Math Academy

Introduction

Adding decimals is no harder than adding whole numbers.

For example, let's find

First, we write one number above the other, lining up the decimal points and placing a decimal point below the others in our answer:

Next, we proceed by adding the two numbers just as we would with whole numbers. After each addition, we write the ones place and carry the tens place of the result.

Therefore,

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Example: Adding Two-Digit and Three-Digit Decimals

What is

EXPLANATION

First, we line up the decimal points:

Next, we proceed by adding the two numbers just as we would with whole numbers:

Therefore,

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Practice: Adding Two-Digit and Three-Digit Decimals

$1.1+2.2 =$

a	$3.6$
b	$3.2$
c	$3.3$
d	$3.4$
e	$3.1$

EXPLANATION

First, we line up the decimal points:

$\begin{array}{cccccccc} & & \!\!\!\! 1 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 1 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! 2 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 2 \!\!\!\! \\ \hline & & & \!\!\!\! . \!\!\!\!& \end{array}$

Next, we proceed by adding the two numbers just as we would with whole numbers:

$\begin{array}{cccccccc} & & \!\!\!\! 1 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 1 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! 2 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 2 \!\!\!\! \\ \hline & & \!\!\!\! 3 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 3 \!\!\!\! \end{array}$

Therefore, $1.1+2.2 = 3.3.$

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Practice: Adding Two-Digit and Three-Digit Decimals

$2.16+4.71 =$

a	$8.87$
b	$4.87$
c	$6.76$
d	$6.87$
e	$4.76$

EXPLANATION

First, we line up the decimal points:

$\begin{array}{cccccccc} & & \!\!\!\! 2 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 1 \!\!\!\!& \!\!\!\! 6 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! 4 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 7 \!\!\!\!& \!\!\!\! 1 \!\!\!\! \\ \hline & & & \!\!\!\! . \!\!\!\!& & \end{array}$

Next, we proceed by adding the two numbers just as we would with whole numbers:

$\begin{array}{cccccccc} & & \!\!\!\! 2 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 1 \!\!\!\!& \!\!\!\! 6 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! 4 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 7 \!\!\!\!& \!\!\!\! 1 \!\!\!\! \\ \hline & & \!\!\!\! 6 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 8 \!\!\!\!& \!\!\!\! 7 \!\!\!\! \end{array}$

Therefore, $2.16+4.71 = 6.87.$

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Example: Adding Decimals Containing Different Numbers of Whole Digits

What is

EXPLANATION

First, we line up the decimal points:

Next, we proceed by adding the two numbers just as we would with whole numbers:

Therefore,

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Practice: Adding Decimals Containing Different Numbers of Whole Digits

$4.95 + 79.22 =$

a	$84.17$
b	$85.33$
c	$83.12$
d	$84.47$
e	$84.28$

EXPLANATION

First, we line up the decimal points:

$\begin{array}{cccccccc} & & & \!\!\!\! \!\!\!\!& \!\!\!\! 4 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 9 \!\!\!\!& \!\!\!\! 5 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! \!\!\!\!& \!\!\!\! 7 \!\!\!\!& \!\!\!\! 9 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 2 \!\!\!\!& \!\!\!\! 2 \!\!\!\! \\ \hline & & & & & \!\!\!\! . \!\!\!\!& & \end{array}$

Next, we proceed by adding the two numbers just as we would with whole numbers:

$\begin{array}{cccccccc} & & & \!\!\!\! \color{blue}\substack{ \\ 1} \!\!\!\!& \!\!\!\! \color{blue}\substack{ \\ 1} \!\!\!\!& & \!\!\!\! \color{blue}\substack{ \\ } \!\!\!\! \\ & & & \!\!\!\! \!\!\!\!& \!\!\!\! 4 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 9 \!\!\!\!& \!\!\!\! 5 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! \!\!\!\!& \!\!\!\! 7 \!\!\!\!& \!\!\!\! 9 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 2 \!\!\!\!& \!\!\!\! 2 \!\!\!\! \\ \hline & & \!\!\!\! \!\!\!\!& \!\!\!\! 8 \!\!\!\!& \!\!\!\! 4 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 1 \!\!\!\!& \!\!\!\! 7 \!\!\!\! \end{array}$

Therefore, $4.95 + 79.22=84.17.$

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Practice: Adding Decimals Containing Different Numbers of Whole Digits

$7.92 +412.34 =$

a	$420.98$
b	$420.18$
c	$420.26$
d	$419.46$
e	$419.24$

EXPLANATION

First, we line up the decimal points:

$\begin{array}{cccccccc} & & & \!\!\!\! \!\!\!\!& \!\!\!\! 7 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 9 \!\!\!\!& \!\!\!\! 2 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! 4 \!\!\!\!& \!\!\!\! 1 \!\!\!\!& \!\!\!\! 2 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 3 \!\!\!\!& \!\!\!\! 4 \!\!\!\! \\ \hline & & & & & \!\!\!\! . \!\!\!\!& & \end{array}$

