underline{ +28.8 {\color{blue}0}}\\
& \ \ 61.27\end{align*}
The numbers being added are both positive so the answer will also be positive.
\begin{align*}(+61.27)+(19.645)\end{align*}
Write the problem using the vertical alignment method.
\begin{align*} 61.27 {\color{blue}0} & \\
\underline{19.645} & \end{align*}
The number of greater magnitude was written above the smaller number. A zero was added to 61.27. The numbers have opposite signs so they will be subtracted and the answer will have the same sign as the larger number – positive.
\begin{align*} 61.27 {\color{blue}0} & \\
\underline{19.645} & \\
41.625 & \end{align*}
3. Amount in her bank account  $59.27
Amount of the written cheque  $62.18
The amount of the cheque is greater than the amount of money in the account.
The account will have a negative value. This means that her account is overdrawn.
Practice
Subtract the following decimal numbers:

\begin{align*}42.3715.32\end{align*}
42.37−15.32 
\begin{align*}37.8917.2827\end{align*}
37.891−7.2827 
\begin{align*}579.23745.68\end{align*}
579.237−45.68 
\begin{align*}4.29350.327\end{align*}
4.2935−0.327 
\begin{align*}16.0747.58\end{align*}
16.074−7.58
Subtract the following signed decimal numbers:

\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 No.12 wire than 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.
Image Attributions
Here you will learn to subtract decimal numbers. You will learn first to subtract decimal numbers that are positive values by applying the vertical alignment method. Then, you will subtract decimal numbers that are both negative and positive values. You will again apply the rules for adding integers since subtracting decimals that are signed numbers is the same as adding the opposite. Mastering these concepts will lead to the formation of rules for subtracting decimal numbers.
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