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Add Whole Numbers

Use addition notation

Use Addition Notation

A college student has a part-time job. Last week he worked $3$ hours on Monday and $4$ hours on Friday. To find the total number of hours he worked last week, he added $3$ and $4.$

The operation of addition combines numbers to get a sum. The notation we use to find the sum of $3$ and $4$ is:

\[3+4\]

We read this as three plus four and the result is the sum of three and four. The numbers $3$ and $4$ are called the addends. A math statement that includes numbers and operations is called an expression.

Example

Try it.

Translate from math notation to words:

ⓐ $7+1$
ⓑ $12+14$

Solution

ⓐ The expression consists of a plus symbol connecting the addends 7 and 1. We read this as seven plus one. The result is the sum of seven and one.
ⓑ The expression consists of a plus symbol connecting the addends 12 and 14. We read this as twelve plus fourteen. The result is the sum of twelve and fourteen.

Model Addition of Whole Numbers

Addition is really just counting. We will model addition with $\text{base-10}$ blocks. Remember, a block represents $1$ and a rod represents $10.$ Let’s start by modeling the addition expression we just considered, $3+4.$

Each addend is less than $10,$ so we can use ones blocks.

We start by modeling the first number with 3 blocks.
Then we model the second number with 4 blocks.
Count the total number of blocks.

There are $7$ blocks in all. We use an equal sign $\text{(=)}$ to show the sum. A math sentence that shows that two expressions are equal is called an equation. We have shown that. $3+4=7.$

Example

Try it.

Model the addition $2+6.$

Solution

$2+6$ means the sum of $2$ and $6$

Each addend is less than 10, so we can use ones blocks.

Model the first number with 2 blocks.
Model the second number with 6 blocks.
Count the total number of blocks	There are $8$ blocks in all, so $2+6=8.$

When the result is $10$ or more ones blocks, we will exchange the $10$ blocks for one rod.

Example

Try it.

Model the addition $5+8.$

Solution

$5+8$ means the sum of $5$ and $8.$

Each addend is less than 10, se we can use ones blocks.
Model the first number with 5 blocks.
Model the second number with 8 blocks.
Count the result. There are more than 10 blocks so we exchange 10 ones blocks for 1 tens rod.
Now we have 1 ten and 3 ones, which is 13.	5 + 8 = 13

Notice that we can describe the models as ones blocks and tens rods, or we can simply say ones and tens. From now on, we will use the shorter version but keep in mind that they mean the same thing.

Next we will model adding two digit numbers.

Example

Try it.

Model the addition: $17+26.$

Solution

$17+26$ means the sum of 17 and 26.

Model the 17.	1 ten and 7 ones
Model the 26.	2 tens and 6 ones
Combine.	3 tens and 13 ones
Exchange 10 ones for 1 ten.	4 tens and 3 ones $40+3=43$
We have shown that $17+26=43$

Add Whole Numbers Without Models

Now that we have used models to add numbers, we can move on to adding without models. Before we do that, make sure you know all the one digit addition facts. You will need to use these number facts when you add larger numbers.

Imagine filling in by adding each row number along the left side to each column number across the top. Make sure that you get each sum shown. If you have trouble, model it. It is important that you memorize any number facts you do not already know so that you can quickly and reliably use the number facts when you add larger numbers.

+	0	1	2	3	4	5	6	7	8	9
0	0	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9	10
2	2	3	4	5	6	7	8	9	10	11
3	3	4	5	6	7	8	9	10	11	12
4	4	5	6	7	8	9	10	11	12	13
5	5	6	7	8	9	10	11	12	13	14
6	6	7	8	9	10	11	12	13	14	15
7	7	8	9	10	11	12	13	14	15	16
8	8	9	10	11	12	13	14	15	16	17
9	9	10	11	12	13	14	15	16	17	18

Did you notice what happens when you add zero to a number? The sum of any number and zero is the number itself. We call this the Identity Property of Addition. Zero is called the additive identity.

Example

Try it.

