Writing GCSE Maths Calculations

Introduction

These notes are about:

To keep this as simple as possible there are no examples of calculations containing algebra. All the examples are numerical. However, the same basic methods apply both to numeric and algebraic calculations.

To read or to write a calculation correctly you must be confident using the rules for the order of operations, sometimes called BODMAS.

In a maths question you may be given a calculation in the form of an expression or an equation. For example:

Evaluate 3 × (25 − 2)

Clearly, to complete the calculation you have to interpret the maths correctly.

3 × (25 − 2) = 3 × (32 − 2) = 3 × (30) = 90

If you are allowed to use a calculator for the above question then just type the expression directly into the calculator. If your calculator has a xy button then use it for 25.

Type these keys:

3×(2xy5−2)=

In another question you may have to write out an expression for yourself. For example:

write a calculation for 20 as a percentage of 50

You can write that as:

$${20 \over 50}$$ × 100  or  20 ÷ 50 × 100

Of course, there are other ways to write the calculation. The standard method, shown above, is to first write the 20 as a fraction of 50. Then multiply by 100 to convert the fraction to a percentage.

* * *

Method Marks

Method marks are for showing the working. In a calculation, the best way to show the working is to write the calculation using standard maths. That may seem obvious but may need some explanation.

In a typical GCSE calculation worth 2 marks there is 1 mark for the method and 1 mark for the answer.

Example: calculate 20% of £60

All you need to write for the method mark is a correct numerical expression, such as:

60 ÷ 100 × 20  or  20 × 60 ÷ 100

Question: calculate 20% of £60 (2 marks)

Method: 60 ÷ 100 × 20 (1 mark)

Answer: £12 (1 mark)

In practice the answer is written something like:

20% of £60 = 60 ÷ 100 × 20 = 0.6 × 20 = £12 (2 marks)

For an explanation of the above method, point at the More Info link below.
More Info The calculation 60 ÷ 100 × 20 is made up of two separate calculations.

The £60 is divided by 100 to work out 1% of 60: 1% of £60 = 60 ÷ 100
60 ÷ 100 is multiplied by 20 to give 20% of 60: 20% of £60 = 60 ÷ 100 × 20

There are a few benefits of writing the method as a single calculation - the most obvious is that it takes less time and needs less explanation.

Writing just the underlined parts in the above would get the 2 marks.

Include the working out, that's the 0.6 × 20 bit in the example, only if you wish to. Most people need to write some working out to do the calculation without a calculator.

If a calculator is allowed then you should still write the method but would not need to show any working out. You just type the calculation, the 60 ÷ 100 × 20, into your calculator.

For practice using your calculator in similar examples go to Calculations, Calculators and BODMAS.

You may prefer to write the calculation in fraction format, as a numerator over denominator, such as:

20% of £60 = $${{60 \times 20}\over 100}$$ = $${12\bcancel{00}\over 1\bcancel{00}}$$ = £12 (2 marks)

If you leave out the method you may still get 2 marks for the correct answer £12. This is because, in some questions, a correct final answer is evidence that you used the correct method. Unfortunately, if you make a simple mistake, maybe write the answer as 120 or 1.2, then you get no marks. Hence the importance of writing the method. With a correct method, even if you do not write a correct final answer, you still get 1 mark.

The message here is do not risk it. Always write the method.

There are other ways of writing the method. Some prefer to do this type of calculation in two steps, first calculating 1% and then multiplying by 20 for 20%. That requires more writing and takes longer. For more detail on percentage questions, see:

Calculating a Fraction Of
Calculating a Percentage Of


Contents

We start with some basic revision on the order of operations and using BODMAS. Alternative words for BODMAS include BIDMAS, BEDMAS and PEMDAS. As you will see later, it does not matter whether you write the D for division or the M for multiplication first. BODMAS could be written BOMDAS.

We include some simple examples and questions that practise writing the method in calculations. All the examples are numerical, for example, 2×3 + 5. The same approach applies to writing algebraic expressions, for example 2x + 5.

There are links to videos, online tests and some revision material. The links are collected together at the end for convenience, under the heading External Links.

A student with basic numeracy skills and confidence in using BODMAS, will not need to work through all this content. For example, the revision on addition and multiplication, in the first topic below, is very basic. You could try the online tests and then decide what you need to revise.


Revision - The Order of Operations in a Calculation - BODMAS

The rules for the order of mathematical operations are summarised using a memory aid, such as BODMAS. The four main operations are Addition, Subtraction, Multiplication and Division, the A, S, M and D in BODMAS. We'll concentrate, for the moment, on addition and multiplication and include a few examples of how to use BODMAS.

Addition and Multiplication

Let's start with addition. A simple calculation: add 2, 3 and 4. You write that as 2+3+4.

