N1–N3Place value, ordering and order of operations

Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Number

Practise Place value, ordering and order of operations. 1 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.

Build a paper on this topic

▶ Watch videos on Place value, ordering and order of operations (Corbettmaths on YouTube) · Practise all of Number

Free downloads:Open in the Notes section

Revision notes

The basic number skills behind every paper: place value, ordering numbers, the order of operations, negative numbers, written methods and reciprocals. Expect short questions, especially on the non-calculator paper, and you need these skills inside almost every other question.

Grade by grade

What you need to be able to do, from the first marks up to the top grade.

1
Order integers and decimalsLine up the decimal points and compare digits from the left, so 0.407 is smaller than 0.47.
1
Use the order of operations (BIDMAS)Brackets, then indices, then × and ÷, then + and −, so 3 + 4 × 2 = 11.
2
Calculate with negative numbersAdd, subtract, multiply and divide with negatives, e.g. −3 − (−8) = 5 and −4 × −6 = 24.
2
Use the inequality symbols correctlyUse \(=\), \(\ne\), \(\lt\), \(\gt\), \(\le\) and \(\ge\) to compare numbers, e.g. \(-7 \lt -2\).
3
Use written methods for all four operationsUse column or grid multiplication and short or long division without a calculator.
3
Multiply and divide with decimalsWork with whole numbers, then place the decimal point, e.g. 0.3 × 0.04 = 0.012 and 7.2 ÷ 0.3 = 24.
4
Use one calculation to find anotherGiven 34 × 26 = 884, adjust the place value to get 3.4 × 2.6 = 8.84 or 884 ÷ 2.6 = 340.
4
Find and use reciprocalsThe reciprocal of a number is 1 divided by it, so the reciprocal of 4 is \(\frac{1}{4}\) and of \(\frac{2}{3}\) is \(\frac{3}{2}\).

Notes

Place value and ordering

Each column is worth 10 times the column to its right: thousands, hundreds, tens, ones, then tenths, hundredths, thousandths after the decimal point.
To order decimals, line up the decimal points and compare digits from the left: 0.47 = 0.470, which is bigger than 0.407.
On a number line, numbers get smaller as you go left, so −7 is less than −2.
Symbols: \(\lt\) less than, \(\gt\) greater than, \(\le\) less than or equal to, \(\ge\) greater than or equal to, \(\ne\) not equal to. The narrow end points at the smaller number.

Order of operations

BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.
Division and multiplication have equal priority, so work from left to right: 12 ÷ 3 × 2 = 4 × 2 = 8. The same goes for addition and subtraction.
Example: \(5 + 3 \times 2^2 = 5 + 3 \times 4 = 5 + 12 = 17\).
A fraction bar and a square root sign work like brackets: work out the whole top, the whole bottom, or everything under the root first. \(\sqrt{9 + 16} = \sqrt{25} = 5\), not 3 + 4.
Inverse operations undo each other (+ and −, × and ÷, squaring and square rooting). Use them to check answers.

Negative numbers

Adding a negative is the same as subtracting: 4 + (−6) = −2.
Subtracting a negative is the same as adding: −3 − (−8) = −3 + 8 = 5.
Multiplying or dividing: same signs give a positive answer, different signs give a negative answer. −4 × −6 = 24 and 20 ÷ (−5) = −4.
\((-3)^2 = 9\), but \(-3^2\) means \(-(3^2) = -9\). Use brackets on your calculator when you square a negative number.

Decimals and reciprocals

Multiplying decimals: multiply as whole numbers, then give the answer as many decimal places as the numbers had in total. 0.3 × 0.04: 3 × 4 = 12, three decimal places, so 0.012.
Dividing by a decimal: multiply both numbers by 10 (or 100) until you are dividing by a whole number. 7.2 ÷ 0.3 = 72 ÷ 3 = 24.
Using a known fact: if 34 × 26 = 884, then 3.4 × 26 = 88.4 and 0.34 × 2.6 = 0.884.
The reciprocal of a number is 1 divided by it: the reciprocal of 5 is \(\frac{1}{5}\) and of \(\frac{3}{4}\) is \(\frac{4}{3}\). A number times its reciprocal is 1. Zero has no reciprocal.

Cheatsheet

BIDMAS: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right)
× and ÷: same signs → positive, different signs → negative
Subtracting a negative is adding: 5 − (−2) = 7
Decimal places in a product = total decimal places in the numbers being multiplied
To divide by a decimal, multiply both numbers by 10, 100, ... first
Reciprocal of \(x\) is \(\frac{1}{x}\); reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\)
\(\lt\) less than, \(\le\) less than or equal to, \(\gt\) greater than, \(\ge\) greater than or equal to, \(\ne\) not equal to

How to answer each type of question

Work out, using the order of operations

1 to 2 marks2

Work out brackets first. Treat the top and the bottom of a fraction as if each were in brackets.
Then powers and roots.
Then × and ÷ from left to right, then + and − from left to right.
Write each stage down so a slip still leaves you with method marks.

Example. Work out \(\frac{2 + 6 \times 3^2}{5 \times 3 - 7}\)

Show the model answer

Top: \(3^2 = 9\), \(6 \times 9 = 54\), \(2 + 54 = 56\)
Bottom: \(15 - 7 = 8\) (M1 for 56 or 8)
\(56 \div 8 = 7\) (A1)

Use a given calculation to write down others

1 mark each4

Compare each number in the new calculation with the given fact: how many times bigger or smaller is it?
For ×, apply both changes to the answer. For ÷, a smaller number to divide by gives a bigger answer.
Check with a rough estimate that the size is sensible.