Find each sum:

ⓐ $0+11$
ⓑ $42+0$

Solution

ⓐ The first addend is zero. The sum of any number and zero is the number.	$0+11=11$
ⓑ The second addend is zero. The sum of any number and zero is the number.	$42+0=42$

Look at the pairs of sums.

$2+3=5$	$3+2=5$
$4+7=11$	$7+4=11$
$8+9=17$	$9+8=17$

Notice that when the order of the addends is reversed, the sum does not change. This property is called the Commutative Property of Addition, which states that changing the order of the addends does not change their sum.

Example

Try it.

Add:

ⓐ $8+7$
ⓑ $7+8$

Solution

ⓐ
Add. $8+7$
$15$
ⓑ
Add. $7+8$
$15$

Did you notice that changing the order of the addends did not change their sum? We could have immediately known the sum from part ⓑ just by recognizing that the addends were the same as in part ⓐ , but in the reverse order. As a result, both sums are the same.

In the previous example, the sum of the ones and the sum of the tens were both less than $10.$ But what happens if the sum is $10$ or more? Let’s use our $\text{base-10}$ model to find out. shows the addition of $17$ and $26$ again.

Condensed — the full section is in OpenStax Prealgebra 2e.

Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation into words. Now we’ll reverse the process. We’ll translate word phrases into math notation. Some of the word phrases that indicate addition are listed in .

Operation	Words	Example	Expression
Addition	plus sum increased by more than total of added to	$1$ plus $2$ the sum of $3$ and $4$ $5$ increased by $6$ $8$ more than $7$ the total of $9$ and $5$ $6$ added to $4$	$1+2$ $3+4$ $5+6$ $7+8$ $9+5$ $4+6$

Example

Try it.

Translate and simplify: the sum of $19$ and $23.$

Solution

The word sum tells us to add. The words of $19$ and $23$ tell us the addends.

	The sum of $19$ and $23$
Translate.	$19+23$
Add.	$42$
	The sum of $19$ and $23$ is $42.$

Example

Try it.

Translate and simplify: $28$ increased by $31.$

Solution

The words increased by tell us to add. The numbers given are the addends.

	$28$ increased by $31.$
Translate.	$28+31$
Add.	$59$
	So $28$ increased by $31$ is $59.$

Add Whole Numbers in Applications

Now that we have practiced adding whole numbers, let’s use what we’ve learned to solve real-world problems. We’ll start by outlining a plan. First, we need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we write a sentence to answer the question.

Example

Try it.

Hao earned grades of $87,93,68,95,\ \text{and}\ 89$ on the five tests of the semester. What is the total number of points he earned on the five tests?

Solution

We are asked to find the total number of points on the tests.

Write a phrase.	the sum of points on the tests
Translate to math notation.	$87+93+68+95+89$
Then we simplify by adding.
Since there are several numbers, we will write them vertically.	$\begin{array}{l} \\ \\ \\ \overset{3}{8}7 \\ 93 \\ 68 \\ 95 \\ \underset{\text{____}}{+89} \\ 432\end{array}$
Write a sentence to answer the question.	Hao earned a total of 432 points.

Notice that we added points, so the sum is $432$ points. It is important to include the appropriate units in all answers to applications problems.

Some application problems involve shapes. For example, a person might need to know the distance around a garden to put up a fence or around a picture to frame it. The perimeter is the distance around a geometric figure. The perimeter of a figure is the sum of the lengths of its sides.

Example

Try it.

Find the perimeter of the patio shown.

Solution

We are asked to find the perimeter.
Write a phrase.	the sum of the sides
Translate to math notation.	$4+6+2+3+2+9$
Simplify by adding.	$26$
Write a sentence to answer the question.
We added feet, so the sum is $26$ feet.	The perimeter of the patio is $26$ feet.

