Maths · 1 Number

1.5 Ordering

Before you start

Place value after the decimal point: tenths, hundredths, thousandths.
Negative numbers on a number line, e.g. $- 3$ is to the left of $1$ .
Changing between fractions, decimals and percentages, e.g. $\frac{3}{8} = 0.375 = 37.5 %$ (subtopic 1.4).
Finding the lowest common multiple of small numbers, e.g. the LCM of 8 and 12 is 24 (subtopic 1.1).
Squares and square roots from 1 to 15, e.g. $\sqrt{169} = 13$ (subtopic 1.3).

Syllabus objectives covered (1)

Order quantities by magnitude and demonstrate familiarity with the symbols =, ≠, >, < , ⩾ and ⩽.

Part 1 of 6

The six symbols and how to read them

To order quantities by magnitude means to put them in order of size. Every method in this lesson does one thing: it turns the numbers into a form where you can see which is bigger.
$=$ means 'is equal to'. $\neq$ means 'is not equal to'. Use $=$ only when two values are exactly the same. $0.25 = \frac{1}{4}$ is true. $\frac{1}{3} \neq 0.33$ , because $\frac{1}{3} = 0.333 \dots$ and $0.33$ is only a rounded value of it.
$<$ means 'is less than'. $>$ means 'is greater than'. The open side of the symbol faces the bigger number, and the point faces the smaller number.
$3 < 8 8 > 3$
Read the statement from left to right. $3 < 8$ reads 'three is less than eight'. $8 > 3$ is the same fact, read the other way round. To turn one into the other, swap the two sides and turn the symbol round. If you swap only the sides, the statement becomes false.
$⩽$ means 'is less than or equal to'. $⩾$ means 'is greater than or equal to'. The line under the symbol is half of an equals sign. It adds the words 'or equal to'.
So $5 ⩽ 5$ is true, but $5 < 5$ is false. The difference between $<$ and $⩽$ matters only at the end value itself. You will use this when a letter stands for a range of numbers, at the end of this lesson.
Write the symbols clearly. A $>$ drawn with a faint extra line can look like $⩾$ , and that is a different statement.

Part 2 of 6

Integers and negative numbers: use the number line

On a number line, numbers get bigger as you move to the right. A number is greater than every number to its left.
graph diagram — not drawn yet
A horizontal number line from -10 to 10 with every integer marked and labelled, the word 'smaller' with an arrow pointing left at the left end and the word 'bigger' with an arrow pointing right at the right end; dots on -8 and -3, with the statement '-8 < -3' written beneath them.
Moving right always means bigger, so -8, which is further left, is smaller than -3.
So every positive number is greater than every negative number, and $0$ sits between them: $- 1 < 0 < 1$ .
For two negative numbers, the one further from zero is the smaller one. $- 8$ is further left than $- 3$ , so $- 8 < - 3$ .
The trap is the digits. $8$ is bigger than $3$ , so it feels as if $- 8$ should be bigger than $- 3$ . The digit tells you the distance from zero, not the position. Think of temperature: $- 8$ °C is colder than $- 3$ °C.
Ascending order means smallest first, going up. Descending order means largest first, going down. A question may say it in plain words instead: 'starting with the smallest' means ascending, and 'starting with the largest' means descending.
To order $- 5, 3, - 12, 0, 7, - 1$ in ascending order, place each one on the number line in your head and read from left to right.
$- 12, - 5, - 1, 0, 3, 7$

Part 3 of 6

Decimals: line up the place values

A longer decimal is not a bigger decimal. $0.5$ has one decimal place and $0.45$ has two, but $0.5$ is bigger: five tenths is more than four tenths.
The reliable method is to write the numbers under each other with the decimal points in a line. Then fill the empty places with zeros, so that every number has the same number of decimal places.
Zeros at the end of a decimal do not change its value: $0.5 = 0.50 = 0.500$ . They only make the columns the same length, so you can compare them fairly. Here are $0.5$ , $0.45$ , $0.405$ and $0.045$ written this way.
$0.500 0.450 0.405 0.045$
Now compare column by column from the left: units, then tenths, then hundredths, then thousandths. The first column where the digits differ decides the order. Here the tenths are $5, 4, 4, 0$ , so $0.5$ is largest and $0.045$ is smallest. For $0.450$ and $0.405$ the tenths match, so look at the hundredths: $5 > 0$ .
$0.045 < 0.405 < 0.45 < 0.5$
Always compare the whole-number part first. $2.09 > 1.99$ , because $2 > 1$ in the units column. Once the units differ, the decimal digits do not matter.
Negative decimals follow the number line. $- 0.35$ is further from zero than $- 0.3$ , so $- 0.35 < - 0.3$ . A quick way: order the numbers as if they were positive, then reverse that order for the negatives.

