# Parentheses and Order of Operations

We use parentheses when we want to do arithmetic operations in a different order than MDAS. This means that the main purpose of the parentheses is to allow you to add and subtract before you multiply and divide.

Rule

When you have parentheses in a problem, you calculate the expression inside the parentheses first as if it was its own problem. When you’ve done that, solve the problem according to MDAS.

Before you get to look at combined problems, I’ll teach you how parentheses work.

• Multiple numbers inside parentheses: Add the numbers to each other:

 $\left(2+3\right)=\left(5\right)=5$
• A number multiplied by parentheses with numbers inside: Solve the expression inside the parentheses, then multiply by the number outside:

 $3\left(4+2\right)=3\left(6\right)=18$
• Parentheses multiplied by parentheses: Solve the expression inside each of the parentheses, then multiply the remaining numbers with each other:

 $\left(3+6\right)\left(5+1\right)=\left(9\right)\left(6\right)=54$
• A number multiplied by parentheses multiplied by another parentheses: Solve the expression inside of each of the parentheses, then multiply the remaining numbers by each other:

 $2\left(4+1\right)\left(7+3\right)=2\left(5\right)\left(10\right)=100$

What do you think the calculations would have looked like if there were no parentheses?

Compare the following calculations with the calculations above:

• In this case, you can see that the answers are the same. These parentheses don’t make any difference:

 $2+3=5$
• In this case, you can see that the answers are different. The parentheses changed the order of operations:

 $3\cdot 4+2=12+2=14\ne 18$
• In this case, you can see that the answers are different. The parentheses changed the order of operations:

$\begin{array}{llll}\hfill 3+6\cdot 5+1& =3+30+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =34\ne 54\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $3+6\cdot 5+1=3+30+1=34\ne 54$

• In this case, you can see that the answers are different. The parentheses changed the order of operations:

$\begin{array}{llll}\hfill 2\cdot 4+1\cdot 7+3& =8+7+3\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =18\ne 100\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $2\cdot 4+1\cdot 7+3=8+7+3=18\ne 100$