Key Concepts

Addition Notation To describe addition, we can use symbols and words.
Operation Notation Expression Read as Result
Addition $+$ $3+4$ three plus four the sum of $3$ and $4$
Identity Property of Addition
- The sum of any number $a$ and $0$ is the number. $a+0=a$ $0+a=a$
Commutative Property of Addition
- Changing the order of the addends $a$ and $b$ does not change their sum. $a+b=b+a$.
Add whole numbers.
1. Write the numbers so each place value lines up vertically.
2. Add the digits in each place value. Work from right to left starting with the ones place. If a sum in a place value is more than 9, carry to the next place value.
3. Continue adding each place value from right to left, adding each place value and carrying if needed.

Add Whole Numbers

Use Addition Notation

In the following exercises, translate the following from math expressions to words.

Try it.

$5+2$

Solution

five plus two; the sum of 5 and 2.

Try it.

$6+3$

Try it.

$13+18$

Solution

thirteen plus eighteen; the sum of 13 and 18.

Try it.

$15+16$

Try it.

$214+642$

Solution

two hundred fourteen plus six hundred forty-two; the sum of 214 and 642

Try it.

$438+113$

Model Addition of Whole Numbers

In the following exercises, model the addition.

Try it.

$2+4$

Solution

$2+4=6$

Try it.

$5+3$

Try it.

$8+4$

Solution

$8+4=12$

Try it.

$5+9$

Try it.

$14+75$

Solution

$14+75=89$

Try it.

$15+63$

Try it.

$16+25$

Solution

$16+25=41$

Try it.

$14+27$

Add Whole Numbers

In the following exercises, fill in the missing values in each chart.

Try it.

Solution

Try it.

Solution

Try it.

Solution

Try it.

In the following exercises, add.

Try it.

ⓐ $0+13$
ⓑ $13+0$

Solution

ⓐ 13
ⓑ 13

Try it.

ⓐ $0+5,280$
ⓑ $5,280+0$

Try it.

ⓐ $8+3$
ⓑ $3+8$

Solution

ⓐ $11$
ⓑ $11$

Try it.

ⓐ $7+5$
ⓑ $5+7$

Try it.

$45+33$

Solution

$78$

Try it.

$37+22$

Try it.

$71+28$

Solution

$99$

Try it.

$43+53$

Try it.

$26+59$

Solution

$85$

Try it.

$38+17$

Try it.

$64+78$

Solution

$142$

Try it.

$92+39$

Try it.

$168+325$

Solution

$493$

Try it.

$247+149$

Try it.

$584+277$

Solution

$861$

Try it.

$175+648$

Try it.

$832+199$

Solution

$1,031$

Try it.

$775+369$

Try it.

$6,358+492$

Solution

$6,850$

Try it.

$9,184+578$

Try it.

$3,740+18,593$

Solution

$22,333$

Try it.

$6,118+15,990$

Try it.

$485,012+619,848$

Solution

$1,104,860$

Try it.

$368,911+857,289$

Try it.

$24,731+592+3,868$

Solution

$29,191$

Try it.

$28,925+817+4,593$

Try it.

$8,015+76,946+16,570$

Solution

$101,531$

Try it.

$6,291+54,107+28,635$

Translate Word Phrases to Math Notation

In the following exercises, translate each phrase into math notation and then simplify.

Try it.

the sum of $13$ and $18$

Solution

$13+18=31$

Try it.

the sum of $12$ and $19$

Try it.

the sum of $90$ and $65$

Solution

$90+65=155$

Try it.

the sum of $70$ and $38$

Try it.

$33$ increased by $49$

Solution

$33+49=82$

Try it.

$68$ increased by $25$

Try it.

$250$ more than $599$

Solution

$599+250=849$

Try it.

$115$ more than $286$

Try it.

the total of $628$ and $77$

Solution

$628+77=705$

Try it.

the total of $593$ and $79$

Try it.

$1,482$ added to $915$

Solution

$915+1,482=2,397$

Try it.

$2,719$ added to $682$

Add Whole Numbers in Applications

In the following exercises, solve the problem.

Try it.

Home remodeling Sophia remodeled her kitchen and bought a new range, microwave, and dishwasher. The range cost $\text{\$1,100},$ the microwave cost $\text{\$250},$ and the dishwasher cost $\text{\$525}.$ What was the total cost of these three appliances?

