math problems written on paper with pencil
Mayukh Saha
Mayukh Saha
December 9, 2021 ·  5 min read

How to Do Math Problems With Parentheses

Even simple math problems with one-digit numbers can seem very confusing when there is more than one action involved. Should you do the addition first or the multiplication? Depending on which is done first, the answer is bound to be different too. Add to that parentheses, and you get a math problem that seemingly makes you want to pull your hair.

But actually, it is not that tough at all if you know some very simple rules when it comes to problems that have parentheses. All you have to do if follow the order of operations.

numbers 123456789 with small coloured plastic cubes beneath each number
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There is a simple acronym to help you decide the order of operations. It is known as PEMDAS. It stands for Parentheses Exponents Multiplication Division Addition Subtraction. Check an equation and find out all the operators that are present in it. Then, solve them according to the order in PEMDAS. In the following sections, we will simplify it even more for you. It may also be knows as BEDMAS if you live in Canada or New Zealand.

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For Simple Math Problems

First off, solve every exponent or root that is there. Keep in mind that every action is done from left to right before you start solving. Then do the multiplications and divisions first. After that, do the additions and subtractions.

So just to summarize: multiplication/division first, addition/subtraction second. That is the MDAS part of PEMDAS.

Let’s try to solve this problem: 5 x 4 – 8 ÷ 2.

So first we will do 5 x 4 = 20 and 8 ÷ 2 = 4.

Then we will subtract* 20 – 4 to get the final answer.

The correct answer is 16.

*Regardless of the subtraction is before the division, it will be done first.

Finally, you do not need to insert parentheses for math problems that only have one kind of action. For example in the case of 4 -2 + 3, there is only addition and subtraction so it doesn’t need parenthesis. There is also no need for parentheses if the actions are already in order. For example, in 4 x 3 + 5, the multiplication will take place first anyway.

For Problems With Parentheses

There are a few more rules that get added when a problem has parentheses in it. Parentheses can change the normal order in which a problem is solved. Because you must always solve what is inside the parenthesis first before you solve anything else in the math problem. That is, you already know about MDAS part of PEMDAS. Here is how it works when there is (P)arentheses present as well:

First, solve everything inside the parenthesis. After that, the rest is the same as in the previous case: multiplication/division first, addition/subtraction second (MDAS). However, if the math problem has exponents or fractions, then their values should be found first before everything else. But this only if it is possible at all to calculate their values.

Here is an example to give you the hang of how parentheses work.

Let’s try to solve: 8 – 2 x (15 – 4 x 3) + (7 + 3 x 2)

First, we begin by solving the parentheses. So in the first parentheses we have:

15 – 4 x 3 = 15 – 12 = 3 (following the rule of multiplication/division first, and addition/subtraction second).

Similarly in the second parentheses, we have: 7 + 3 x 2 = 7 + 6 = 13

When we replace the parentheses with these values in the original problem we get: 8 – 2 x 3 + 13

So now, we have to follow the multiplication first rule again. Following that, we get: 8 – 6 + 13 = 15.

The correct answer is 15.

If there are more parentheses inside a parenthesis then don’t worry! The rules remain the same. Start by solving the innermost parentheses first and slowly work your way outwards.

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Common Mistakes When Solving Parenthetical Equations

Sometimes, there is no symbol between a number and a parenthesis. This means that there is a multiplication symbol there. For example in 8 ÷ 4 (3 – 1), it actually means 8 ÷ 4 x (3 – 1). See if you can figure out what the answer is to this on your own. It should be 4. Here is the working:

Let’s try to solve: 8 ÷ 4(3 – 1)

So first off, according to the rule, we will do the parentheses. if we do that, we get 3 – 1 = 2

Then the equation becomes 8 ÷ 4 (2).

Remember that if there are no symbols between a number and parentheses, then that means there is a multiplication sign there.

So 8 ÷ 4 (2) can also be written as 8 ÷ 4 x 2.

The rest should be easy. Since there is no special hierarchy between multiplication and division, we only need to do the actions from left to right. So…

8 ÷ 4 x 2
= 2 x 2
= 4

The correct answer is 4.

Note: If you punch this into a calculator as 8 ÷ 4(3 – 1), you may get the wrong answer of 1. However if you punch it in as 8 ÷ 4 x (3 – 1), you should get 4.

Another mistake that often happens is when there is a minus symbol before the parentheses. Here, remember that all minus and plus inside the parentheses will become their opposite if there is a minus before the parentheses when you remove the parentheses. Ok, it might sound too complicated, so here is how it works:

Take this problem for example: 6 + 5 – (4 + 3 – 2)

There are two ways you can approach this, both are correct and both approaches will give you the same answer.

Either solve what’s inside the parentheses first. So we solve 4 + 3 – 2 = 5. This simplifies the original equation to 6 + 5 – 5 = 6. This is the recommended method as it follows PEMDAS.

Or, you can directly remove the parentheses and rewrite the math problem. As we remove the parentheses, notice how all the plus and minuses became their opposite. So, we have:

6 + 5 – (4 + 3 – 2). If we remove the parentheses we get 6 + 5 – 4 – 3 + 2. This is just simple addition and subtraction, and the result is 6. This method works because there addition and subtraction have the same hierarchy.

The correct answer is 6.

This rule is especially important in algebra where there is an unknown number ‘y’ and you have to solve the equation. For example:

Let’s try to solve: 3 + 2(y + 1) – 2(y – 1).

We don’t know the value of ‘y’ so we cannot carry out the actions within the parentheses first. So we move on to the next step and remove the parentheses. To make it simpler you can write the equation as:

3 + 2 x (y + 1) – 2 x (y – 1)

You can open each parenthesis separately to make it easier. First off, we have:

+ 2 x (y + 1)
= (+2 x y) + (+2 x 1)
= + 2y + 2

Be careful with the second one though. Here is how that works out:

– 2 x (y – 1)
= (-2 x y) – (-2 x 1).
= -2y + – (-2)
= -2y + 2

So the equation becomes 3 + 2y + 2 – 2y + 2 = 7.

The correct answer is 7.

Finally, not all calculators follow this rule, but some do. To know if your one can solve problems according to this rule, type in 1 + 5 x 7. If it returns 42, then your calculator cannot solve such math problems.

We hope this short math lesson helps!  

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