Order of Operations and Parentheses

A calculation can have more than one job. Before you start, you need to know which job comes first.

A numerical expression uses numbers and operation signs to name an amount. For example, $2+3\times 4$ is an expression. It has no equals sign. To evaluate it means to find its value.

Two ways to start

Look at $2+3\times 4$. If you add first, you get 5. Multiplying that by 4 gives 20. If you multiply first, three fours make 12. Adding 2 gives 14. Those are different answers to the same writing.

The order of operations is the agreement that settles this. An operation is a math job: adding, subtracting, multiplying, or dividing. Everyone who reads an expression follows the same agreed order. That makes the expression's meaning clear.

Here the multiplication comes first:

See the equal groups

Picture two loose counters beside three bags of four counters. The bags hold $3\times 4=12$ counters altogether. The two loose counters bring the total to 14.

Adding the 2 to the 3 first would mix loose counters with bags. It would then give five bags of four. That describes a different collection. Multiplying first keeps the three groups of four together.

A money story works the same way. You have $2 and earn three tips of $4 each. The tips total $12, so you now have $14. The starting $2 is added once.

A name to remember

You may know PEMDAS from “Please Excuse My Dear Aunt Sally.” The letters stand for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Parentheses group numbers. An exponent can show a square, such as $4^2$. You will meet both soon.

Read the last four letters in pairs: MD, then AS. Multiplication and division share a rank. Addition and subtraction share another. The letters do not give multiplication a turn ahead of division.

Other books may say GEMDAS, BODMAS, or BIDMAS. These are other reminders for the same order. Their grouping words differ, but the math rule stays the same.

Before you calculate, look across the whole expression. A multiplication or division on the right may need your attention first.

With no grouping symbols or squares, do multiplication and division before addition and subtraction. Keep every untouched number and sign when you rewrite a step.

Take away a whole product

Evaluate $28-4\times 5$. The amount being taken away is four groups of five. Find that amount first.

Now subtract:

Starting with $28-4$ would give 24, then 120 after multiplying. That treats the entire difference as five groups. There are no parentheses asking you to do that.

Division comes early too

Evaluate $14+18÷6$. The division finds the amount added to 14.

Adding 14 and 18 first would make 32 the amount being divided. In the expression, only 18 is divided by 6.

More than one part

A class has five trays of six cups and four trays of seven cups. Each multiplication gives one collection's total:

First find the cups on the five trays:

Then find the cups on the four trays:

Both products must be ready before you add them. You can lightly underline each product on paper to see the two amounts.

The same method works when one part is a division:

When the leftmost job is ready

In $7\times 8-6$, multiplication is already on the left. You can start there and get $56-6=50$. You followed the same rule as before.

The position alone does not choose your first job. In $9+6÷3$, the division is farther right but comes first. It gives $9+2=11$. Reading the expression straight across would give 5 and change its meaning.

Multiplication and division share one rank. When they appear together, work through that part from left to right. Neither operation always comes first.

Divide, then multiply

Evaluate $36÷6\times 3$. The division is on the left, so start there:

Multiplying 6 by 3 first would give $36÷18=2$. That wrongly turns the multiplier into part of the amount you divide by.

You can read the expression as an instruction: “Divide 36 by 6, then multiply the result by 3.” Keep each operation attached to its own number.

Subtract, then add

Addition and subtraction also share one rank. In $23-8+4$, subtract first because it appears first:

The 8 is taken away, and the 4 is added. Adding 8 and 4 first would take away both amounts. That gives 11, which changes the instructions.

Use both ranks

Try $42÷7\times 3-5+2$. First finish division and multiplication from left to right:

Now finish subtraction and addition from left to right:

The plus sign near the end does not jump ahead of subtraction. The A in PEMDAS is paired with the S.

Easier mental math

When a calculation has only addition, you can choose a helpful order. In $25+38+75$, add 25 and 75 to make 100. Then add 38 to get 138.

When it has only multiplication, you can also choose a helpful order:

Mixed subtraction or division needs more care. For expressions like the earlier examples, the left-to-right method keeps every instruction in place.

Parentheses are the curved symbols ( ). They keep an amount together and tell you to work inside that group first.

Same numbers, different instructions

Compare $3+2\times 5$ and $(3+2)\times 5$. Without parentheses, multiply first:

With parentheses, add the grouped numbers first:

The parentheses make the entire sum the amount multiplied by 5.

Keep the usual order inside

Parentheses do not mean “add first.” They mean “work out this group first.” Inside the group, multiplication and division still come before addition and subtraction.

For $(5+3\times 4)-6$, start with the multiplication inside:

Finish that group, then subtract:

Adding 5 and 3 first would ignore the multiplication inside.