The order of operations means the order in which you carry out a calculation. You can do addition in the order in which it is written, working from left to right. The addition 2+3+4 means start with 2, add 3 and then add 4 to give 9. Of course, for addition you can add the numbers in any order - you always get the answer 9. You do not have to follow the written order, although you may prefer to.

Now for multiplication. Multiply 2, 3 and 4. You write that as 2×3×4. As with addition, you can start the calculation with the 2 and multiply by 3 to give 6. Then multiply the 6 by 4. That gives 24. You get the same answer if you use a different order.

Provided the calculation is addition only or multiplication only, the order does not matter. This is called the commutative law. You can write the calculation in any order and you can do the calculation in any order.
More Info The above is slightly simplified. Both the commutative law and the associative law together show that the order does not matter. For addition, the commutative law states that:
a+b = b+a. For example, 2+3 = 3+2.

For multiplication the commutative law states that:
a×b = b×a. For example, 2×3 = 3×2.

If you multiply just two numbers, a and b, the commutative law states that the order does not matter. For more numbers, such as 2×3×4, you also need the associative law to show the written order does not matter. For GCSE maths the important thing is that you know the result - order does not matter. Those laws are mentioned here just for completeness. This page refers to the commutative law a few times, as a shorthand way of saying "order does not matter".

* * *

Addition and Multiplication Combined

Take care with calculations such as 2+3×4. There is a mix of two different operations. They are traps for the unwary. Add the 2 and the 3. What could be more simple? That gives 5. What next? Easy. Just multiply the 5 by 4. Answer: 20. Gotcha! What went wrong?

The cruel thing is that you simply followed the written order - that was the fatal error. The main message here is:

"You Cannot Do ALL Calculations in the Order that they are Written"

BODMAS

If a calculation contains a mix of addition and multiplication, as in 2+3×4, the rule is that you must always complete the multiplication before the addition. Calculate the 3×4 first. That gives 12. Add the 12 to 2. The answer is 2+3×4 = 2+12 = 14.

This rule is shown by writing Multiplication before Addition in BODMAS.
More Info If the calculation is a mix of operations then doing the calculation in the written order may give a wrong answer. In the calculation 2×3+4 there are two different operations, multiplication and addition. This calculation can be done in the written order, starting with 2×3 to give 6. Then add 4 to the 6 to give 10. That is written: 2×3+4 = 6+4 = 10.

The calculation cannot be done in any order. The rule is that, where you have a mix of addition and multiplication, you must calculate the multiplication part first.

The calculation 2×3+4 can be written as 4+2×3. But, if you start with the 4 and do the calculation in the written order, you get 4+2×3 = 6×3 = 18. That is wrong. Calculate the multiplication first and you get 10 - the correct answer. That is written:  4+2×3 = 4+6 = 10.


Revision - Brackets in Calculations - BODMAS

Brackets have a special meaning in the language of maths. Brackets are an instruction that you include in a calculation to change the order of a calculation. Here's an addition without brackets: 2+3+4. That means first add 3 to 2 and then add 4 to the result. Of course, you already know that you can add those numbers in any order. The result is the same. So, in the case of addition you can ignore the written order and choose your own order.

Video: Commutative Law of Addition. Click here.

Suppose you wish to specify a different order. You can use brackets as in 2+(3+4). This says "first add the numbers in brackets, the 3 and the 4. That gives 7. Then add 7 to the 2". That may seem to be a pointless example - because, in the case of addition the order does not matter.

Brackets are normally used only when the order does matter. For example, when the calculation contains both addition and multiplication. In the calculation 2×(3+4), the brackets mean first calculate the 3+4 to give 7. You then multiply the 2 by 7 to give 14.

Didn't we say that multiplication should be done first? Correct - multiplication is ahead of addition but brackets are top of the order. Brackets are always calculated first. If there is an addition inside brackets and a multiplication outside the brackets, the addition jumps ahead of the multiplication. In the calculation 2×(3+4), the 2 is outside the brackets. Only the term, the 3+4, in brackets is calculated first.

Try this one:
5×(4+2×3) = 5×(4+6) = 5×10 = 50

That may need some explanation. The expression 4+2×3 in brackets is calculated first. It contains an addition and a multiplication. Calculate the multiplication first to give 2×3=6. You now have (4+6) in brackets. Calculate the bracket and then multiply by 5.

The BODMAS rule begins with B because whatever is inside brackets is calculated first. Another word for brackets is Parentheses - hence the alternative rule PEMDAS.


Revision - Combined Multiplication and Division and BODMAS

Combined multiplication and division is a calculation that contains just multiplication and division: example, multiply 12 by 6, then divide by 3. It can contain any number of multiplications and divisions.