Next, we proceed by adding the two numbers just as we would with whole numbers:

$\begin{array}{cccccccc} & & & \!\!\!\! \color{blue}\substack{ \\ 1 } \!\!\!\!& \!\!\!\! \color{blue}\substack{ \\ 1} \!\!\!\!& & \!\!\!\! \color{blue}\substack{ \\ } \!\!\!\! \\ & & & \!\!\!\! \!\!\!\!& \!\!\!\! 7 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 9 \!\!\!\!& \!\!\!\! 2 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! 4 \!\!\!\!& \!\!\!\! 1 \!\!\!\!& \!\!\!\! 2 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 3 \!\!\!\!& \!\!\!\! 4 \!\!\!\! \\ \hline & & \!\!\!\! 4 \!\!\!\!& \!\!\!\! 2 \!\!\!\!& \!\!\!\! 0 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 2 \!\!\!\!& \!\!\!\! 6 \!\!\!\! \end{array}$

Therefore, $7.92 +412.34=420.26.$

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Example: Adding Four-Digit, Five-Digit and Six-Digit Decimals

What is

EXPLANATION

First, we line up the decimal points:

Next, we proceed by adding the two numbers just as we would with whole numbers:

Therefore,

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Practice: Adding Four-Digit, Five-Digit and Six-Digit Decimals

$8.135 + 7.901 =$

a	$16.046$
b	$16.036$
c	$16.136$
d	$15.036$
e	$17.036$

EXPLANATION

First, we line up the decimal points:

$\begin{array}{cccccccc} & & \!\!\!\! 8 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 1 \!\!\!\!& \!\!\!\! 3 \!\!\!\!& \!\!\!\! 5 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! 7 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 9 \!\!\!\!& \!\!\!\! 0 \!\!\!\!& \!\!\!\! 1 \!\!\!\! \\ \hline & & & \!\!\!\! . \!\!\!\!& & & \end{array}$

Next, we proceed by adding the two numbers just as we would with whole numbers:

$\begin{array}{cccccccc} & & \!\!\!\! \color{blue}\substack{ \\ 1 } \!\!\!\!& & & \\ & & \!\!\!\! 8 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 1 \!\!\!\!& \!\!\!\! 3 \!\!\!\!& \!\!\!\! 5 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! 7 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 9 \!\!\!\!& \!\!\!\! 0 \!\!\!\!& \!\!\!\! 1 \!\!\!\! \\ \hline & \!\!\!\! 1 \!\!\!\!& \!\!\!\! 6 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 0 \!\!\!\!& \!\!\!\! 3 \!\!\!\!& \!\!\!\! 6 \!\!\!\! \end{array}$

Therefore, $8.135 + 7.901 = 16.036.$

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Practice: Adding Four-Digit, Five-Digit and Six-Digit Decimals

$72.112 + 52.303 =$

a	$124.315$
b	$124.415$
c	$124.425$
d	$125.415$
e	$124.515$

EXPLANATION

First, we line up the decimal points:

$\begin{array}{cccccccc} & & \!\!\!\! 7 \!\!\!\!& \!\!\!\! 2 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 1 \!\!\!\!& \!\!\!\! 1 \!\!\!\!& \!\!\!\! 2 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! 5 \!\!\!\!& \!\!\!\! 2 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 3 \!\!\!\!& \!\!\!\! 0 \!\!\!\!& \!\!\!\! 3 \!\!\!\! \\ \hline & & & & \!\!\!\! . \!\!\!\!& & & \end{array}$

Next, we proceed by adding the two numbers just as we would with whole numbers:

$\begin{array}{cccccccc} & & \!\!\!\! 7 \!\!\!\!& \!\!\!\! 2 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 1 \!\!\!\!& \!\!\!\! 1 \!\!\!\!& \!\!\!\! 2 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! 5 \!\!\!\!& \!\!\!\! 2 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 3 \!\!\!\!& \!\!\!\! 0 \!\!\!\!& \!\!\!\! 3 \!\!\!\! \\ \hline & \!\!\!\! 1 \!\!\!\!& \!\!\!\! 2 \!\!\!\!& \!\!\!\! 4 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 4 \!\!\!\!& \!\!\!\! 1 \!\!\!\!& \!\!\!\! 5 \!\!\!\! \end{array}$

Therefore, $72.112+52.303 = 124.415.$

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Example: Word Problems

Brandon stirs liters of milk with liters of water in a pot. How much liquid is in the pot?

EXPLANATION

To find how much liquid is in the pot, we need to calculate

First, we line up the decimal points:

Next, we proceed by adding the two numbers just as we would with whole numbers:

Therefore, there are liters of liquid in the pot.