Part 4 of 6

Fractions: make the denominators the same

If two fractions have the same denominator, compare the numerators. $\frac{5}{9} > \frac{4}{9}$ , because five ninths is more than four ninths.
If two fractions have the same numerator, the larger denominator gives the smaller fraction. $\frac{1}{8} < \frac{1}{5}$ , because cutting a whole into 8 equal pieces makes each piece smaller than cutting it into 5.
If the denominators are different, rewrite every fraction over a common denominator. The lowest common multiple of the denominators keeps the numbers small. To compare $\frac{3}{4}$ and $\frac{5}{7}$ , use 28, because $4 \times 7 = 28$ and 4 and 7 have no common factor except 1.
$\frac{3}{4} = \frac{21}{28} \frac{5}{7} = \frac{20}{28} so \frac{3}{4} > \frac{5}{7}$
Multiply the top and the bottom by the same number. That is what keeps the value the same: $\frac{3}{4} = \frac{3 \times 7}{4 \times 7}$ . Multiplying only the bottom changes the fraction.
With a calculator, you can instead divide the numerator by the denominator and compare the decimals. On the non-calculator paper, the common denominator lets you avoid long division.
For mixed numbers, compare the whole-number parts first. $2 \frac{1}{3} > 1 \frac{7}{8}$ , because $2 > 1$ . Only when the whole numbers are the same do you need to compare the fraction parts.
For negative fractions, order the sizes first and then reverse. $\frac{3}{4} > \frac{2}{3}$ , so $- \frac{3}{4} < - \frac{2}{3}$ .

Part 5 of 6

Mixed forms and units: change everything to one form

When a list mixes fractions, decimals, percentages, powers and roots, you cannot compare them as they stand. Change each one to a decimal and write the decimal beside the original, so you can see both.
Percentages: divide by 100. $37 % = 0.37$ and $4.5 % = 0.045$ .
Powers and roots: work out the value. ${0.6}^{2} = 0.36$ . Notice that squaring a number between 0 and 1 makes it smaller, and taking the square root of a number between 0 and 1 makes it bigger: $\sqrt{0.14} = 0.3742$ to 4 decimal places.
Use enough decimal places to separate the values. $\sqrt{0.14} = 0.3741 \dots$ and $\frac{3}{8} = 0.375$ both begin $0.37$ , so the first two decimal places cannot tell them apart. The third place can: $0.374 < 0.375$ , so $\sqrt{0.14} < \frac{3}{8}$ . If two values begin with the same digits, look at the next decimal place before you decide.
To compare a square root with a number without a calculator, square both. Is $\sqrt{60}$ bigger than $7.7$ ? ${7.7}^{2} = 59.29$ , which is less than 60, so $7.7 < \sqrt{60}$ . This works because both numbers are positive.
Your final answer is the original numbers, in order. The decimals are your working. If the list gave you $\frac{3}{8}$ , the answer says $\frac{3}{8}$ , not $0.375$ .
Quantities with units work the same way: change them all to one unit first. You need $1$ km $= 1000$ m, $1$ m $= 100$ cm, $1$ kg $= 1000$ g and $1$ litre $= 1000$ ml. For example, $1.2$ kg $= 1200$ g, so $1.2$ kg $> 1150$ g, even though $1.2$ looks smaller than $1150$ .