The calculation, written in the order of the question, is:

12×6÷3  or  12×6/3  where / is used for the division sign.

What is the correct order for this calculation? The original question stated multiply 12 by 6, and so the order is clear. But, suppose you had not seen the original and were just given the expression 12×6÷3. Should you start with the multiplication or the division? Does it matter? Does BODMAS help?

Do not interpret the word BODMAS too literally. It appears, according to BODMAS, that division, comes before multiplication. Does that mean that you should first divide 6 by 3 to give 2 and then multiply the 12 by 2 to give 24? The answer, 24, is correct but NO - you do not have to carry out the division first. That is a wrong use of BODMAS.

The D is, of course, written in front of the M but, in the order of operations, division and multiplication have equal priority, meaning that they are at the same level in the order list. Division is not done before multiplication. Multiplication is not done before division. BODMAS does not say divide before multiplying. In fact, BODMAS does not say anything about the relative order of division and multiplication. Do NOT use BODMAS.

Perhaps we should write BODMAS as BODorMAS, meaning D or M. When you use BODMAS or BIDMAS or PEMDAS or whatever you call it, think of it as DorMAS.

If there is only division and multiplication and no brackets, then forget BODMAS.

So, what do you do? Do division and multiplication in the order they are written, working from left to right. If the multiplication is written before the division then do the multiplication first. If the division is written before the multiplication then do the division first.

In the case of 12×6/3, do the multiplication 12×6 first.

12×6/3 = 72/3 = 24

Do You Have to Work in the written Order?

For multiplication and division it is safer to work in the written order. Remember, though, that the same calculation can be written in different orders.

Compare the calculation:

12×6/3 = 72/3 = 24. The 12×6 is calculated first, to give 72.

with this calculation:

12/3×6 = 4×6 = 24. The 12/3 is calculated first, to give 4.

12×6/3 is the same calculation as 12/3×6. You can see that it does not matter whether you first multiply the 12 by 6 or first divide the 12 by 3. Dividing first is often better because it simplifies the calculation.

If this is confusing, just remember that:

Test Yourself - online questions on combined multiplication and division. Click here

* * *

Revision - Addition and Subtraction

As with multiplication and division, addition and subtraction are also equal when using BODMAS. Think of BODMAS as BODMAorS. If a calculation contains only addition and subtraction, with no brackets, calculate the additions and subtractions from left to right, in the written order.
More Info It should not surprise you that multiplication and division are equal in the order of operations. Dividing by 2 is the same as multiplying by one-half. Multiplication and division are similar operations.

What about addition and subtraction? In the word BODMAS, Addition is ahead of Subtraction. In the example 5 - 3 + 2 = 4, should you do the addition first? That would give: 5 - 3 + 2 = 5 - 5 = 0. That's the wrong answer. Addition and subtraction are equal in the order of operations. The calculation should be done in the written order, from left to right. You should ignore BODMAS. 5 - 3 + 2 = 2 + 2 = 4.

Addition and subtraction are similar operations. Subtracting 3 is the same as adding -3. You can write 5 - 3 + 2 as an addition: 5 + (-3) + 2. The three terms of the addition can be written in any order, beginning with the 5 or the -3 or the 2. For example, you can write it as: (-3) + 2 + 5 = -3 + 2 + 5 = -1 + 5 = 4. Now you can see that you can do the calculation in any order provided you move the sign along with the number when you change the order.

Examples

There are more examples of combined multiplication and division in the rest of this page, starting with Example 4.

The video provides a few examples of combined multiplication and division. The first half is more relevant. The second half practises negative numbers in multiplication and division.

Video: Multiplication and Division. Click here


Revision - Addition, Subtraction, Multiplication and Division and BODMAS

BODMAS means:

if there is a division and an addition/subtraction then do the division first.

if there is a multiplication and an addition/subtraction then do the multiplication first.

If the calculation contains a mix of addition, subtraction, multiplication and division - and no brackets - work from left to right. Complete the first multiplication or division. Then continue along to the next multiplication or division, and so on to the end. Then start on the left again and this time do the additions or subtractions, whichever comes first.

If there are brackets the process is similar, but evaluate the bracketed terms first.

Try this: 5-12÷6×3×(4+2×8÷4)

Calculate the brackets first:

(4+2×8÷4)=4+16÷4=4+4=8

Note that within the brackets, work from left to right, do the multiplication 2×8=16 first, then divide the 16 by 4.

