Ordering Numbers Worksheets, Questions and Revision

Ordering Numbers Worksheets, Questions and Revision

GCSE 1 - 3KS3AQAEdexcelOCRWJECFoundationAQA 2022Edexcel 2022OCR 2022WJEC 2022

Ordering Numbers

When ordering numbers, they can either go in ascending order (smallest to largest) or descending order (largest to smallest). There are 5 main ways that you will be asked to order numbers.

Make sure you are happy with the following topics before continuing.

Level 1-3 GCSE KS3 AQA Edexcel OCR WJEC

Type 1: Ordering decimals

You may be asked to order decimals, to do this we need to look at each number in turn and compare starting with the first digit.

Example: Put the following numbers in ascending order.

1.101, \,\,\, 0.11, \,\,\, 1.10,  \,\,\, 0.01

Step 1:  look at the first digit for each number and compare.

\xcancel{\textcolor{blue}{1}.101}, \,\,\, \overset{\downarrow}{\textcolor{blue}{0}}.11, \,\,\, \xcancel{\textcolor{blue}{1}.10},  \,\,\, \overset{\downarrow}{\textcolor{blue}{0}}.01

We can see 2 numbers start with 1 and 2 numbers start with 0, as we are looking for the smallest first, we can eliminate the two largest.

Step 2: look at the second digits for the remaining numbers.

\xcancel{0.\textcolor{blue}{1}1}, \,\,\, 0.\overset{\downarrow}{\textcolor{blue}{0}}1

We can see that the second digit for the 1st number is a 1 and for the 2nd number it is a 0, so as we are looking for the smallest first, we can eliminate the 1st number.

So we have 0.01 in position 1 and 0.11 in position 2.

Step 3: look at the other numbers that we previously crossed out, to see which of them is the smallest.

\xcancel{1.{\textcolor{blue}{1}} {\textcolor{red}{0}}{\textcolor{limegreen}{1}}}, \,\,\, 1.{\textcolor{blue}{1}}{\textcolor{red}{0}}

We look at the second digit for each number and compare, but these are the same. Comparing the 3rd digit, but again these are the same. We can see that the 1st number has a fourth digit, but the 2nd number doesn’t, so we can eliminate the 1st number as it is larger.

So we have 1.10 in position 3 and 1.101 in position 4.

Step 4: write down the correct order, which is

0.01, \,\,\, 0.11, \,\,\, 1.10, \,\,\, 1.101

Level 1-3 GCSE KS3 AQA Edexcel OCR WJEC

Type 2: Ordering Large Numbers

You may be asked to order large numbers, to do this we need to look at each number in turn and compare starting with the first digit.

Example: Put the following numbers in descending order.

3456, \,\,\, 5364, \,\,\, 3465, \,\,\ 5634

Step 1: we look at the first digit for each number and compare.

\xcancel{\textcolor{blue}{3}456}, \,\,\, \overset{\downarrow}{\textcolor{blue}{5}}364, \,\,\, \xcancel{\textcolor{blue}{3}465},  \,\,\, \overset{\downarrow}{\textcolor{blue}{5}}634

2 of the number start with 3. As we are looking for the largest first, we can eliminate the 2 smallest.

Step 2: look at the second digits.

5 \xcancel{\textcolor{blue}{3}64}, \,\,\,  5\overset{\downarrow}{\textcolor{blue}{6}}34

We know 6>3 so we can eliminate the first number.

So we have 5634 in position 1 and 5364 in position 2.

Step 3: look at the other numbers.

3 \xcancel{\textcolor{blue}{4}\textcolor{red}{5}6}, \,\,\,  3 \textcolor{blue}{4}\overset{\downarrow}{\textcolor{red}{6}} 5

The second digit for each number is the same, so we compare the third digit. We know 6>5 so we can eliminate the first number.

So we have 3465 in position 3 and 3456 in position 4.

Step 4: write down the correct order, which is

5634, \,\,\, 5364, \,\,\, 3465, \,\,\, 3456

Level 1-3 GCSE KS3 AQA Edexcel OCR WJEC
Level 1-3 GCSE KS3 AQA Edexcel OCR WJEC

Type 3: Ordering Fractions, Decimals and Percentages

We may be asked to order fractions, decimals and percentages, which can be tricky. Revise fractions, decimals and percentages here.

Example: Put the following numbers in ascending order.

\dfrac{1}{5}, \,\,\, 21 \%, \,\,\, 0.205, \,\,\, \dfrac{1}{4}

Step 1: First, we need to convert all of these numbers into the same format. For this example we will convert them all to decimals.

0.205 is already in decimal form.

Fractions to decimals:

\dfrac{1}{5} = 0.2 \,\,\, and \,\,\, \dfrac{1}{4} = 0.25

Percentage to a decimal:

21 \% \div 100 = 0.21

Step 2: We now rewrite the list using our conversions to decimals:

0.2, \,\,\, 0.21, \,\,\, 0.205, \,\,\, 0.25

Step 3: We now use what we have already learnt from Type 1 to order our list in ascending order.

0.2, \,\,\, 0.205, \,\,\, 0.21, \,\,\, 0.25

Step 4: Finally, write down the correct order, converting back to the original values, which is

\dfrac{1}{5}, \,\,\, 0.205, \,\,\, 0.21\%, \,\,\, \dfrac{1}{4}

Level 1-3 GCSE KS3 AQA Edexcel OCR WJEC

Type 4: Ordering Negative Numbers

Remember that for negative numbers, the bigger the number after the minus sign, the smaller the value of the negative number.