Solution

The total cost was $1,875.

Try it.

Sports equipment Aiden bought a baseball bat, helmet, and glove. The bat cost $\text{\$299},$ the helmet cost $\text{\$35},$ and the glove cost $\text{\$68}.$ What was the total cost of Aiden’s sports equipment?

Try it.

Bike riding Ethan rode his bike $14$ miles on Monday, $19$ miles on Tuesday, $12$ miles on Wednesday, $25$ miles on Friday, and $68$ miles on Saturday. What was the total number of miles Ethan rode?

Solution

Ethan rode 138 miles.

Try it.

Business Chloe has a flower shop. Last week she made $19$ floral arrangements on Monday, $12$ on Tuesday, $23$ on Wednesday, $29$ on Thursday, and $44$ on Friday. What was the total number of floral arrangements Chloe made?

Try it.

Apartment size Jackson lives in a $7$ room apartment. The number of square feet in each room is $238,120,156,196,100,132,$ and $225.$ What is the total number of square feet in all $7$ rooms?

Solution

The total square footage in the rooms is 1,167 square feet.

Try it.

Weight Seven men rented a fishing boat. The weights of the men were $175,192,148,169,205,181,$ and $225$ pounds. What was the total weight of the seven men?

Try it.

Salary Last year Natalie’s salary was $\text{\$82,572}.$ Two years ago, her salary was $\text{\$79,316},$ and three years ago it was $\text{\$75,298}.$ What is the total amount of Natalie’s salary for the past three years?

Solution

Natalie’s total salary is $237,186.

Try it.

Home sales Emma is a realtor. Last month, she sold three houses. The selling prices of the houses were $\text{\$292,540},\text{\$505,875},$ and $\$423,699.$ What was the total of the three selling prices?

In the following exercises, find the perimeter of each figure.

Try it.

Solution

The perimeter of the figure is 44 inches.

Try it.

Solution

The perimeter of the figure is 56 meters.

Try it.

Solution

The perimeter of the figure is 71 yards.

Try it.

Solution

The perimeter of the figure is 62 feet.

Try it.

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

What is the number modeled by the $\text{base-10}$ blocks?

If you missed this problem, review .

Откройте ответ.

$215$
Write the number three hundred forty-two thousand six using digits?
If you missed this problem, review .

Откройте ответ.

$342,006$
Translate from math notation to words:
1. ⓐ $7+1$
2. ⓑ $12+14$
Откройте ответ.
- ⓐ The expression consists of a plus symbol connecting the addends 7 and 1. We read this as seven plus one. The result is the sum of seven and one.
- ⓑ The expression consists of a plus symbol connecting the addends 12 and 14. We read this as twelve plus fourteen. The result is the sum of twelve and fourteen.
Translate from math notation to words:
1. ⓐ $8+4$
2. ⓑ $18+11$
Откройте ответ.
- ⓐ eight plus four; the sum of eight and four
- ⓑ eighteen plus eleven; the sum of eighteen and eleven
Translate from math notation to words:
1. ⓐ $21+16$
2. ⓑ $100+200$
Откройте ответ.
1. ⓐ twenty-one plus sixteen; the sum of twenty-one and sixteen
2. ⓑ one hundred plus two hundred; the sum of one hundred and two hundred
Model the addition $2+6.$

Откройте ответ.

$2+6$ means the sum of $2$ and $6$
Each addend is less than 10, so we can use ones blocks.
Model the first number with 2 blocks.
Model the second number with 6 blocks.
Count the total number of blocks
There are $8$ blocks in all, so $2+6=8.$
Model: $3+6.$

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Model: $5+1.$

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Model the addition $5+8.$

Откройте ответ.

$5+8$ means the sum of $5$ and $8.$

Each addend is less than 10, se we can use ones blocks.
Model the first number with 5 blocks.
Model the second number with 8 blocks.
Count the result. There are more than 10 blocks so we exchange 10 ones blocks for 1 tens rod.
Now we have 1 ten and 3 ones, which is 13.	5 + 8 = 13

Model the addition: $5+7.$

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Model the addition: $6+8.$

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Model the addition: $17+26.$

Откройте ответ.