That gives:

5-12÷6×3×(4+2×8÷4)=5-12÷6×3×8

With the brackets completed, start on the left and complete the first multiplication or division - the 12÷6=2. Then proceed to the next, the 2×3 and so on:

5-12÷6×3×8=5-2×3×8=5-6×8=5-48=-43

Video: Clear Intro to Order of Operations

The Maths Is Fun website has a clear summary and examples on BODMAS. Click here


How to Write Calculations - Example Questions 1 to 4

So far we have focused on how to interpret a calculation already written as a mathematical expression. In an exam question the calculation may be given in words - not as an expression. You may then have to write out the calculation. The purpose of these simple examples is to practise how to write calculations. A calculation question could be worth 1 mark for the method and 1 mark for the result of the calculation. These examples focus on the method mark - how to write the calculation as an expression as concisely and as clearly as possible.

Example 1 - Addition: add the numbers 6, 12 and 4

Write the calculation as 6+12+4, preferably with the numbers in the same order as the question.

It is best to write any calculation with the numbers in the order in which they were given in the question. That is not always possible and is not absolutely necessary - but is worth remembering. It can help avoid errors and may help the examiner to follow your explanation.

Example 2 - Multiplication: multiply the numbers 2, 3, 4 and 5

Write or type the calculation as 2×3×4×5, preferably with the numbers in the same order as the question.

For typing, use the x keyboard character for multiplication.

Example 3 - Division: divide 6 by 3

You can write this simple calculation using the ÷ sign or the / sign for division:

either 6÷3 or 6/3

or you can write it as a fraction, using numerator over denominator, as in $${\it{6 \over 3}}$$

Example 4 - Combined Multiplication and Division: multiply 6 by 3 and then divide the result by 2

You may prefer to write this calculation using a horizontal fraction bar, as in $${\it{{6 \times 3} \over 2}}$$

The product 6×3 is the numerator. The numerator is divided by the denominator 2.

You can write or type the same calculation in one line, as 6×3÷2  or as  6×3 ⁄ 2

The horizontal bar has been replaced, in the above, with a division sign to show division. This method of writing maths is called in-line maths because the numerator and denominator are written on the same line.

How do you interpret the in-line version of this calculation? The calculation contains only multiplication and division. There is no addition or subtraction. There are no brackets. You can carry out this type of in-line calculation working from left to right, in the order it is written - in this case start with the 6, then multiply by 3 and then divide the result of that calculation by 2. But you do not have to.

The calculation 6×3 ⁄ 2 is really the same as 6 ⁄ 2×3. You can multiply the 6 by 3 to give 18 and then divide by 2 to give 9, OR you can divide the 6 by 2 to give 3 and then multiply by 3 to give 9. The order is different but you get the same answer 9.
More Info Can you write an in-line combined multiplication and division in any order? - a good question. The simple answer is 'yes'. The only restriction is that you cannot start the calculation with a division. Writing ÷3×9 does not make sense. You would write it as 9÷3. Here's an example:

multiply 2 by 3, then divide by 4 and then multiply by 5. That is written 2×3/4×5.

There is no addition, no subtraction and no brackets - but there is a division by 4. Division by 4 is really a multiplication by ¼ and so you can apply the commutative law of multiplication - which means you can write a multiplication calculation in any order.

2×3/4×5 can be written 2×3×¼×5. You can now, for example, swap the 2 with the 3 or the 3 with the ¼. The latter gives: 2×3/4×5 = 2×3×¼×5 = 2×¼×3×5 = 2/4×3×5.

The /4 has changed places with the ×3. Therefore, changing the order does not affect the result. But take care. Try this - swap the 2 with the ¼:

2×3/4×5 = 2×3×¼×5 = ¼×3×2×5 = 1/4×3×2×5. The calculation now starts with 1 and reads 1 divide by 4 times 3 etc.

In an expression such as 6×3 ⁄ 2, terms that multiply, such as the 3 in ×3, are in the numerator. Terms that divide, in this case the 2 in  ⁄ 2, are in the denominator.

To make this clearer, compare the in-line format with the 'numerator over denominator' format of the calculation:

6×3 ⁄ 26 ⁄ 2×3$${\it{{6 \times 3} \over 2}}$$

Question how do you write this expression in 'numerator over denominator' format?

2×3 ⁄ 4×5 ⁄ 6

Answer

$${\it{{2 \times 3 \times 5} \over {4 \times 6}}}$$

The numerator is 2×3×5. The denominator is 4×6.

In the numerator the 2, 3 and 5 are multiplied. In the denominator the 4 and 6 are also multiplied. You almost certainly know why - but here's a reminder. First a mathematical 'proof'.