After the group is finished

Evaluate $32-(3+2)\times 4$. Replace the parentheses with their value:

The multiplication still comes before the subtraction:

Do not switch to straight left-to-right calculation after a group disappears.

Multiply the whole amount

Four children each choose a $3 snack and a $2 drink. Each child spends $3+2=5$ dollars. The total is:

Writing $4\times 3+2$ would count four snacks but only one drink. The parentheses keep one child's complete purchase together.

Sometimes a multiplication sign is left out: $4(3+2)$ means $4\times (3+2)$. Writing the sign makes the instruction easier to see.

Two separate groups can appear side by side:

Finish both groups before dividing. Parentheses around an already-early product, such as $6+(2\times 5)$, simply make the grouping clear. Its value is the same as $6+2\times 5$.

A group can sit inside another group. These are nested groups. Start at the innermost group and work outward.

You may see parentheses ( ), brackets [ ], and braces { }. All three group an amount. Different shapes help you spot which opening and closing symbols belong together.

Work outward one group at a time

Evaluate $3\times [14-(2+5)]$. The sum sits inside both the parentheses and the brackets. It is the first group you can finish.

Next finish the brackets:

Do not start with $3\times 14$. The multiplier acts on the entire bracketed amount.

Three layers

Now evaluate $\{50-[4\times (3+5)]\}÷3$.

1. The innermost sum is $3+5=8$. Replace it to get $\{50-[4\times 8]\}÷3$.
2. Finish the brackets: $4\times 8=32$. You now have $\{50-32\}÷3$.
3. Finish the braces: $50-32=18$. You now have $18÷3$.
4. Divide to get 6.

Each group disappears after its inside becomes one number.

Keep the order inside each layer

In $[24-3\times (2+4)]$, first replace the inner sum:

Now multiply inside the brackets:

Doing $24-3$ next would give 21, then 126. The outer group still contains more than one operation, so its multiplication comes first.

Match the pairs

Every opening symbol needs a closing symbol of the same shape. Follow a pair from its opening to its matching closing symbol. Everything between them belongs to that group.

You may see only round parentheses, as on many calculators:

The inner sum gives 7. The outer group becomes $7\times 3=21$. Then subtract 4 to get 17. The position of a group decides when it is ready, even when every symbol has the same shape.

Adding parentheses can change what an expression tells you to do. Try the new grouping and calculate its value to see whether it works.

Change a value

Without parentheses, $7+3\times 4$ equals 19. Grouping the sum gives:

Grouping the product gives $7+(3\times 4)=19$. That grouping keeps the same value because the multiplication already came first.

Suppose you want $18-6÷3$ to have a value of 4. Without parentheses, its value is 16. Try grouping the subtraction:

You kept the numbers and signs in their original order. The new parentheses moved the subtraction ahead of division.

Regroup for mental math

The associative property lets you regroup addition or regroup multiplication without changing the result:

Choose a group that is easy to work out. This rule does not carry over to subtraction or division. For example, $(18-7)-3=8$, but $18-(7-3)=14$.

Division can change too: $(32÷8)÷2=2$, but $32÷(8÷2)=8$.

Multiply every part

The distributive property gives another way to multiply a sum:

Each of the seven groups contains a ten and a four. You need seven of each part. Multiplying only the 10 would give $70+4=74$ and miss six fours.

This works with subtraction too. For $6\times 99$, think of 99 as $100-1$:

Share a total in parts

You can divide parts of a total by the same number:

You share all 42 items among three groups. But you cannot split the number you divide by this way. In $30÷(3+2)$, you divide by 5 and get 6. Dividing 30 by 3 and by 2, then adding, gives $10+15=25$.

To square a number, multiply it by itself. A small raised 2 tells you to do this. You read $6^2$ as “six squared.”

The raised 2 is an exponent. It does not mean “multiply by 2.” Six doubled is 12, but six squared is 36.

Picture a square

Imagine a square made from six rows of six unit squares. It contains 36 unit squares. That is why numbers made by squaring whole numbers are called square numbers.

The squares from one through twelve are:

You can rebuild any one by multiplying its side length by itself.

Fit squares into the order

Work out groups first, then squares, then multiplication and division, then addition and subtraction.

For $5+3^2\times 4$, first square the 3:

Then multiply, and finally add:

Squaring comes before subtraction too: $14-3^2=14-9=5$.

Which amount is squared?

An exponent acts on the number just before it. After a closing parenthesis, it acts on the whole group.

Compare $2\times 4^2$ with $(2\times 4{)}^2$. In the first, square 4 to get 16, then multiply by 2 to get 32. In the second, finish the group to get 8, then square 8 to get 64.

The same care matters with a sum:

Squaring the separate numbers would give $2^2+5^2=4+25=29$. That is a different expression. A square after a group applies to the group's complete value.