$17+26$ means the sum of 17 and 26.
Model the 17. 1 ten and 7 ones
Model the 26. 2 tens and 6 ones
Combine. 3 tens and 13 ones
Exchange 10 ones for 1 ten. 4 tens and 3 ones
$40+3=43$
We have shown that $17+26=43$
Model the addition: $15+27.$

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Model the addition: $16+29.$

Откройте ответ.
Find each sum:
1. ⓐ $0+11$
2. ⓑ $42+0$
Откройте ответ.

ⓐ The first addend is zero. The sum of any number and zero is the number. $0+11=11$
ⓑ The second addend is zero. The sum of any number and zero is the number. $42+0=42$
Find each sum:
1. ⓐ $0+19$
2. ⓑ $39+0$
Откройте ответ.
1. ⓐ $0+19=19$
2. ⓑ $39+0=39$
Find each sum:
1. ⓐ $0+24$
2. ⓑ $57+0$
Откройте ответ.
1. ⓐ $0+24=24$
2. ⓑ $57+0=57$
Add:
1. ⓐ $8+7$
2. ⓑ $7+8$
Откройте ответ.
- ⓐ
  Add. $8+7$
  $15$
- ⓑ
  Add. $7+8$
  $15$
Did you notice that changing the order of the addends did not change their sum? We could have immediately known the sum from part ⓑ just by recognizing that the addends were the same as in part ⓐ , but in the reverse order. As a result, both sums are the same.
Add: $9+7$ and $7+9.$

Откройте ответ.

$9+7=16;7+9=16$
Add: $8+6$ and $6+8.$

Откройте ответ.

$8+6=14;6+8=14$

Add: $28+61.$

Откройте ответ.

To add numbers with more than one digit, it is often easier to write the numbers vertically in columns.

Write the numbers so the ones and tens digits line up vertically.	$\begin{array}{l}28\ \\ \\ \underset{\text{____}}{+61}\end{array}$
Then add the digits in each place value. Add the ones: $8+1=9$ Add the tens: $2+6=8$	$\begin{array}{l}28\ \\ \\ \underset{\text{____}}{+61} \\ 89\ \end{array}$

Add: $32+54.$

Откройте ответ.

$32+54=86$
Add: $25+74.$

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$25+74=99$

Add: $43+69.$

Откройте ответ.

Write the numbers so the digits line up vertically.	$\begin{array}{l}43\ \\ \\ \underset{\text{____}}{+69}\end{array}$
Add the digits in each place. Add the ones: $3+9=12$
Write the $2$ in the ones place in the sum. Add the $1$ ten to the tens place.	$\begin{array}{l}\overset{1}{4}3\ \\ \underset{\text{____}}{+69} \\ 2\ \end{array}$
Now add the tens: $1+4+6=11$ Write the 11 in the sum.	$\begin{array}{l}\overset{1}{4}3\ \\ \underset{\text{____}}{+69} \\ 112\ \end{array}$

Add: $35+98.$

Откройте ответ.

$35+98=133$
Add: $72+89.$

Откройте ответ.

$72+89=161$

Add: $324+586.$

Откройте ответ.

Write the numbers so the digits line up vertically.
Add the digits in each place value. Add the ones: $4+6=10$ Write the $0$ in the ones place in the sum and carry the $1$ ten to the tens place.
Add the tens: $1+2+8=11$ Write the $1$ in the tens place in the sum and carry the $1$ hundred to the hundreds
Add the hundreds: $1+3+5=9$ Write the $9$ in the hundreds place.

Add: $456+376.$

Откройте ответ.

$456+376=832$
Add: $269+578.$

Откройте ответ.

$269+578=847$

Add: $1,683+479.$

Откройте ответ.