Division by 4 is the same as multiplication by $${\it{1 \over 4}}$$. Division by 6 is the same as multiplication by $${\it{1 \over 6}}$$. The calculation can be written:

2×3 ⁄ 4×5 ⁄ 6 = 2×3×$${\it{1 \over 4}}$$×5×$${\it{1 \over 6}}$$ = 2×3×5×$${\it{1 \over 4}}$$×$${\it{1 \over 6}}$$ = 2×3×5× $${\it{{1 \times 1} \over {4 \times 6}}}$$ = $${\it{2 \over 1}}$$ × $${\it{3 \over 1}}$$ × $${\it{5 \over 1}}$$ × $${\it{{1 \over {4 \times 6}}}}$$ = $${\it{{2 \times 3 \times 5} \over {4 \times 6}}}$$

If you prefer a simpler and more natural explanation, point at the More Info link below.
More Info Division is the reverse of multiplication. Start with 2, multiply by 6 and then multiply by 4. The answer is 2×6×4 = 12×4 = 48.

Multiplication by 6 and then by 4 is clearly equivalent to multiplying by 6×4.

The reverse process is dividing 48 by 4 and then by 6. That is equivalent to dividing by 4×6.

To simplify, what this says is that because multiplying by 4 and then 6 is the same as multiplying by 4×6, then dividing by 4 and then 6 is the same as dividing by 4×6.

Write in the Same Order

In a question where you are given the numbers, as in multiply 6 by 3 and then divide the result by 2, it is best to write the calculation in the order given. In this case you would write:

either $${\it{{6 \times 3} \over 2}}$$  where the numerator is written 6×3 and not 3×6

or 6×3 ⁄ 2 where all the numbers are in the original order.

Writing in the same order may not always be possible - it depends on the type of question and how the question is worded.


Symbols for Maths Operators

Use these symbols to type calculations:

For the minus sign use the ordinary keyboard hyphen symbol –
More Info There are at least two symbols for multiplication and two for division. Alternative symbols are a complication for online tests and so only one symbol is supported. Reasons for the choice include the ease of use and the versatility of the symbol.

For example, the slash / division sign is easier to type than the ÷ sign. It is used to represent division in calculations and in fractions, such as 1/3, meaning both 1 divided by 3 and one-third.

Therefore, for division in online answers use the slash / division sign and not the ÷ sign. Example: for divide 15 by 3, type 15/3.

11/3 means both 11 divided by 3 and eleven-thirds. Do not confuse it with the fraction 1⅓ - some people type 1⅓ with a space, as in  1 1/3  but that can be confused with 11/3.

For multiplication, as in multiply 15 by 3, type 15×3. Use the ordinary keyboard character x for the multiplication sign. Do not use the * symbol. The * is used in spreadsheets and other computer programs but it is used only by a minority in GCSE maths exams. A * for multiplication is not supported in these GCSE maths online tests.

The use of the ^ symbol, as in 2^3 for 23 to represent an index (a power of), is widely accepted. The test questions rarely require you to type an index but it cannot be avoided completely.


Test Your Skill - writing maths calculations

In these three questions do not calculate the result.

For example, if the calculation is:
divide 12 by 3 and then multiply by 4

write the calculation as: 12/3x4

Use x for multiply and / for divide. Do not include any brackets.
Do not include an = sign. Do not include the result of the calculation.

Click the Check button to submit. Click Reset for another try.

Q1 Type the calculation for:
multiply 6 by 2 and divide the result by 3



 
 

Q2 Type the calculation for:
divide 6 by 3 and multiply the result by 2



 
 

Q3 The input of a number machine is multiplied by 5. The ouput is the result of the multiplication, plus 3. Type the calculation for the output when the input is 4.



 
 


Calculations, Calculators and BODMAS

BODMAS or BIDMAS

BODMAS, BIDMAS and BEDMAS summarise the rules for the order of operations. The letter after the B for brackets refers to exponents, also called indices or powers. In the calculation 2 + (3 + 4) × 52, the bracketed term (3 + 4) is calculated first, followed by the power 52, then the multiplication and then the addition.

2 + (3 + 4) × 52 = 2 + (7) × 52 = 2 + 7 × 25 = 2 + 175 = 177

A few more examples. Check you get the same answer. If not, you should revise.

2 + 3 × 4 = 14
30 / 5 × 3 = 18
30 / (5 × 3) = 2
3 × (7 - 3) = 12

BBC Bitesize and Schoolsnet have examples, explanation and a few test questions.

How to Use Maths Expressions with a Scientific Calculator

You know that when you calculate  2 + 3 × 4, you must calculate the product 3 × 4 first. Scientific calculators also ‘know’ that multiplication is done before addition. Check that your calculator works that way. To enter the calculation, press:

2 and then + and then 3 and then × and then 4.

Then press = to complete the calculation.

If your calculator gives the correct answer 24, then it is programmed to follow the BODMAS rules for the order of operations. It is a scientific calculator. If it gives the wrong answer 20, then you are strongly recommended to use a scientific calculator instead.