Part 6 of 6

Inequalities with a letter, and words into symbols

$x > 3$ means ' $x$ is any number greater than 3'. It describes a range of values, not one number. $3.01$ is in the range. $3$ itself is not, because 3 is not greater than 3.
$x ⩾ 3$ includes 3. So if $x$ must be an integer, $x > 3$ gives $4, 5, 6, \dots$ and $x ⩾ 3$ gives $3, 4, 5, \dots$
A double inequality such as $- 2 < x ⩽ 3$ says two things at once: $x$ is greater than $- 2$ , and $x$ is less than or equal to $3$ . Read it from the middle outwards: ' $x$ is between $- 2$ and $3$ , and it can equal 3 but not $- 2$ '.
To list the integers in a double inequality, check each end on its own. The left end has $<$ , so $- 2$ is not included and the list starts at $- 1$ . The right end has $⩽$ , so $3$ is included.
$- 2 < x ⩽ 3 \Rightarrow x = - 1, 0, 1, 2, 3$
Words into symbols: 'more than', 'over' and 'exceeds' mean $>$ . 'Less than', 'under' and 'fewer than' mean $<$ . 'At least', 'no less than' and 'a minimum of' mean $⩾$ . 'At most', 'no more than' and 'a maximum of' mean $⩽$ .
'At least', 'at most', 'no less than', 'no more than', 'a minimum of' and 'a maximum of' all allow the end value. A ride for people at least 120 cm tall lets in someone who is exactly 120 cm, so $h ⩾ 120$ . A coach for no more than 52 passengers can take exactly 52, so $p ⩽ 52$ .
Put the letter on the left when you can, as in $h ⩾ 120$ . $120 ⩽ h$ says the same thing, but with the letter first you can read it straight off: ' $h$ is at least 120'.
$\neq$ also works with a letter. $x \neq 0$ means $x$ can be any number except 0. You see it where 0 would break a calculation, such as the reciprocal $\frac{1}{x}$ , because $\frac{1}{0}$ has no value.

Worked examples

Example 1. Write these numbers in order of size, starting with the smallest: $0.6, 0.06, 0.606, 0.66, 0.066$ .

Show the working

The longest has 3 decimal places, so pad every number to 3 decimal places. Zeros at the end do not change the values.
$0.600 0.060 0.606 0.660 0.066$
Look at the tenths column first. $0.060$ and $0.066$ have 0 tenths, so they are the two smallest. Their hundredths match (both 6), so the thousandths decide: $0 < 6$ , so $0.060 < 0.066$ .
The other three have 6 tenths. Compare their hundredths: $0.600$ and $0.606$ have 0 and $0.660$ has 6, so $0.660$ is the largest. Then thousandths: $0.600 < 0.606$ .
Write the answer with the numbers as they were given, without the extra zeros.

Answer: $0.06, 0.066, 0.6, 0.606, 0.66$

Example 2. Without using a calculator, write these fractions in order of size, starting with the smallest: $\frac{5}{8}, \frac{2}{3}, \frac{7}{12}, \frac{3}{4}$ .

Show the working

Find a common denominator. The denominators are 8, 3, 12 and 4. Multiples of 12 are $12, 24, \dots$ and 24 is the first one that 8 and 3 also divide, so use 24.
Rewrite each fraction over 24. Multiply top and bottom by the same number each time: by 3, by 8, by 2 and by 6.
$\frac{5}{8} = \frac{15}{24} \frac{2}{3} = \frac{16}{24} \frac{7}{12} = \frac{14}{24} \frac{3}{4} = \frac{18}{24}$
Now the denominators are the same, so order by the numerators: $14 < 15 < 16 < 18$ .
Write the answer using the original fractions, not the twenty-fourths.

Answer: $\frac{7}{12}, \frac{5}{8}, \frac{2}{3}, \frac{3}{4}$

Example 3. Write these in order of size, starting with the smallest: $\frac{3}{8}, 0.38, 37 %, {0.6}^{2}, \sqrt{0.14}$ .