Write the numbers so the digits line up vertically.	$\begin{array}{l}1,683\ \\ \\ \underset{\text{______}}{+\ 479}\end{array}$
Add the digits in each place value.
Add the ones: $3+9=12.$ Write the $2$ in the ones place of the sum and carry the $1$ ten to the tens place.	$\begin{array}{l}1,6\overset{1}{8}3\ \\ \\ \underset{\text{______}}{+\ 479} \\ 2\ \end{array}$
Add the tens: $1+7+8=16$ Write the $6$ in the tens place and carry the $1$ hundred to the hundreds place.	$\begin{array}{l}1,\overset{1}{6}\overset{1}{8}3\ \\ \\ \underset{\text{______}}{+\ 479} \\ 62\ \end{array}$
Add the hundreds: $1+6+4=11$ Write the $1$ in the hundreds place and carry the $1$ thousand to the thousands place.	$\begin{array}{l}1,\overset{1}{6}\overset{1}{8}3\ \\ \\ \underset{\text{______}}{+\ 479} \\ 162\ \end{array}$
Add the thousands $1+1=2$. Write the $2$ in the thousands place of the sum.	$\begin{array}{l}\overset{1}{1,}\overset{1}{6}\overset{1}{8}3\ \\ \\ \underset{\text{______}}{+\ 479} \\ 2,162\ \end{array}$

When the addends have different numbers of digits, be careful to line up the corresponding place values starting with the ones and moving toward the left.

Add: $4,597+685.$

Откройте ответ.

$4,597+685=5,282$
Add: $5,837+695.$

Откройте ответ.

$5,837+695=6,532$

Add: $21,357+861+8,596.$

Откройте ответ.

Write the numbers so the place values line up vertically.	$\begin{array}{l}21,357 \\ 861 \\ \\ \underset{\text{_______}}{+\ 8,596}\end{array}$
Add the digits in each place value.
Add the ones: $7+1+6=14$ Write the $4$ in the ones place of the sum and carry the $1$ to the tens place.	$\begin{array}{l}21,3\overset{1}{5}7 \\ 861 \\ \\ \underset{\text{_______}}{+\ 8,596} \\ 4\end{array}$
Add the tens: $1+5+6+9=21$ Write the $1$ in the tens place and carry the $2$ to the hundreds place.	$\begin{array}{l}21,\overset{2}{3}\overset{1}{5}7 \\ 861 \\ \\ \underset{\text{_______}}{+\ 8,596} \\ 14\end{array}$
Add the hundreds: $2+3+8+5=18$ Write the $8$ in the hundreds place and carry the $1$ to the thousands place.	$\begin{array}{l}2\overset{1}{1,}\overset{2}{3}\overset{1}{5}7 \\ 861 \\ \\ \underset{\text{_______}}{+\ 8,596} \\ 814\end{array}$
Add the thousands $1+1+8=10$. Write the $0$ in the thousands place and carry the $1$ to the ten thousands place.	$\begin{array}{l}\overset{1}{2}\overset{1}{1,}\overset{2}{3}\overset{1}{5}7 \\ 861 \\ \\ \underset{\text{_______}}{+\ 8,596} \\ 0814\end{array}$
Add the ten-thousands $1+2=3$. Write the $3$ in the ten thousands place in the sum.	$\begin{array}{l}\overset{1}{2}\overset{1}{1,}\overset{2}{3}\overset{1}{5}7 \\ 861 \\ \\ \underset{\text{_______}}{+\ 8,596} \\ 30,814\end{array}$

This example had three addends. We can add any number of addends using the same process as long as we are careful to line up the place values correctly.

Add: $46,195+397+6,281.$

Откройте ответ.

$46,195+397+6,281=52,873$
Add: $53,762+196+7,458.$

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$53,762+196+7,458=61,416$
Translate and simplify: the sum of $19$ and $23.$

Откройте ответ.

The word sum tells us to add. The words of $19$ and $23$ tell us the addends.
The sum of $19$ and $23$
Translate. $19+23$
Add. $42$
The sum of $19$ and $23$ is $42.$
Translate and simplify: the sum of $17$ and $26.$

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Translate: $17+26$; Simplify: $43$
Translate and simplify: the sum of $28$ and $14.$

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Translate: $28+14$; Simplify: $42$
Translate and simplify: $28$ increased by $31.$

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The words increased by tell us to add. The numbers given are the addends.
$28$ increased by $31.$
Translate. $28+31$
Add. $59$
So $28$ increased by $31$ is $59.$
Translate and simplify: $29$ increased by $76.$

Откройте ответ.