In a calculator question, provided you can write a correct BODMAS expression for the method, as in 30 / (5 × 3), then you get the method mark. You can then type the same expression into your calculator to complete the calculation. This is because the expression is written in the language of mathematics and the calculator can interpret that language.

Take the trouble to learn how to write maths expressions and how to use a calculator correctly. It will improve your accuracy using a calculator. It should save you time transferring the hand-written calculation to the calculator.

Check out this Calculator Tutorial. Click here

Example Questions 5 to 7

Examples 5 to 7 show how a correctly formatted written expression can be used with a scientific calculator. Each example shows first how to write the expression in an exam question and then how the same expression is typed into the calculator.

When writing the calculation use ÷ or / for division. Most calculators use ÷ for division.

Example 5: divide 30 by 5 and then multiply the result by 3

Write or type the calculation as: 30 ⁄ 5×3

Using a calculator you press these buttons:

30÷5×3=

That gives: 6×3=18

Example 6: divide 30 by the product 5 times 3

Write or type the calculation as: 30 ⁄ (5×3)

Using the bracket buttons on a scientific calculator you enter:

30÷(5×3)=

From the BODMAS rule, the brackets mean:

  1. first evaluate the product 5×3, to give 15
  2. then divide 30 by the 15
The full calculation is written or typed in-line as:

30 ⁄ (5×3) = 30 ⁄ 15 = 2   or more simply   30 ⁄ (5×3) = 2

In a written answer you may prefer to use a horizontal fraction bar as in:

$${\it{{30}\over 5 \times 3} = {{30}\over 15} =\small 2}$$   or just   $${\it{{30}\over 5 \times 3} = \small 2}$$

Example 7: Calculate the average value of 6, 12 and 4

This is a slightly more complicated example. You add the numbers and divide the total by 3.

You may prefer to first calculate the sum: $${\it{6 + 12 + 4}}$$ and then divide by 3 for the average.

Alternatively you can write the complete calculation:

$${\it{{6 + 12 + 4}\over 3}}$$  or  (6+12+4) ⁄ 3

or  type it as:

(6+12+4) ⁄ 3

Notice that the three numbers, 6, 12 and 4 are written in the same order as given.

Using a scientific calculator you enter:

(6+12+4)÷3=

The brackets are important - they show that the total is divided by 3. Writing 6+12+4 ⁄ 3 without the brackets is not correct.

Practice Quiz

The online Practice Quiz includes three questions, 6, 7 and 8, that practise typing simple maths calculations. Typing even the most simple calculation needs practice. You have to follow the correct order of operations - the BODMAS rule. To do the quiz you can log in as a Visitor as explained on the login screen.
Click Practice Quiz


The Calculator Fraction Button

This web page is about how to write calculations in a test or exam question to get the method mark. It is not about how to complete the calculation. However, it may be useful to revise the use of the fraction button.

Example 8: calculate the fraction two-thirds of 66.

The method for this is very similar to Example 4. To complete the calculation using a calculator, you could enter the two-thirds as a fraction - using the fraction button. You would press these buttons: 2F3×66= where F is the fraction button. The answer is 44.

How would you write your working for the method mark? You cannot just say "I used the fraction button on the calculator". For the method mark the examiner wants you to write the calculation - and is not interested in whether you can press the correct calculator buttons. The next section on Fraction Calculations explains the quick way to write this type of fraction calculation. It also shows how to do the calculation without using the fraction button.


Fraction Calculations - calculating a 'fraction of'

To calculate a ' fraction of ', such as two-thirds of 66, you multiply the fraction two-thirds and the 66.

2 ⁄ 3 of 66  means  2 ⁄ 3 × 66

This is explained in detail below but, for the moment, think of it this way. Work out the cost of 2 items at £66 each. To calculate for ' 2 of ' you multiply by 2. The cost is 2 × 66. For ' 2 ⁄ 3 of ' you multiply by 2 ⁄ 3. The cost is 2 ⁄ 3 × 66.

In your pre-GSCE days when you learned basic numeracy, you were probably taught to first calculate one-third and then multiply by 2 for two-thirds. You did something like this, writing the calculation in two steps:

(1) one-third =  66÷3 =  22
(2) two-thirds =  2×22 =  44

Video: calculate a fraction of a number. The video uses a similar example to 2 ⁄ 3 of  66. Click here.

The above method is exactly right when you first learn fractions - but it is not necessarily the best way to write out the method now. In a GCSE exam, to save time, you should try to combine those two lines of working into a single calculation. After all, you only get 1 mark for writing the calculation. Ask yourself - can you do the calculation and get the correct answer without showing the working in two steps? You are now beyond the initial learning stage and so may be happy to write the calculation in one step.

We'll write a single calculation to combine steps 1 and 2.