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Practice: Word Problems

Emmy spent $\$1.91$ on fruits and $\$2.29$ on vegetables in a grocery store. How much money did Emmy spend in total?

a	$\$3.19$
b	$\$3.10$
c	$\$4.20$
d	$\$4.10$
e	$\$3.20$

EXPLANATION

To find Emmy's total spend, we need to calculate the value of $1.91+2.29.$

First, we line up the decimal points:

$\begin{array}{cccccccc} & & \!\!\!\! 1 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 9 \!\!\!\!& \!\!\!\! 1 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! 2 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 2 \!\!\!\!& \!\!\!\! 9 \!\!\!\! \\ \hline & & & \!\!\!\! . \!\!\!\!& & \end{array}$

Next, we proceed by adding the two numbers just as we would with whole numbers:

$\begin{array}{cccccccc} & & \!\!\!\! \color{blue}\substack{ \\ 1 } \!\!\!\!& & \!\!\!\! \color{blue}\substack{ \\ 1 } \!\!\!\! \\ & & \!\!\!\! 1 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 9 \!\!\!\!& \!\!\!\! 1 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! 2 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 2 \!\!\!\!& \!\!\!\! 9 \!\!\!\! \\ \hline & & \!\!\!\! 4 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 2 \!\!\!\!& \!\!\!\! 0 \!\!\!\! \end{array}$

So, $1.91+2.29=4.20.$

Therefore, Emmy spent $\$4.20.$

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Practice: Word Problems

Martha downloaded two apps for her smartphone. One was $21.54 \, \textrm{mb}$ and the other was $24.05 \, \textrm{mb}.$ What size was the combined download?

a	$48.45 \, \textrm{mb}$
b	$50.54 \, \textrm{mb}$
c	$47.46 \, \textrm{mb}$
d	$56.54 \, \textrm{mb}$
e	$45.59 \, \textrm{mb}$

EXPLANATION

To find the size of the combined download, we need to calculate the value of $21.54 + 24.05$

First, we line up the decimal points:

$\begin{array}{cccccccc} & & \!\!\!\! 2 \!\!\!\!& \!\!\!\! 1 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 5 \!\!\!\!& \!\!\!\! 4 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! 2 \!\!\!\!& \!\!\!\! 4 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 0 \!\!\!\!& \!\!\!\! 5 \!\!\!\! \\ \hline & & & & \!\!\!\! . \!\!\!\!& & \end{array}$

Next, we proceed by adding the two numbers just as we would with whole numbers:

$\begin{array}{cccccccc} & & \!\!\!\! 2 \!\!\!\!& \!\!\!\! 1 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 5 \!\!\!\!& \!\!\!\! 4 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\! 2 \!\!\!\!& \!\!\!\! 4 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 0 \!\!\!\!& \!\!\!\! 5 \!\!\!\! \\ \hline & & \!\!\!\! 4 \!\!\!\!& \!\!\!\! 5 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 5 \!\!\!\!& \!\!\!\! 9 \!\!\!\! \end{array}$

Therefore, the size of the combined download was $45.59\, \textrm{mb}.$

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Practice: Word Problems

If Jane has $\$8.75$ in her purse and $\$ 264.45$ in her bank account, how much money does she have in total?

a	$211.30$ dollars
b	$251.25$ dollars
c	$273.20$ dollars
d	$262.40$ dollars
e	$233.70$ dollars

EXPLANATION

To find the amount of money Jane has, we need to calculate the value of $264.45+8.75$

First, we line up the decimal points:

$\begin{array}{cccccccc} & & \!\!\!\! 2 \!\!\!\!& \!\!\!\! 6 \!\!\!\!& \!\!\!\! 4 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 4 \!\!\!\!& \!\!\!\! 5 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & & & \!\!\!\! 8 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 7 \!\!\!\!& \!\!\!\! 5 \!\!\!\! \\ \hline & & & & & \!\!\!\! . \!\!\!\!& & \end{array}$

Next, we proceed by adding the two numbers just as we would with whole numbers:

$\begin{array}{cccccccc} & & & \!\!\!\! \color{blue}\substack{ \\ 1 } \!\!\!\!& \!\!\!\! \color{blue}\substack{ \\ 1 } \!\!\!\!& & \!\!\!\! \color{blue}\substack{ \\ 1 } \!\!\!\! \\ & & \!\!\!\! 2 \!\!\!\!& \!\!\!\! 6 \!\!\!\!& \!\!\!\! 4 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 4 \!\!\!\!& \!\!\!\! 5 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & & & & \!\!\!\! 8 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 7 \!\!\!\!& \!\!\!\! 5 \!\!\!\! \\ \hline & & \!\!\!\! 2 \!\!\!\!& \!\!\!\! 7 \!\!\!\!& \!\!\!\! 3 \!\!\!\!& \!\!\!\! . \!\!\!\!& \!\!\!\! 2 \!\!\!\!& \!\!\!\! 0 \!\!\!\! \end{array}$

Therefore, Jane has a total of $273.20$ dollars.

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