This article is written considering that the students (learners) are in grade five or higher and they have very good knowledge of operations with integers. It is also considered that readers have good knowledge of basic math operations of addition, subtraction and multiplication. After reading this article grade 5 or higher students should be able to do the following:
Order of operations involving multiplication, addition and subtraction.
Order of operations involving open brackets, multiplication, addition and subtraction.
Preview:
Before we start the order of operations, I like to review the rules involving brackets (parenthesis). Following are the basic methods to solve the open brackets;
(3+7) = 10 2(3+7) = 2(10) = 20
Remember that if there is no sign between a bracket and a number; then multiply (times) the numbers outside and inside the bracket.
-3(7- 4) = -3(3) = - 9 -3(-7+4) = -3(-3) = 9
If you have problems understanding the above examples; read my articles on integers. That will help you to understand all operations on integers and above examples.
Basic order of operations:
At very basic (grade 5) levels, students need to understand how to solve problems involving two operations at a time. Two operations can be any combinations of four basic operations, such as addition and subtraction or addition and multiplication.
1. Problems involving addition and subtraction:
If you see only positive and negative signs in the problem, I highly recommend that add all the positive numbers to get one positive number. Same way add all the negative numbers to get one negative number.
Now you have two numbers with opposite signs, subtract them and write the bigger number sign in front of the answer and you done.
Let's do the following example to understand the above concepts.
9 - 11 - 5 + 7 - 20 + 17
We have three positive numbers; 9, 7 and 17. Add these numbers together to get 33.
Also, there are three negative numbers; - 11, -5 and - 20. Add these numbers together to get - 36.
This way we have converted six numbers into two numbers with opposite signs. You already know that two numbers with opposite signs are subtracted, so subtract 33 from 36 to get 3 and 3 will take the negative sign which is the bigger number's sign. The given problem turns to be as given below;
9 -11 - 5 + 7 - 20 +17 = 33 - 36 = -3
2. Problems involving addition and multiplication or subtraction and multiplication or all three.
If there is multiplication symbol involves together with positive or negative signs in the given question. Always solve the multiplication before addition and subtraction. Consider the following example to understand this.
2 + 3 x 5
Never ever start adding 2 and 3 in this problem. Multiplication has the priority over add and subtract, therefore multiple 3 and 5 to get 15 first, then add 2 and 15 to get 17 as the answer. The solution is shown below;
2 + 3 x 5 = 2 + 15 = 17
Let's do the following example problems to clear the above concepts further.
- 6 + 4 x 5
12 - 3 x 7 - 1
9 - 2 x 16 + 50 - 33
- 21 - 20 x (- 3) + 5 x 7
8 x 2 x 3 + 12(-2) - 3(-6)
Solutions:>
1.
There are three signs involving in this problem, the negative, positive and multiply (times). Multiplication has the priority over addition and subtraction. Therefore, do 4 times 5 to get 20.
Now, you have two numbers - 6 and 20. As these numbers have opposite signs, subtract them to get 14 and number 14 is positive because bigger number 20 is positive. Mathematically, the problem is solved as below;
- 6 + 4 x 5
= - 6 + 20
= 14
2.
As multiplication has the priority over add and subtract, solve all the multiplication signs first of all. In the given problem - 3 is getting multiplied with 7. Solve - 3 x 7 to get -21. After this we get three numbers in the problem, 12, - 21 and -1.
Add both the negative numbers -21 and -1 to get -22. Now 12 and -22 have the opposite signs, subtract them to get - 10 as your answer. Mathematically;
12 - 3 x 7 - 1
= 12 - 21 -1
= 12 - 22
= -10
3.
In first step, solve - 2 x 16 to get - 32. Now there are four numbers in the question, 9, - 32, 50, -33. Add both the positive numbers 9 and 50 to get 59 and both negative numbers - 32 and - 33 to get - 65.
So, 59 and -65, with opposite signs, need to be subtracted to get -6 as answer to the problem. Mathematically;
9 - 2 x 16 + 50 - 33
= 9 - 32 + 50 - 33
= 59 - 65
= -6
4.
In this problem, there are two multiply signs. Solve both of multiplications first, do - 20 x (- 3) to get +60 (Note that,60 is not front of the problem but in the middle, so we have to include the plus sign with 60). Also do, + 5 x 7 to get +35 (as 35 is not in the beginning of the problem, so use the plus sign with 35)
Now we end up with three numbers, - 21, +60 and + 35. Add both the positive numbers 60 and 35 to get +95. Next, - 21 and +95 have opposite signs, therefore subtract them to get 74. Notice here, the answer is +74 but we need not to show the plus sign with our final answer. Always write the plus sign in front of the number which is not at the starting of the problem as we did in case of 60 and 35 above. Mathematically;
- 21 - 20 x (- 3) + 5 x 7
= - 21 + 60 + 35
= - 21 + 95
= 74
5.
Let's do this problem step by step with explanations.
8 x 2 x 3 + 12(-2) - 3(-6)
Multiply 8, 2 and 3 to get 48. Note that, there is just a bracket between +12 and - 2 and between - 3 and - 6, which means multiply as well.
Therefore multiply +12 and - 2 to get - 24 and multiply - 3 and - 6 to get +18 as shown below;
8 x 2 x 3 + 12(-2) - 3(-6)
= 48 - 24 +18
Now, we got three numbers with plus and minus signs only.
Add the positive numbers, 48 and 18 to get 66. After this we have 66 and - 24 with opposite signs, subtract them to get 42 as the answer as shown below;
8 x 2 x 3 + 12(-2) - 3(-6)
= 48 - 24 +18
= 66 - 24
= 42
This is very much the coverage of the basic order of operations. Keep in mind, I wrote this article considering that the reader have sound knowledge of the integers and operations with them. If you want to learn integers, read my previous articles, explaining integers and operations on integers.
The summary of today's lesson is;
If there are only positive and negative signs in the problem, add all the positive numbers to get one positive number and add all the negative numbers to get one negative number. Now subtract the two numbers you obtained above to get the answer.
Multiplications have the priority over addition and subtraction. Never ever add or subtract the numbers until there are multiplication signs in the problem. Always multiply the numbers before you add or subtract.
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