Step1: divide 66 by 3 to get one-third. Step2: multiply the result 66÷3 by 2 for two-thirds.

The calculation is 66÷3×2  or  66 ⁄ 3×2

If you did not follow the above, all we are saying is that two-thirds of 66 is the same as 66 divided by 3 and then multiplied by 2.

The same calculation can be written as $${{66 \over 3} \times \small{2}}$$

We'll now change the order:

$${{66 \over 3} \times \small{2}}={{66 \over 3} \times {2 \over 1}}={{66 \times 2} \over {3 \times 1}}={{2 \times 66} \over {3 \times 1}}={{2 \over 3} \times {66 \over 1}}={{2 \over 3} \times \small{66}}$$

The above mathematical 'proof' is overly complicated. All it says is that, because the 66 and the 2 multiply, the 66×2 can be written 2×66. In other words the 66 and the 2 can be swapped. It follows that:

$${{66 \over 3} \times \small{2}}={{2 \over 3} \times \small{66}}$$

That may be easier to see if the calculation is written in-line:

two-thirds of 66 = 66÷3×2

Swap the 66 and the 2 to give:

two-thirds of 66 = 2÷3×66  or  2 ⁄ 3×66

* * *

We have shown that the calculation 2 ⁄ 3 of 66 is the same as 2 ⁄ 3 × 66

There are several ways of writing this fraction calculation - you can decide which you prefer. Here are just three of them:

$${{2 \over 3} \times \small{66}}$$  or   $${{2 \times 66}\over 3}$$  or   $${{66 \over 3} \times \small{2}}$$

The same calculations written in-line are:

2 ⁄ 3×66  or   2×66 ⁄ 3  or   66 ⁄ 3×2

Questions what is the easiest way to write these fraction calculations:

1) 2/3 of 66   2) 3/4 of 144   3) 4/3 of 9   4) 75/100 of 144

Answers

1) 2/3 x 66   2) 3/4 x 144   3) 4/3 x 9   4) 75/100 x 144  or 3/4 x 144 after simplifying.

Without a Calculator - to complete the calculation for  2 ⁄ 3 of  66  without a calculator, simplify the numerator and denominator. Of course, in another question the numbers may not be so friendly as 2, 3 and 66.

If you prefer to write the calculation in the form $${{66} \over 3}$$ × 2 then write the complete calculation something like:

$${{66} \over 3}$$ × 2  =  $${{22} \over 1}$$ × 2  =  44 (2 marks)

If you prefer to write the calculation in the form $${{{2 \times 66}\over 3}}$$ then write the complete calculation something like:

$${{{2 \times 66}\over 3}={{2 \times 22}\over 1}}$$  =  44 (2 marks)

In the question calculate two-thirds of 66 the quick way to write the calculation is:

$${2 \over 3}$$ × 66

The complete calculation showing the calculation (1 mark) and the result of the calculation (1 mark) is

either $${2 \over 3}$$ × 66  =  2 × 22  =  44     or just   $${2 \over 3}$$ × 66  =  44

Show the cross-cancellation if it helps you to complete the calculation:

$${2 \over 3}$$ × $${66}$$  =  $${2 \over \bcancel{3}_1}$$ × $${\bcancel{66}^{22}}$$  =  2 × 22  =  44

Using a Calculator - you have a choice. If you are 'hooked on' the fraction button then you'll probably use that. First, however, consider these alternatives.

If you prefer the two step method for a 'fraction of a number', where you divide by 3 for one-third and multiply by 2 for two-thirds, then you may prefer to write the method as:

two-thirds of 66 = 66 ⁄ 3×2 (1 mark)

Then, to complete the calculation with a calculator you just key in that calculation - remembering to replace the slash / with the division sign ÷.

Press these calculator buttons for one-third: 66÷3

Then, for two-thirds you multiply by 2. For the complete calculation on a calculator, press 66÷3×2=

This website recommends the 'multiply by the fraction' method to calculate a 'fraction of'. You write the method as:

two-thirds of 66 = 2 ⁄ 3×66 (1 mark)

Then, to complete the calculation with a calculator you just key in that calculation - remembering to replace the slash / with the division sign ÷.

For the complete calculation on a calculator, press 2÷3×66=

You can see from the above that you simply key into your calculator the same calculation that you wrote for your method. It's a 'no-brainer'.

To use the fraction button, the method is similar - but instead of pressing 2÷3×66= you press 2F3×66= where F is the fraction button. Of course, unless the answer is an integer, if you use the fraction button the answer will be in fraction format.


Video: calculate a fraction of a number. This video uses a method similar to the single calculation method. Click here.


Percentage Calculations - calculating a 'percentage of'

A percentage is really a fraction, as in 20%, which is the same as the fraction 20 ⁄ 100. Calculating a 'percentage of' is the same as calculating a 'fraction of'.