Show the working

Change every value to a decimal and write it beside the original. Use the calculator for the root.
$\frac{3}{8} = 0.375$ . $0.38 = 0.380$ . $37 % = 0.370$ . ${0.6}^{2} = 0.360$ . $\sqrt{0.14} = 0.3742$ to 4 decimal places.
Check that you have enough decimal places to tell the values apart. $\sqrt{0.14} = 0.3741 \dots$ and $\frac{3}{8} = 0.375$ both begin $0.37$ , so look at the third decimal place: $0.374 < 0.375$ .
Order the decimals.
$0.360 < 0.370 < 0.374 < 0.375 < 0.380$
Replace each decimal with the original value it came from.

Answer: ${0.6}^{2}, 37 %, \sqrt{0.14}, \frac{3}{8}, 0.38$

Example 4. (a) List the integers $n$ such that $- 3 ⩽ n < 2$ . (b) Write $<$ or $>$ between each pair of numbers to make a true statement: $- 5 □ - 8$ and $\frac{2}{3} □ 0.67$ . (c) A coach has seats for no more than 52 passengers. Write an inequality for the number of passengers, $p$ .

Show the working

(a) Check each end on its own. The left end has $⩽$ , so $- 3$ is included. The right end has $<$ , so $2$ is not. Count up from $- 3$ and stop before 2.
(b) On the number line, $- 5$ is to the right of $- 8$ , so $- 5$ is greater: $- 5 > - 8$ .
For the second pair, change $\frac{2}{3}$ to a decimal: $0.6666 \dots$ . Compare it with $0.6700$ . The tenths match, and in the hundredths $6 < 7$ , so $\frac{2}{3} < 0.67$ . It also shows that $\frac{2}{3} \neq 0.67$ : $0.67$ is only a rounded value.
(c) 'No more than 52' means 52 is allowed, but nothing above it. That is 'less than or equal to'.

Answer: (a) $- 3, - 2, - 1, 0, 1$ (b) $- 5 > - 8$ and $\frac{2}{3} < 0.67$ (c) $p ⩽ 52$

Where people go wrong

Mistake: Thinking the decimal with more digits is bigger, e.g. putting $0.45$ above $0.5$ .

Instead: Line up the decimal points and pad with zeros: $0.50$ against $0.45$ . The tenths decide first: $5 > 4$ , so $0.5 > 0.45$ .

With whole numbers, more digits does mean bigger ( $45 < 450$ ). After the decimal point it does not, because each extra digit is in a smaller place value.

Mistake: Ordering negative numbers by their digits, e.g. writing $- 8 > - 3$ .

Instead: Use the number line. $- 8$ is to the left of $- 3$ , so $- 8 < - 3$ .

The digit gives the distance from zero, not the position. For negative numbers, further from zero means further left, and further left means smaller.

Mistake: Thinking a larger denominator makes a larger fraction, e.g. $\frac{1}{8} > \frac{1}{5}$ .

Instead: With the same numerator, the larger denominator gives the smaller fraction: $\frac{1}{8} < \frac{1}{5}$ . With different numerators, use a common denominator.

The denominator is the number of equal pieces the whole is cut into. More pieces means each piece is smaller.

Mistake: Including the end value for $<$ or $>$ , e.g. listing $- 2$ as a solution of $- 2 < x ⩽ 3$ .

Instead: $<$ and $>$ leave the end value out. Only $⩽$ and $⩾$ let it in. Check each end separately: here the list is $- 1, 0, 1, 2, 3$ .

The line under $⩽$ is the 'or equal to' part. Without that line, being equal to the end value is not allowed.

Mistake: Giving the final order as the converted decimals, or in the wrong direction.

Instead: Write the original numbers, in the direction the question asks: 'starting with the smallest' or 'starting with the largest'.

The question asks you to order the numbers it gave you. The decimals are your working, and a list in the other direction is not the order that was asked for.

