1.3: Addition of Decimals
Stephen went shopping to buy some new school supplies. He bought a backpack that cost $28.67 and a scientific calculator for $34.88. How much money did Stephen spend altogether?
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Guidance
When you add decimal numbers you are adding whole numbers and like fraction parts. To make this process simpler, you should add decimal numbers using the vertical alignment method. The decimal points must be kept directly under each other and 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 addition process is the same as adding whole numbers. If the decimal numbers are signed numbers, the rules for adding integers are applied to the problem.
Example A
\begin{align*}87.296+48.6\end{align*}
Begin by writing the question using the vertical alignment method.
\begin{align*} 87.296 & \\
\underline{+48.6\phantom{00}}\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 zeros to 48.6.
\begin{align*} 87.296 & \\
\underline{+48.6{\color{blue}00}} & \end{align*}
Add the numbers.
\begin{align*} 87.296 & \\
\underline{+ 48.6{\color{blue}00}} & \\
135.896 & \end{align*}
Example B
\begin{align*}(97.38)+(-45.17)\end{align*}
The first step is to write the problem using the vertical alignment method. 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*} 97.38 & \\
\underline{- 45.17} & \\
\quad 52.21 & \end{align*}
The larger decimal number is 97.38 and it has a positive sign. This means that the sign of the answer will also be a positive value.
Example C
\begin{align*}(-168.8)+(-217.4536)\end{align*}
The first step is to write the problem using the vertical alignment method. The two decimal numbers that are being added have the same signs. Apply the same rule that you used when adding integers that had same signs – add the numbers and use the sign of the numbers in the answer.
\begin{align*} -168.8\phantom{000} & \\
\underline{+-217.4536} & \end{align*}
To ensure that the digits are aligned correctly, add zeros to 168.8. Add the numbers.
\begin{align*} -168.8{\color{blue}000} & \\
\underline{+-217.4536} & \end{align*}
Add the numbers.
\begin{align*} -168.8{\color{blue}000} & \\
\underline{+-217.4536} & \\
-386.2536 & \end{align*}
The decimal numbers being added both had negative signs. This means that the sign of the answer is also a negative value.
Concept Problem Revisited
Stephen went shopping to buy some new school supplies. He bought a backpack that cost $28.67 and a scientific calculator for $34.88.
Stephen bought two items. To determine the total amount of money he spent, add the prices of the items.
The numbers and the decimal points have been correctly aligned. Now add the numbers.
Stephen spent $63.55 altogether.
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.
Guided Practice
1. \begin{align*}45.36+15+137.692+32.8\end{align*}
2. \begin{align*}(53.69)+(-33.7)+(6.298)\end{align*}
3. \begin{align*}14.68+39.217\end{align*}
Answers:
1. \begin{align*}45.36+15+137.692+32.8\end{align*}
Write the decimal numbers using the vertical alignment method.
\begin{align*} 45.36\phantom{0} & \\
15\phantom{.000} & \\
137.692 & \\
\underline{+32.8\phantom{00}}\end{align*}
Attach zeros to provide the same number of decimal digits in all of the addends. In a whole number, the decimal point is not written but it is understood as being at the end of the number. \begin{align*}15=15\end{align*}
\begin{align*} 45.36{\color{blue}0} & \\
15.{\color{blue}000} & \\
137.692 & \\
\underline{+32.8{\color{blue}00}} & \end{align*}
Add the numbers in each vertical column.
\begin{align*}& \overset{2 1 \ \ 1}{\quad 45.36{\color{blue}0}}\\ & \quad 15.{\color{blue}000}\\ & \ 137.692\\ & \underline{+ 32.8{\color{blue}00}}\\ & \ 230.852\end{align*}
2. \begin{align*}(53.69)+(-33.7)+(6.298)\end{align*}
Add the two positive decimal numbers. The answer will have a positive value – add the numbers with the same sign and the answer will have the same sign as the number being added.
\begin{align*}53.69+6.298\end{align*}
Write the numbers using the vertical alignment method and add zeros so that all addends will have the same number of decimal digits. Add the numbers in each vertical column.
\begin{align*} 53.69{\color{blue}0} & \\ \underline{+ \ \ 6.298} & \\ 59.988 & \end{align*}
Add the negative decimal number to this answer. When adding numbers with opposite signs, subtract the numbers and the answer will have the sign of the larger number. In this case, the larger number is 59.988, so the answer will have a positive value. Don’t forget the zeros.
\begin{align*} 59.988 & \\ \underline{-33.7{\color{blue}00}} & \\ 26.288 & \end{align*}
3. \begin{align*}14.68+39.21=53.897\end{align*}
Practice
Add the following decimal numbers by using the expanded fraction form:
- \begin{align*}14.36+9.42\end{align*}
- \begin{align*}52.72+27.163\end{align*}
- \begin{align*}0.26+4.5+1.137\end{align*}
- \begin{align*}37.231+14.567\end{align*}
- \begin{align*}78.32+6.2+19.46\end{align*}
Add the following decimal numbers:
- \begin{align*}65.23+12.75\end{align*}
- \begin{align*}148.067+53.78+6.9\end{align*}
- \begin{align*}56.75+14.9294+17.854\end{align*}
- \begin{align*}18+26.87+65.358\end{align*}
- \begin{align*}23.067+268.93+9.4\end{align*}
Add the following signed decimal numbers:
- \begin{align*}(-24.69)+(-39.87)\end{align*}
- \begin{align*}(76.35)+(-36.68)\end{align*}
- \begin{align*}(-12.5)+(47.97)+(-21.653)\end{align*}
- \begin{align*}(62.462)+(254.69)+(-427.9)\end{align*}
- \begin{align*}(-37.76)+(-45.8)+(53.92)\end{align*}
Determine the answer to the following problems.
- When the owners of the Finest Fixer Co. completed a small construction job, they found that the following expenses had been incurred: labour, $975.75; gravel, $88.79; sand, $43.51; cement, $284.96; and bricks $2214.85. What bill should they give the customer if they want to make a profit of $225 for the job?
- A tile setter purchases the following supplies for the day: One bag of thin-set mortar - 5.67 per bag 44 sq ft of tile - 107.80 for 44 sq ft of tile One gallon of grout - 17.97 per gallon One container of grout sealer - 32.77 per container 3 containers of grout and tile cleaner - 5.99 per container 4 scrub pads - 2.78 each One trowel - @ 3.95 each 2 packages of tile spacers - @2.27 each One grout bag - @2.79 each One grout float - @10.45 each What is the cost of these items before tax is added?
- The four employees of the Broken Body Shop earned the following amounts last week: $815.86, $789.21, $804.18 and $888.35. What is the average weekly pay for the employees?
- Jennifer bought the following school supplies: 1000 sheets of paper - 14.67 36 pencils - @ 6.55 1 binder - 18.48 1 backpack - @ $22.74 1 lunch bag - @ 4.64 How much did Jennifer spend on these supplies before taxes?
- A local seamstress needs to purchase fabric to sew curtains for the local theatre. She needs 123.75 yd. of black cotton for a backdrop, 217.4 yd. of white linen for stage curtains, 75 yd. for accessory curtains and 98.5 yd. for costumes. How many yards of fabric must be purchased to fill this order?
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Image Attributions
Here you will learn to add decimal numbers. You will learn first to add decimal numbers that are positive values. Then, you will add decimal numbers that are both negative and positive values. Mastering these concepts will lead to the formation of rules for adding decimal numbers.
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