Example. Using the information that \(47 \times 23 = 1081\), write down the value of
(a) \(4.7 \times 2.3\)
(b) \(0.47 \times 230\)
(c) \(1081 \div 230\)

Show the model answer

(a) Both numbers are 10 times smaller, so the answer is 100 times smaller: 10.81 (B1)
(b) ÷ 100 and × 10, so the answer is 10 times smaller: 108.1 (B1)
(c) \(1081 \div 23 = 47\), and dividing by a number 10 times bigger gives 4.7 (B1)

Calculate with decimals without a calculator

2 marks each3

Multiplying: remove the decimal points, multiply the whole numbers with a written method, then put back the total number of decimal places.
Dividing: multiply both numbers by the same power of 10 so you divide by a whole number, then use short division.
Estimate to check the position of the decimal point.

Example. (a) Work out 2.4 × 0.35
(b) Work out 9.36 ÷ 0.4

Show the model answer

(a) 24 × 35 = 840 (M1 for a complete written method)
3 decimal places in total, so 0.840 = 0.84 (A1)
(b) 9.36 ÷ 0.4 = 93.6 ÷ 4 (M1)
= 23.4 (A1)

Use your calculator: write down all the figures, then round

2 to 3 marks3

Work out the top and the bottom separately (write them down), or put brackets round the whole top and whole bottom.
Write down every figure on the display.
Round only in the part that asks you to, e.g. to 3 significant figures.

Example. (a) Use your calculator to work out \(\frac{\sqrt{19.6 + 3.2^2}}{4.1 - 0.75}\)
Write down all the figures on your calculator display.
(b) Write your answer to part (a) correct to 3 significant figures.

Show the model answer

(a) Top: \(19.6 + 10.24 = 29.84\), \(\sqrt{29.84} = 5.4626...\) Bottom: 3.35 (B1 for 29.84 or 5.4626... or 3.35 seen)
1.630626899... (B1; your display may show it rounded, e.g. 1.6306269)
(b) 1.63 (B1)

Shortcuts and memory tricks

BIDMAS: D and M are equal partners, and so are A and S. When two are equal, go left to right.
For × and ÷ with negatives: same signs, positive; different signs, negative.
Two minus signs side by side, as in 5 − (−2), become a plus.
Estimate first: 4.7 × 2.3 is about 5 × 2 = 10, so 10.81 is sensible and 108.1 is not.
On a calculator, put brackets round the whole top and whole bottom of a fraction, or use the fraction key.

Where marks are lost

Working from left to right whatever the operation: 3 + 4 × 2 is 11, not 14.
Doing every multiplication before any division: 12 ÷ 3 × 2 is 8, not 2.
Typing −3² into a calculator (which gives −9) when you mean \((-3)^2 = 9\).
Putting the decimal point in the wrong place: 0.3 × 0.04 = 0.012, not 0.12.
Thinking a decimal with more digits is bigger: 0.47 is bigger than 0.407.
Rounding in part (a) when it asks for all the figures on the calculator display.

Exam technique

On the non-calculator paper, show your written method (column, grid or bus stop). A correct method earns marks even if you slip.
When a question gives you a fact such as 47 × 23 = 1081, use it. 'Write down' means little or no working is needed.
When asked for all the figures on the display, copy every digit; round only when a later part tells you to.
Check answers with inverse operations or a quick estimate.

Sample questions

Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.

Question 1Easy2 marks

Sam works out \(20 - 6 \div 2 + 1\).
Sam's answer is 8.

(a) Work out the correct value of \(20 - 6 \div 2 + 1\).[1]

(b) Explain the mistake that Sam made.[1]

Show the answer and mark scheme

(a) Answer: 18

B1 for 18

Worked solution: \(6 \div 2 = 3\) is worked out first, then \(20 - 3 + 1 = 18\).

(b) Answer: Sam subtracted first (20 − 6 = 14, then 14 ÷ 2 = 7, then 7 + 1 = 8). The division must be done before the subtraction.

C1 for explaining that Sam worked out 20 − 6 before dividing, when the division must be done first

Worked solution: Sam's 8 comes from \((20 - 6) \div 2 + 1 = 7 + 1\). Division comes before subtraction, so \(6 \div 2\) should be worked out first.

Question 2Medium2 marks

Write these numbers in order of size.
Start with the smallest number.
\(-0.3\), \(-0.9\), \(0.39\), \(-\frac{22}{25}\), \(\frac{3}{8}\)[2]

Show the answer and mark scheme

Answer: \(-0.9\), \(-\frac{22}{25}\), \(-0.3\), \(\frac{3}{8}\), \(0.39\)

M1 for converting at least three of the fractions and decimals to the same form (e.g. \(-\frac{22}{25} = -0.88\) and \(\frac{3}{8} = 0.375\))
A1 for \(-0.9\), \(-\frac{22}{25}\), \(-0.3\), \(\frac{3}{8}\), \(0.39\) (the numbers may be written in their original form)

Worked solution: As decimals: \(-0.3\) = −0.3, \(-0.9\) = −0.9, \(0.39\) = 0.39, \(-\frac{22}{25}\) = −0.88, \(\frac{3}{8}\) = 0.375.
In order: \(-0.9\), \(-\frac{22}{25}\), \(-0.3\), \(\frac{3}{8}\), \(0.39\).

Related subtopics

Stuck? Get 1-to-1 help. Chhetri Academy tutors GCSE and A level Maths and Science online, with a free 30-minute trial lesson.

Book a free trial