Ordering numbers and inequality symbols - Edexcel GCSE Maths (Calculator) Exam Questions

Practise 18 exam-style questions on Ordering numbers and inequality symbols for Edexcel GCSE Maths (Calculator). Write your answer as you would in the exam and get it marked in seconds against the mark scheme, point by point, with an AI tutor on hand if you get stuck.

18 questions
1–5 marks each
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18 questions, marked as you go

Sample questions with model answers

Here are 4 of the questions in this topic, each with a model answer, an explanation of why it earns the marks and the mark scheme an examiner would use.

Question 1

2 marks

Write the correct symbol, <, > or =, between each pair of numbers.

\(\sqrt{50}\) ... 7.07
\(\dfrac{22}{7}\) ... \(\pi\)
\(\dfrac{5}{12}\) ... \(0.41\dot{6}\)
\((-0.9)^3\) ... \(-0.9^2\)

Model answer

\(\sqrt{50} = 7.0710...\), so \(\sqrt{50} > 7.07\)
\(\dfrac{22}{7} = 3.142857...\) and \(\pi = 3.141592...\), so \(\dfrac{22}{7} > \pi\)
\(\dfrac{5}{12} = 0.416666... = 0.41\dot{6}\), so \(\dfrac{5}{12} = 0.41\dot{6}\)
\((-0.9)^3 = -0.729\) and \(-0.9^2 = -(0.9^2) = -0.81\); −0.729 is nearer to zero, so \((-0.9)^3 > -0.9^2\)

Explanation

Turn each number into a decimal with your calculator and compare digit by digit. Take care with \(-0.9^2\): the power is worked out before the minus sign, so it means \(-(0.9 \times 0.9) = -0.81\), whereas \((-0.9)^3\) cubes the negative number. With negative numbers, the one closer to zero is the larger.

Mark scheme

2 marks
B2 for all four correct: \(\sqrt{50} > 7.07\), \(\dfrac{22}{7} > \pi\), \(\dfrac{5}{12} = 0.41\dot{6}\), \((-0.9)^3 > -0.9^2\)
cao; B1 for two or three correct

Question 2

2 marks

Leo says,

"\(\dfrac{4}{7} = 0.57\), so \(\dfrac{4}{7}\) and 0.57 are equal."

Is Leo correct?
You must give a reason for your answer.

Model answer

\(\dfrac{4}{7} = 4 \div 7 = 0.571428...\)
0.571428... is greater than 0.57
So \(\dfrac{4}{7} ≠ 0.57\). Leo is not correct: 0.57 is \(\dfrac{4}{7}\) rounded to 2 decimal places, and \(\dfrac{4}{7} > 0.57\).

Explanation

Work out \(4 \div 7\) on your calculator: the decimal goes on, 0.571428571... Rounding to 2 decimal places gives 0.57, but a rounded value is not equal to the exact value, so the correct symbol is ≠ (in fact \(\frac{4}{7} > 0.57\)).

Mark scheme

1 mark
B1 for \(\dfrac{4}{7} = 0.5714...\)
accept 0.571 or better, or \(\frac{4}{7} = \frac{400}{700}\) and \(0.57 = \frac{399}{700}\) oe
1 mark
C1 for No, with a correct reason, e.g. \(\dfrac{4}{7} = 0.5714...\) so \(\dfrac{4}{7} > 0.57\) (or \(\dfrac{4}{7} ≠ 0.57\)); 0.57 is only \(\dfrac{4}{7}\) rounded to 2 decimal places

Question 3

3 marks

Find all the integers \(n\) such that

\[\dfrac{2}{7} < \dfrac{n}{40} < \dfrac{\sqrt{3}}{5}\]

Model answer

Multiply each part by 40:
\(\dfrac{2}{7} \times 40 = 11.428...\)
\(\dfrac{\sqrt{3}}{5} \times 40 = 13.856...\)
So \(11.428... < n < 13.856...\)
\(n = 12, 13\)

Explanation

Multiplying every part of the inequality by 40 (a positive number) keeps the inequality signs the same and leaves \(n\) on its own in the middle. Then list the integers strictly between 11.43 and 13.86. Check: \(\frac{12}{40} = 0.3\) and \(\frac{13}{40} = 0.325\), which both lie between 0.2857 and 0.3464.

Mark scheme

1 mark
M1 for converting both ends to decimals, \(\frac{2}{7} = 0.2857...\) and \(\frac{\sqrt{3}}{5} = 0.3464...\), or for multiplying each by 40
or for a correct trial of at least two values of \(\frac{n}{40}\) as decimals; correct answer with no working scores full marks
1 mark
A1 for \(11.4... < n < 13.8...\)
oe; accept 11.42 to 11.43 and 13.85 to 13.86
1 mark
A1 for 12 and 13 only
cao

Question 4

4 marks

\(\dfrac{n}{24}\) is a fraction in its simplest form, where \(n\) is a positive integer.

\[\dfrac{\sqrt{2}}{4} < \dfrac{n}{24} < \dfrac{4}{7}\]

Find all the possible values of \(n\).

Model answer

\(\dfrac{\sqrt{2}}{4} \times 24 = 6\sqrt{2} = 8.485...\)
\(\dfrac{4}{7} \times 24 = 13.714...\)
So \(8.485... < n < 13.714...\), giving \(n = 9, 10, 11, 12, 13\)
\(\frac{9}{24} = \frac{3}{8}\), \(\frac{10}{24} = \frac{5}{12}\) and \(\frac{12}{24} = \frac{1}{2}\) are not in simplest form.
\(\frac{11}{24}\) and \(\frac{13}{24}\) cannot be simplified.
\(n = 11\) or \(n = 13\)

Explanation

First find the range of \(n\) by multiplying every part of the inequality by 24. Then check each fraction: it is in its simplest form only when \(n\) and 24 have no common factor other than 1, so \(n\) cannot be even or a multiple of 3. A common mistake is to stop at the list 9 to 13.

Mark scheme

1 mark
M1 for \(\frac{\sqrt{2}}{4} = 0.3535...\) and \(\frac{4}{7} = 0.5714...\), or for multiplying each by 24
1 mark
A1 for \(8.48... < n < 13.7...\)
oe; accept 8.48 to 8.49 and 13.71 to 13.72
1 mark
M1 for considering \(n = 9, 10, 11, 12, 13\) and rejecting those that simplify, e.g. \(\frac{9}{24} = \frac{3}{8}\), \(\frac{10}{24} = \frac{5}{12}\), \(\frac{12}{24} = \frac{1}{2}\)
may be implied by the correct answer
1 mark
A1 for 11 and 13 only
cao

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