Translate: $29+76$; Simplify $105$

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Add Whole Numbers

Use addition notation
Model addition of whole numbers
Add whole numbers without models
Translate word phrases to math notation
Add whole numbers in applications
Write the numbers so each place value lines up vertically.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Попробуй сам.

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Больше Arithmetic

Order of operations Fractions Percentages Powers and roots

\(2+3=5\)	\(3+2=5\)
\(4+7=11\)	\(7+4=11\)
\(8+9=17\)	\(9+8=17\)

	The sum of \(19\) and \(23\)
Translate.	\(19+23\)
Add.	\(42\)
	The sum of \(19\) and \(23\) is \(42.\)

	\(28\) increased by \(31.\)
Translate.	\(28+31\)
Add.	\(59\)
	So \(28\) increased by \(31\) is \(59.\)

Write the numbers so the ones and tens digits line up vertically.	\(\begin{array}{l}28\ \\ \\ \underset{\text{____}}{+61}\end{array}\)
Then add the digits in each place value. Add the ones: \(8+1=9\) Add the tens: \(2+6=8\)	\(\begin{array}{l}28\ \\ \\ \underset{\text{____}}{+61} \\ 89\ \end{array}\)

Write the numbers so the digits line up vertically.	\(\begin{array}{l}43\ \\ \\ \underset{\text{____}}{+69}\end{array}\)
Add the digits in each place. Add the ones: \(3+9=12\)
Write the \(2\) in the ones place in the sum. Add the \(1\) ten to the tens place.	\(\begin{array}{l}\overset{1}{4}3\ \\ \underset{\text{____}}{+69} \\ 2\ \end{array}\)
Now add the tens: \(1+4+6=11\) Write the 11 in the sum.	\(\begin{array}{l}\overset{1}{4}3\ \\ \underset{\text{____}}{+69} \\ 112\ \end{array}\)

Operation	Notation	Expression	Read as	Result
Addition	\(+\)	\(3+4\)	three plus four	the sum of \(3\) and \(4\)

+	0	1	2	3	4	5	6	7	8	9
0	0	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9	10
2	2	3	4	5	6	7	8	9	10	11
3	3	4	5	6	7	8	9	10	11	12
4	4	5	6	7	8	9	10	11	12	13
5	5	6	7	8	9	10	11	12	13	14
6	6	7	8	9	10	11	12	13	14	15
7	7	8	9	10	11	12	13	14	15	16
8	8	9	10	11	12	13	14	15	16	17
9	9	10	11	12	13	14	15	16	17	18

ⓐ The first addend is zero. The sum of any number and zero is the number.	\(0+11=11\)
ⓑ The second addend is zero. The sum of any number and zero is the number.	\(42+0=42\)

Write the numbers so the digits line up vertically.
Add the digits in each place value. Add the ones: \(4+6=10\) Write the \(0\) in the ones place in the sum and carry the \(1\) ten to the tens place.
Add the tens: \(1+2+8=11\) Write the \(1\) in the tens place in the sum and carry the \(1\) hundred to the hundreds
Add the hundreds: \(1+3+5=9\) Write the \(9\) in the hundreds place.