Check yourself

Write $<$ or $>$ between each pair: (a) $- 4 □ - 9$ (b) $0.3 □ 0.29$ (c) $\frac{3}{7} □ \frac{4}{9}$
Answer
(a) $- 4 > - 9$ (b) $0.3 > 0.29$ , because $0.30$ against $0.29$ has more tenths (c) $\frac{3}{7} < \frac{4}{9}$ , because over 63 they are $\frac{27}{63}$ and $\frac{28}{63}$ .
Write these in descending order: $- 2.5, - 2.05, - 2.55, - 2$ .
Answer
$- 2, - 2.05, - 2.5, - 2.55$ . Descending means largest first, and the largest negative is the one nearest zero.
Write these in order of size, starting with the smallest: $0.7, \frac{7}{11}, 69 %, \frac{2}{3}$ .
Answer
$\frac{7}{11}, \frac{2}{3}, 69 %, 0.7$ . As decimals: $0.636$ , $0.667$ , $0.690$ , $0.700$ .
Write these lengths in order, starting with the shortest: $1.5$ km, $1450$ m, $155 000$ cm.
Answer
$1450$ m, $1.5$ km, $155 000$ cm. In metres they are $1450$ , $1500$ and $1550$ .
List the integers $x$ such that $- 1 < x ⩽ 4$ .
Answer
$0, 1, 2, 3, 4$ . $- 1$ is left out because of the $<$ , and $4$ is included because of the $⩽$ .
Write each statement using a symbol: (a) $t$ is at least 18 (b) the mass, $m$ kg, is less than 20 kg (c) $n$ is not equal to 5.
Answer
(a) $t ⩾ 18$ (b) $m < 20$ (c) $n \neq 5$
Without a calculator, decide which is larger: $\sqrt{90}$ or $9.5$ .
Answer
$9.5$ . Square both: ${9.5}^{2} = 90.25$ , which is more than 90, so $\sqrt{90} < 9.5$ .
Which is the largest and which is the smallest of $2^{5}, 5^{2}, 3^{3}, 6^{2}$ ?
Answer
The values are $32$ , $25$ , $27$ and $36$ . Largest: $6^{2} = 36$ . Smallest: $5^{2} = 25$ .

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1.

(a) Write these numbers in order of size, starting with the smallest: $0.304, 0.34, 0.3, 0.043, 0.403$ . 2 marks No calculator

Mark scheme

B2 0.043, 0.3, 0.304, 0.34, 0.403 cao; or B1 for one number out of place, or for the correct list from largest to smallest

(b) Write $<$ , $>$ or $=$ in each box to make a true statement. (i) $- 7 □ - 11$ (ii) $0.25 □ \frac{1}{4}$ (iii) $- \frac{1}{2} □ - 0.45$ 2 marks No calculator

Mark scheme

B2 (i) > (ii) = (iii) < all correct; or B1 for two correct

(c) Explain why $\frac{2}{3} \neq 0.6$ . 1 mark No calculator

Mark scheme

B1 2/3 = 0.666… (recurring), which is greater than 0.6; or 0.6 = 3/5 = 9/15 and 2/3 = 10/15 oe

Question 2.

(a) Without using a calculator, write these fractions in order of size, starting with the largest: $\frac{7}{9}, \frac{13}{18}, \frac{5}{6}, \frac{3}{4}$ . Show your working. 3 marks No calculator

Mark scheme

M2 all four correctly written over a common denominator, e.g. 28/36, 26/36, 30/36, 27/36, or as decimals to 2 decimal places or better (0.78, 0.72, 0.83, 0.75); or M1 for at least two correct
A1 5/6, 7/9, 3/4, 13/18 cao, using the original fractions

Question 3.

(a) Write these in order of size, starting with the smallest: $0.84, \frac{5}{6}, 83.5 %, \sqrt{0.7}, {0.91}^{2}$ . 2 marks

Mark scheme

M1 at least three of 0.833(3…), 0.835, 0.836(6…) or 0.837, 0.828(1) seen, matched to the correct values
A1 0.91², 5/6, 83.5%, √0.7, 0.84 cao

(b) Ana says that $1.25$ litres is more than $1240$ ml. Explain whether she is correct. 2 marks

Mark scheme

M1 1.25 litres = 1250 ml, or 1240 ml = 1.24 litres
A1 she is correct, with 1250 > 1240 or 1.25 > 1.24 stated; dep on M1

Question 4.