Use squares for area

A square mat has sides 5 feet long. Its area is $5^2=25$ square feet. Two such mats and a 6-square-foot strip cover:

The answer is 56 square feet. The 2 counts the mats; the raised 2 tells you how to find each mat's area.

A long expression gets easier when you shorten one part at a time. Copy everything else exactly, including signs and grouping symbols.

Use four passes: groups, squares, multiplication and division, then addition and subtraction. Within the last two pairs, move from left to right.

Follow a complete calculation

Evaluate $64-(2+4{)}^2+18÷3\times 2$.

First finish the group:

Next square the 6:

Division comes before the multiplication here because it is on the left:

Finally subtract and add from left to right:

Notice how the 64 and its subtraction sign stayed in place until needed. Each line names the same value, 40.

What an equals sign says

The sign $=$ means “has the same value as.” It does not mean “and then I did this.” You can join steps only when the amounts on both sides are equal.

For example, $3\times 7+5=21+5=26$ is a true chain. Every part has a value of 26.

Writing $3\times 7=21+5=26$ is wrong. The first expression equals 21, while the next equals 26. This is a run-on equals sign. Keep the untouched $+5$ from the start, or write separate calculations with words between them.

Decimals follow the same order

For $1.2+3\times 0.2$, multiply first:

A decimal point changes a number's value, not its place in the order.

Find a slip

If your final answer seems wrong, compare neighboring lines. Check the part you calculated, then the parts you copied. A dropped number, changed sign, or missing parenthesis can change the whole expression. Keep a grouping pair until its inside has become one number.

Math words help you choose operation signs. A sum is the result of addition. A difference comes from subtraction. A product comes from multiplication, and a quotient comes from division.

“The product of 7 and 8” means $7\times 8$. “The quotient of 45 and 5” means $45÷5$.

Watch the starting number

The order of words can differ from the order of numbers you write. “Subtract 7 from 19” starts with 19 and takes away 7:

“Seven less than 19” means the same thing. Writing $7-19$ would reverse the action. Read “from” as a clue to the amount you start with.

Keep a result together

“Add 6 and 9, then multiply by 4” asks you to use the whole sum in the next step:

The parentheses make addition happen first. Without them, $6+9\times 4$ multiplies just the 9 by 4.

“Four times the sum of 6 and 9” describes the same calculation. “Three times the difference of 16 and 5” becomes $3\times (16-5)$.

A phrase does not always need extra grouping. “Multiply 7 by 6, then add 4” is $7\times 6+4$. Multiplication already happens first.

Build from the inside

For a longer instruction, write the first amount, then build around it. Try “Subtract 3 from 11, multiply the result by 5, then add 2.”

1. Start with the difference: $11-3$.
2. Multiply that whole result: $(11-3)\times 5$.
3. Add the last amount: $(11-3)\times 5+2$.

To check, read your expression back. It tells you to subtract, multiply the difference, then add. Its value is $8\times 5+2=42$.

Read two similar expressions

$6\times (14-4)$ says “six times the difference of 14 and 4.” You subtract first.

$6\times 14-4$ says “multiply 6 by 14, then subtract 4.” Here only 4 is taken from the product. Parentheses show whether the subtraction belongs inside the amount being multiplied.

You can learn something about an expression without finding its exact value. Look for its last operation. That shows how its main amounts join together.

One whole amount multiplied

In $3\times (\mathrm{18,932}+921)$, the sum is one grouped amount. The last operation multiplies that entire amount by 3. So the expression is three times as large as $\mathrm{18,932}+921$. You do not need to calculate the sum to know this.

Picture three identical boxes. Each box holds the same combined collection. Three boxes have three times one box's contents, whatever that count is.

A sum or a product?

In $8+5\times 7$, multiplication happens first and addition comes last. You can call it “8 plus the product of 5 and 7.” The whole expression is a sum.

In $(8+5)\times 7$, addition happens inside the group and multiplication comes last. The expression is a product: seven times the sum of 8 and 5.

Calling the second expression “8 plus five sevens” would leave the 8 outside the multiplication. That describes the first expression instead.

Which amount is halved?

$(\mathrm{1,400}+260)÷2$ means half of the combined amount. Both parts are inside the group.

$\mathrm{1,400}+260÷2$ means 1,400 plus half of 260. Only 260 is divided by 2. The first expression shares the whole total; the second shares just one part.

Compare without finding totals

Compare $4\times (300+17)$ with $4\times (300+71)$. Both multiply a sum by 4. The second sum is larger because 71 is larger than 17. Multiplying both positive sums by 4 keeps the second result larger.

Structure can also reveal equal expressions:

Twelve groups of 19 plus one more group make thirteen groups. You can explain the equality without finding either product.

Reading the structure also helps you check size. The sum $\mathrm{18,932}+921$ is close to 20,000. Three times that is close to 60,000. A much smaller calculator result would be a reason to check the entry.