Write the numbers so the digits line up vertically.	\(\begin{array}{l}1,683\ \\ \\ \underset{\text{______}}{+\ 479}\end{array}\)
Add the digits in each place value.
Add the ones: \(3+9=12.\) Write the \(2\) in the ones place of the sum and carry the \(1\) ten to the tens place.	\(\begin{array}{l}1,6\overset{1}{8}3\ \\ \\ \underset{\text{______}}{+\ 479} \\ 2\ \end{array}\)
Add the tens: \(1+7+8=16\) Write the \(6\) in the tens place and carry the \(1\) hundred to the hundreds place.	\(\begin{array}{l}1,\overset{1}{6}\overset{1}{8}3\ \\ \\ \underset{\text{______}}{+\ 479} \\ 62\ \end{array}\)
Add the hundreds: \(1+6+4=11\) Write the \(1\) in the hundreds place and carry the \(1\) thousand to the thousands place.	\(\begin{array}{l}1,\overset{1}{6}\overset{1}{8}3\ \\ \\ \underset{\text{______}}{+\ 479} \\ 162\ \end{array}\)
Add the thousands \(1+1=2\). Write the \(2\) in the thousands place of the sum.	\(\begin{array}{l}\overset{1}{1,}\overset{1}{6}\overset{1}{8}3\ \\ \\ \underset{\text{______}}{+\ 479} \\ 2,162\ \end{array}\)

Write the numbers so the place values line up vertically.	\(\begin{array}{l}21,357 \\ 861 \\ \\ \underset{\text{_______}}{+\ 8,596}\end{array}\)
Add the digits in each place value.
Add the ones: \(7+1+6=14\) Write the \(4\) in the ones place of the sum and carry the \(1\) to the tens place.	\(\begin{array}{l}21,3\overset{1}{5}7 \\ 861 \\ \\ \underset{\text{_______}}{+\ 8,596} \\ 4\end{array}\)
Add the tens: \(1+5+6+9=21\) Write the \(1\) in the tens place and carry the \(2\) to the hundreds place.	\(\begin{array}{l}21,\overset{2}{3}\overset{1}{5}7 \\ 861 \\ \\ \underset{\text{_______}}{+\ 8,596} \\ 14\end{array}\)
Add the hundreds: \(2+3+8+5=18\) Write the \(8\) in the hundreds place and carry the \(1\) to the thousands place.	\(\begin{array}{l}2\overset{1}{1,}\overset{2}{3}\overset{1}{5}7 \\ 861 \\ \\ \underset{\text{_______}}{+\ 8,596} \\ 814\end{array}\)
Add the thousands \(1+1+8=10\). Write the \(0\) in the thousands place and carry the \(1\) to the ten thousands place.	\(\begin{array}{l}\overset{1}{2}\overset{1}{1,}\overset{2}{3}\overset{1}{5}7 \\ 861 \\ \\ \underset{\text{_______}}{+\ 8,596} \\ 0814\end{array}\)
Add the ten-thousands \(1+2=3\). Write the \(3\) in the ten thousands place in the sum.	\(\begin{array}{l}\overset{1}{2}\overset{1}{1,}\overset{2}{3}\overset{1}{5}7 \\ 861 \\ \\ \underset{\text{_______}}{+\ 8,596} \\ 30,814\end{array}\)

ⓐ
Add.	\(8+7\)
	\(15\)

ⓑ
Add.	\(7+8\)
	\(15\)

ⓐ
Add.	\(8+7\)
	\(15\)

ⓑ
Add.	\(7+8\)
	\(15\)

+	0	1	2	3	4	5	6	7	8	9
0	0	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9	10
2	2	3	4	5	6	7	8	9	10	11
3	3	4	5	6	7	8	9	10	11	12
4	4	5	6	7	8	9	10	11	12	13
5	5	6	7	8	9	10	11	12	13	14
6	6	7	8	9	10	11	12	13	14	15
7	7	8	9	10	11	12	13	14	15	16
8	8	9	10	11	12	13	14	15	16	17
9	9	10	11	12	13	14	15	16	17	18

+	0	1	2	3	4	5	6	7	8	9
0	0	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9	10
2	2	3	4	5	6	7	8	9	10	11
3	3	4	5	6	7	8	9	10	11	12
4	4	5	6	7	8	9	10	11	12	13
5	5	6	7	8	9	10	11	12	13	14
6	6	7	8	9	10	11	12	13	14	15
7	7	8	9	10	11	12	13	14	15	16
8	8	9	10	11	12	13	14	15	16	17
9	9	10	11	12	13	14	15	16	17	18