(a) A lift can carry at most 8 people. Write an inequality for the number of people, $n$ , in the lift. 1 mark No calculator

Mark scheme

B1 n ⩽ 8 oe, e.g. 8 ⩾ n; accept 0 ⩽ n ⩽ 8

(b) The lift does not move unless there is at least 1 person in it. Write a double inequality for the number of people, $n$ , in the lift when it moves. 1 mark No calculator

Mark scheme

B1 1 ⩽ n ⩽ 8 oe, e.g. 0 < n ⩽ 8

(c) Write down all the integers $x$ such that $- 3 < x ⩽ 2$ . 2 marks No calculator

Mark scheme

B2 −2, −1, 0, 1, 2 and no others; or B1 for one error or omission, e.g. −3 included, or 2 left out

(d) $x$ is an integer and $- 5 ⩽ x < 1.5$ . Write down the smallest possible value of $x$ and the largest possible value of $x$ . 2 marks No calculator

Mark scheme

B1 smallest −5
B1 largest 1

How this is examined

Ordering fractions without a calculator by putting them over one denominator: A non-calculator 'Write these fractions in order of size' with four or five fractions whose denominators share a small common multiple, worth about 3 marks Every fraction rewritten over a common denominator (e.g. $\frac{15}{24}, \frac{16}{24}, \frac{14}{24}, \frac{18}{24}$ for $\frac{5}{8}, \frac{2}{3}, \frac{7}{12}, \frac{3}{4}$ ) or as a decimal shows the method. The last mark is for the original fractions listed in the direction the question asks.
Putting a mixed list of fractions, decimals, percentages, powers and roots in order: A calculator-paper list of four to six values in different forms that are close in size, worth about 2 marks Each value changed to a decimal, with enough places to tell them apart (e.g. $0.360, 0.370, 0.374, 0.375, 0.380$ ), written beside the original. Then the answer in the original forms, e.g. ${0.6}^{2}, 37 %, \sqrt{0.14}, \frac{3}{8}, 0.38$ .
Ordering decimals with different numbers of digits, and negative numbers: A 1- or 2-mark 'Write in order of size, starting with the smallest' list of decimals such as $0.6$ , $0.606$ and $0.66$ , or of negative values The complete list in the direction asked. Decimals compared column by column after padding with zeros ( $0.600 < 0.606 < 0.660$ ); negatives placed by their position on the number line ( $- 8 < - 3$ ).
Listing the integers that satisfy an inequality, and knowing which ends are included: A 'Write down the integers' part on a double inequality such as $- 2 < x ⩽ 3$ , worth about 2 marks Exactly the integers in the range and no others. The end with $⩽$ or $⩾$ is in the list; the end with $<$ or $>$ is not. For $- 2 < x ⩽ 3$ that is $- 1, 0, 1, 2, 3$ .
Choosing the right symbol between two values, or turning words into an inequality: 1-mark parts: write $<$ , $>$ or $=$ in a box between two numbers, or write a condition such as 'at most 40 people' with a symbol The symbol whose point faces the smaller value. $=$ only when the values are exactly equal ( $0.5 = \frac{1}{2}$ , but $\frac{2}{3} \neq 0.67$ ). 'At least' is $⩾$ ; 'at most' and 'no more than' are $⩽$ .

Words to know

Magnitude: The size of a quantity. Ordering by magnitude means putting quantities in order of size.
Ascending order: Smallest first, going up to the largest.
Descending order: Largest first, going down to the smallest.
$=$ and $\neq$: 'Is equal to' and 'is not equal to'. Use $=$ only when two values are exactly the same.
$<$ and $>$: 'Is less than' and 'is greater than'. The point of the symbol faces the smaller number. The end value is not included.
$⩽$ and $⩾$: 'Is less than or equal to' and 'is greater than or equal to'. The end value is included.
Inequality: A statement that uses $<$ , $>$ , $⩽$ or $⩾$ to compare two quantities, e.g. $x ⩾ 3$ .
Double inequality: Two inequalities written as one, e.g. $- 2 < x ⩽ 3$ , meaning $x > - 2$ and $x ⩽ 3$ .
Common denominator: A denominator shared by two or more fractions after rewriting them, e.g. $\frac{3}{4} = \frac{21}{28}$ and $\frac{5}{7} = \frac{20}{28}$ .
Number line: A line with numbers marked in order. Numbers get bigger to the right.

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