Start by finding what the question asks. Then decide what each amount and action means. Write one expression whose steps match the story.

Packs, then a gift

Nora has four packs of nine stickers. She gives seven stickers away. How many are left?

The four packs give $4\times 9$ stickers. Giving seven away means subtracting 7:

Nora has 29 stickers left. The sentence tells you what the number 29 counts.

Take away the whole cost

Jules buys three cards at $4 each and one stamp for $2. Jules pays with $20. How much change comes back?

The cost in dollars is $3\times 4+2$. Put that whole amount in parentheses before subtracting it from the payment:

Inside the group, multiply and add:

Jules gets $6 in change. Without the group, $20-3\times 4+2$ would add the stamp price back. It would give $10, even though the stamp costs money.

Share the complete total

Five children share three trays of ten crackers and five loose crackers. The total is $3\times 10+5$. Every cracker is shared, so group that total:

Each child gets seven crackers. Check backward: five children with seven crackers each have 35 altogether.

You could also share each part separately: $3\times 10÷5+5÷5=6+1=7$. Both expressions match the same sharing story.

Compare two collections

One team has six boxes of 18 cans. Another has five boxes of 23 cans. The second team has more. Its extra cans are:

The multiplications already come before subtraction. No extra parentheses are needed.

Include an area

A square rug has sides 3 feet long. A rectangular rug is 3 feet by 2 feet. Their total area is:

Together they cover 15 square feet. Read your expression against the story: one part finds each rug's area, and the final addition combines them.

Calculators do not all handle mixed operations in the same way. Test yours before using it to check a whole expression.

Try a short test

Enter $2+3\times 4$ and press equals. A calculator following the order of operations gives 14. A calculator working straight through each key press gives 20: it adds 2 and 3 before multiplying by 4.

A scientific calculator follows operation order. Many simple four-function calculators work step by step. A calculator app can use either approach, so try the test there too.

On a step-by-step calculator, work out the early jobs separately. For $9+4\times 6$, calculate $4\times 6=24$ first. Write 24 down, then calculate $9+24=33$.

Enter every group

On a calculator with parentheses, enter the groups in the expression. Use round parentheses for brackets and braces too.

For $3\times [14-(2+5)]$, the displayed entry should read $3\times (14-(2+5))$. The inner sum is still inside the outer subtraction. Its value is 21.

Match every opening parenthesis with a closing one. Some calculators supply a missing closing parenthesis at the end. That may close a group in the wrong place. Check the display before pressing equals.

A fraction bar groups two amounts

A fraction bar groups everything above it and everything below it. For example:

To enter this with a division sign, write:

Entering $25+11÷9-3$ would divide only 11 by 9. It would not divide the complete top by the complete bottom.

Check what gets squared

The expressions $3+4^2$ and $(3+4{)}^2$ have values of 19 and 49. In the first, only 4 is squared. In the second, the entire sum is squared.

Use the square key and inspect which amount the display shows under the square. To square a whole sum, include its parentheses. You can also calculate the sum first, then multiply that result by itself. For the second expression, find 7 and then calculate $7\times 7=49$.

Before accepting a result, compare the screen with your written expression: numbers, signs, groups, and squares all need to match.

An estimate is a quick value close to the exact answer. Use easy nearby numbers to judge the size of a result.

Keep the same operations and groups when you estimate. Rounding should change the numbers, not the calculation's meaning.

Estimate before calculating

For $39\times 22+198÷2$, use 40, 20, and 200:

Multiply and divide before adding:

The exact result is $858+99=957$. It is reasonably close to 900. A result of 9,570 would suggest an extra digit or a misplaced decimal point.

Do not change the order

Estimate $298+19\times 4$ with $300+20\times 4=380$. Grouping the rounded addition would give $(300+20)\times 4=\mathrm{1,280}$. That estimate multiplies the 300 too, so it checks a different expression.

The exact answer is $298+76=374$, close to 380.

When answers disagree

After working by hand, enter the expression on a calculator that follows operation order. If the answers differ, retype it carefully. Then compare your written steps one line at a time.

A wrong value can point toward the mistake. For $26-3\times 5$, a result of 115 comes from subtracting first, then multiplying: $23\times 5=115$. The correct calculation is $26-15=11$.

An estimate catches large errors quickly. A close answer can still contain a small mistake, so check the exact work too.

Check the story backward

Suppose shopping costs $17 and you pay $25. Your change is $8. Add the cost and change:

This rebuilds the payment. For ordinary shopping paid with a bill, change cannot exceed the bill's value. An answer of $28 in change would not fit this purchase.

For equal sharing, multiply each person's share by the number of people. If six friends each get eight beads, they have $6\times 8=48$ beads altogether. Compare that with the starting total.