Course:
Mathematics Form 1
Imagine this:
Study hard 🧠 ➡️ Take the Junior Certificate Exams 📚 ➡️ Pass with flying colors 🎉
Now, picture taking the exams without preparing. The outcome? Not so great! But if you put in the effort and study hard, your chances of success skyrocket. Just like in life, the order in which we do things can make all the difference.
In this lesson, you’re about to discover the magic behind the order of operations—the golden rule that ensures every math problem is solved correctly, every time. From solving mixed operations to creating your own, you’ll unlock the secrets of mastering math one step at a time!
By the end of this lesson, you’ll be able to:
Test your skills with our Self-Assessment Exercise #3. It’s time to put your knowledge to the test and see how much you’ve mastered!
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In Lesson 2, you learned about the four basic operations, namely, addition, subtraction, multiplication and division in whole numbers (especially in directed numbers).
So far you used the operations to combine only two numbers. For example,
5+2= 7
10 – 6 = 4
3 x 4 = 12
18 ÷ 6 = 3
Sometimes an operation may be written more than once in an expression, that is combining more than two numbers. For example, 3 + 4 + 2.
This is the kind of situation we are going to look at in this lesson, for each of the operations.
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Let us start with an example,
Example 1📝
Work out 3 + 4 + 2
Solution: ✅
This may be worked out in two ways as shown below:
(i) If we want to do the first addition first, we put brackets around 3 + 4 so it becomes,
(3 + 4) + 2 = 7 + 2 = 9
(ii) If we want to do the second addition first, we put brackets around 4 + 2 so it becomes
3+ (4 + 2) = 3 + 6 = 9
Since (3 + 4) + 2 = 3 + (4 + 2), this means that when we have two additions to perform, the order in which we do the addition does not matter. So when there is only addition, you may work out from left to right or from right to left.
For example, 📝
(a) 6 + 10 + 1 = (6 + 10) + 1
= 16 + 1
= 17
(b) 2 + 3 + 5 + 10 = (2 + 3) + 5 + 10
= (5 + 5) + 10
= 10 + 10
= 20
Remember to do what is inside the bracket first.
Example 2 📝
Work out 17 – 6 – 5
Solution: ✅
(i) Working the first subtraction first
(17 – 6) – 5 = 11 – 5 = 6
(ii) Working the second subtraction first
17 – (6 – 5) = 17 – 1 = 16
The answers in this case are different. This means that when we have two subtractions, the order in which we perform the operation matters. So in working out 17 – 6 – 5, we do the subtractions in the order they are written, that is, you must work from left to right.
Therefore,
17 – 6 – 5 = (17 – 6) – 5
= 11 – 5
=6
Like in addition, you must work what is inside the brackets first.
For example, ✅
(a) 8 – 5 – 2 = (8 – 5) – 2
=3–2
=1
(b) 16 –10 – 2 = (16 –10) –2
=6–2
=4
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Consider the following example,
Example 3
Work out 6 x 2 x 4
(i) Working the first multiplication first
6 x 2 x 4 = (6 x 2) x 4
= 12 x 4
= 48
(ii) Working out the second multiplication first
6 x 2 x 4 = 6 x (2 x 4)
= 6×8
= 48
Like in addition, in multiplication the order does not matter.So when there is only multiplication, you may also work out from left to right or right to left.
For example,
(a) 3 x 7 x 2 = (3 x 7) x 2 OR 3 x 7 x 2 = 3 x (7×2)
= 21 x 2 = 3×14
= 42 = 42
(b) 4 x 2 x 5 = (4x 2) x 5 OR 4 x 2 x 5 = 4 x (2×5)
= 8x 5 = 4 x 10
= 40 = 40
Example 4
Work out 24 ÷ 4 ÷ 2
(i) Working the first division first
24 ÷ 4 ÷ 2 = (24 ÷ 4) ÷ 2
= 6÷2
=3
(ii) Working the second division first
24 ÷ 4 ÷ 2 = 24 ÷ (4 ÷ 2)
= 24÷2
= 12
The answers are different like in subtraction, therefore, the order of division does matter. So we must do the divisions in the written order.
For example,
(a) 18 ÷ 6 ÷ 3 = (18 ÷ 6) ÷ 3
= 3÷3
=1
(b) (36 ÷ 9) ÷ 2 = (36 ÷ 9) ÷ 2
= 4÷2
=2
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Having worked with each of the operations separately, we are now going to work with mixed operations. This is where there are different operations in one expression.
When there are mixed operations, the order in which the operations are done is very important. Otherwise, we obtain different answers, some of which do not make much sense. Let us do the following examples starting with addition and subtraction.
Example 5
Work out
(i) (10 – 3) + 5 = 7 + 5 = 12
(ii) 10 – (3 + 5) = 10 – 8 = 2
The answers above are different, therefore, when there is only addition and subtraction you must work from left to right or the written order. In the above example, case (i) is therefore, the correct working. Remember to work brackets first if there are any.