Example: calculate 20% of 66

20% of 66 = 20 ⁄ 100×66 = 20×0.66 = 2×6.6 = 13.2

The working out above is detailed and you can probably spot some short cuts. 20% can be written as:

20 ⁄ 100   or   2 ⁄ 10   or the decimal   0.2

The calculation can be simplified to:

20% of 66 = 2 ⁄ 10×66 = 2×6.6 = 13.2

Just writing  20 ⁄ 100×66   or   2 ⁄ 10×66  will get the method mark. Of course, you could use alternatives, such as  66 ⁄ 100×20   or   66 ⁄ 10×2.


An Example Exam Question

An item in a sale is reduced by 25% of the normal price, £80. Calculate the price reduction. The question is worth 2 marks, 1 mark for the method and 1 mark for the answer.

Questions about price reductions and price increases always cause confusion. To avoid confusion, this question means that there is 25% off the normal price. Some students wrongly interpret that to mean the sale price is only 25% of the normal price. In fact the sale price is 100%-25% or 75% of the normal price.

In the written exam you can explain the calculation 'in words'. Remember that often there is just one method mark for the calculation and your explanation of the method does not have to be detailed to get that mark.

Acceptable explanations are:

1% of 80 is 80÷100. 25% is 25×80÷100=20 or

25% is one-quarter of 80 = 80/4 = 20 or

Even simpler, you could just write: 25% is 25 ⁄ 100×80=20.

All three answers above get the 'method' mark for the explanation/working and the 'accuracy' mark for a correct answer. The three methods are explained clearly and so would also get any additional mark for quality of communication.

The last method 25 ⁄ 100×80=20 is concise and uses a correctly written calculation. It saves time in the exam and shows good understanding of maths calculations.

An online marking system does not understand explanations written 'in words'. Apart from the last method 25 ⁄ 100×80 it is impossible for online marking to grade the above answers.

To provide the opportunity for maximum marks in an online test, questions similar to this example question are divided into two parts, with 1 mark for each part. The first part deals with the method used to calculate the answer, as follows.

Calculation Part (1 method mark)
An item costs £80. The price is reduced by 25%. Type a single calculation, in the Calculation box below, for the price reduction. Do not include the £ and % symbols and do not include the result of the calculation. Do not attempt to include more than one calculation. If possible type the numbers in the same order as given in the question.

Calculation: price reduction =  25x80/100     £

Answer Part (1 mark)
Now complete the calculation for the price reduction and enter the answer in the Answer box. Do not include the £ symbol.

Answer: price reduction =  20            £

The calculation box shows a correct answer: 25×80 ⁄ 100. Note that it was not written as  25×80 ⁄ 100= with an equal sign at the end. The = sign is entered when using a calculator simply to tell the calculator to carry out the calculation. In the calculation part of a test question you are showing just the method used in the calculation. If the question is not asking you how to do the calculation using a calculator, do not include the = sign.

You can see that for online marking of calculations, your answer has to be precise. You may not like that but there are a few benefits. An answer such as  25×80 ⁄ 100 is much more concise than an answer 'in words'. It is quicker to write because it is written in the language of mathematics. Using that language will improve the quality of your answers.

Other possible calculation answers are:

25 ⁄ 100×80(25 ⁄ 100)×8080 ⁄ 100×25

Notice that the brackets in (25 ⁄ 100)×80 are not needed, although they do avoid ambiguity. Without the brackets you may think that  25 ⁄ 100×80 means divide 25 by the product 100×80. The BODMAS rule does not help because, although the D is before the M in BODMAS, division and multiplication are equal in the rule. When there is doubt then it is accepted that the first operation, division or multiplication, is carried out first. Therefore, 25 ⁄ 100×80 means divide 25 by 100 because 25 ⁄ 100 is written first. Then you multiply the result by 80.  

External Links

Video - the Commutative Law of Addition
OnlineMathLearning - Online Questions on Combined Multiplication and Division
The next video provides a few examples of combined multiplication and division. The first half is more relevant. The second half practises negative numbers in multiplication and division.
Video - Combined Multiplication and Division
Video - Intro to Order of Operations - Clear Explanations
MathsIsFun - BODMAS - Clear Summary, Examples and Online Questions
BBC Bitesize and Schoolsnet - examples, explanation and a few test questions on BODMAS.
BBC Bitesize - Order of Operations
Schoolsnet - BODMAS Revision
Calculator Tutorial
Video - Calculate a Fraction of a Number using a two-step calculation
Video - Calculate a Fraction of a Number using a one-step calculation method


© Edu-Sol.co.uk 2007-2012

eXTReMe Tracker