Study these examples carefully. Work out the following:
(a) 3 + 4 – 3 = (3 + 4) – 3
= 7–3
=4
(b) 6 – 5 + 2 = (6 – 5) + 2
= 1+2
=3
(c) 8 – 2 + 1 = (8 – 2) + 1
= 6+1
=7
Example 6
Work out:
(i) (40 ÷ 5) x 2 = 8 x 2 = 16
(ii) 40 ÷ (5 x 2) = 40 ÷ 10 = 4
The answers are not the same. Therefore, multiplication and division combined requires that you work in the written order or from left to right.
Further examples
(a) 6 x 3 ÷ 9 = (6 x 3) ÷ 9
= 18÷9
=2
(b) 12 ÷ 3 x 4 = (12 ÷ 3) x 4
= 4×4
= 16
Having worked with addition and subtraction combined and also multiplication and division combined, we have found out that in both cases you work in the written order. Now we are going to combine any two or more of the four basic operations.
The order of operation is as follows:
Brackets ———–>Multiplication / Division ———–>Addition ———–>Subtraction (which ever comes first)
Brackets – must always be worked first.
Multiplication and division – to be performed before Addition and Subtraction
(which ever comes first)
Addition – to be performed before subtraction
Subtraction – to be performed last.
Remember that you can workout calculations in the written order only when addition and subtraction are combined, and only when multiplication and division are combined.
Example 7
Work out the following:
(a) 10 x 3 + 8 = (10 x 3) + 8 Multiplication
= 30 + 8 Addition
=38
(b) 17 – 5 x 3 = 17 – (5 x 3) Multiplication
= 30 + 8 Subtraction
=2
(c) 15 – (8 + 13) = 15 – (8 + 13) Brackets first
= 15 – 21 Subtraction
= -6
(d) 9 + 16 ÷ 4. = 9 + (16 ÷ 4) Division
= 9+4 Addition
= 13
(e) 4 x (5 + 1) x 2 = 4 x (5 + 1) x 2 Brackets first
= (4 x 6) x 2 Multiplication
= 24 x 2 Multiplication
= 48
(f) 8 x 5 – 8 ÷2 + 7 = (8 x 5) – 8 ÷ 2 +7 Multiplication
= 40 – (8 ÷ 2) + 7 Division
= (40 – 4) + 7 Written order
= 36+7 Addition
= 43
(g) 7 + 4 x (6 – 3) = 7 + 4 x (6 – 3) Brackets first
= 7 + (4 x 3) Multiplication
= 7 + 12 Addition
= 19
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In self-defined operations, you are not given the operation to perform. Instead, symbols are used to represent an operation and you are supposed to investigate and find out what the operation is. Different symbols such as * and Ø may be used to represent an operation.
Example,
(a) If 10 * 2 = 5, what does * represent?
Since 10 ÷ 2 = 5, * therefore represent division.
(b) If a * b = 2a – b, calculate 4 * 3
In this case, a = 4 and b = 3
Therefore, 4 * 3 = (2 x 4) – 3 (substituting in 2a – b)
=8–3
=5
(c) If 2 * 3 = 7 and 3 * 3 = 9, what does * represent?
In this case, you have to find out how you may combine 2 and 3 to get 7 and in the same way combine 3 and 3 to get 9.
2 * 3; (2 x 2) + 3 = 4 + 3 = 7
3 * 3; (3 x 2) + 3 = 6 + 3 = 9
Therefore, * represents,
Multiply the first number by 2 and then add the second number.
(d) If Ø represents subtract the second number from twice the first number, work out
5 Ø 9 5 Ø 9 = (5 x 2) –9
= 10 – 9
=1
(e) If 6 * (1 x 3) = 2 and24* (-2 x 4) = -3
work out 12 * (4 x 3)
6 * (1 x 3) = 6*3 = 2
= 6 ÷ 3 = 2
24 * (-2 x 4) = 24 * -8 = -3
= 24 ÷ -8 = -3
Therefore, 12 * (4 x 3) = 12*12
= 12 ÷ 12
= 1
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In this lesson, you have learnt that:
In performing additions, subtractions, multiplications and divisions only, work from left to right or in the written order.
For example,
2 + 5 + 6 = 7 + 6 = 13
10 – 6 – 2 = 4 – 2 = 2
2 x 3 x 6 = 6 x 6 = 36
24 ÷ 8 ÷ 3 = 3 ÷ 3 = 1
When performing combined addition and subtraction, work from left to right.
For example,
8 + 6 – 3 = 14 – 3 = 11.
When performing combined multiplication and division, work from left to right.
For example,
24 ÷ 8 x 4 = 3 x 4 = 12.
The order of operations is as follows, Brackets, Multiplication, Division, Addition and Subtraction.
In self-defined operations, symbols are used to represent operations.
For example,
10 * 2 = 5, the * represents division.
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