Example: Put the following numbers in descending order.

2.3, \,\,\, -2.3, \,\,\, 3.2, \,\,\, -3.2

Step 1: We first look at the positive numbers and compare their first digits.

\xcancel{\textcolor{blue}{2}.3}, \,\,\,  \overset{\downarrow}{\textcolor{blue}{3}}.2

3>2, this gives 3.2 in position 1 and 2.3 in position 2.

Step 2: We now look at the negative numbers and compare their first digits.

- \overset{\downarrow}{\textcolor{blue}{2}}.3, \,\,\,  - \xcancel{\textcolor{blue}{3}.2}

-3 is more negative than - 2

So we have -2.3 in position 3 and -3.2 in position 4.

Step 3: Finally, we write down the correct order, which is

3.2, \,\,\, 2.3, \,\,\, -2.3, \,\,\, -3.2

Level 1-3 GCSE KS3 AQA Edexcel OCR WJEC

Type 5: Ordering Algebraic Terms

We may also be asked to order numbers that are algebraic, in terms of x for example, with a given condition on its value.

Example: Put the following numbers in ascending order, given that x is a positive integer greater than 1.

x^2, \,\,\, \dfrac{1}{x}, \,\,\, x, \,\,\, x+1

Step 1: Substitute a value in for x that is greater than 1. We will substitute in x=2 as this will be easiest.

x^2 = 2^2 = 4 \,\,\,, \,\,\, \dfrac{1}{x} = \dfrac{1}{2}\,\,\,, \,\,\, x = 2\,\,\,, \,\,\, x+1 = 2+1 = 3

Step 2: We now rewrite the list using the substituted values.

4, \,\,\, \dfrac{1}{2}, \,\,\, 2, \,\,\, 3

Step 3: We now use what we have already learnt to order our list in ascending order.

\dfrac{1}{2}, \,\,\, 2, \,\,\, 3, \,\,\, 4

Step 4: Finally, write down the correct order, writing in terms of x, which is

\dfrac{1}{x}, \,\,\, x, \,\,\, x+1, \,\,\, x^2

Level 4-5 GCSE AQA Edexcel OCR WJEC
Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC

Example 1: Ascending Order

Write these numbers in ascending order of size.

460,\,\,\, 346,\,\,\,64,\,\,\,46,\,\,\,476

[1 mark]

We need to start with the smallest and increase in size.

Now, the biggest numbers here are hundreds.

460,\,\,\, 346\,\,\,476

Then, numbers in the 400‘s

460<476

This gives the final answer to be,

46,\,\,\,64,\,\,\,346,\,\,\,460,\,\,\,476

Level 1-3 GCSE KS3 AQA Edexcel OCR WJEC

Example 2: Descending Order

Write these numbers in descending order of size.

-3,\,\,\,12,\,\,\,18,\,\,\,-21,\,\,\,4

[1 mark]

Firstly, deal with the positives. Going in descending order – from largest to smallest –  we get

18,\,\,\,12,\,\,\,4

Now, recall that negative numbers get smaller when the number after the minus sign gets bigger.

The completed list is

18,\,\,\,12,\,\,\,4,\,\,\,-3,\,\,\,-21

Level 1-3 GCSE KS3 AQA Edexcel OCR WJEC

Example 3: Ascending Order

Write the following decimals in order from smallest to largest.

0.6,\,\,\,0.31,\,\,\,0.07,\,\,\,1.04,\,\,\,0.998

[1 mark]

Firstly, 1.04 is the only value bigger than 1 here so it must be the largest.

Secondly, 0.07 is the only value that’s smaller than 0.1, so it must be the smallest.

Continuing this we get,

0.07,\,\,\,0.31,\,\,\,0.6,\,\,\,0.998,\,\,\,1.04

Level 1-3 GCSE KS3 AQA Edexcel OCR WJEC

Example 4: Ordering Numbers of Different Types

Put the numbers listed below in order from smallest to largest.

\dfrac{2}{5},\,\,\,0.45,\,\,\,44.5\%,\,\,\,\dfrac{7}{20}

[2 marks]

In this example, we will convert all numbers to decimals.

0.45 is already in decimal form.

44.5\% =44.5\div 100 = 0.445

\dfrac{2}{5} = 0.4

\dfrac{7}{20} = \dfrac{7\times 5}{20\times 5}=\dfrac{35}{100}=35\div 100 = 0.35

Now we have our 4 decimals, we can put them in order from smallest to largest:

0.35,\,\,\,0.4,\,\,\,0.445,\,\,\,0.45

Finally, we must write the numbers in order in their original forms. This looks like

\dfrac{7}{20},\,\,\,\dfrac{2}{5},\,\,\,44.5\%,\,\,\,0.45

Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC

Example Questions

Descending order means from largest to smallest, so let’s first place the positive numbers in order,

 

23,\,\,\,4,\,\,\,1

 

Recall that for negative numbers, the bigger the number after the minus sign, the smaller the value of the negative number.

 

42 is bigger than 23.5, which means -42 is smaller than -23.5, so the order should be,

 

23,\,\,\,4,\,\,\,1,\,\,\,-23.5,\,\,\,-42

Ascending order means from smallest to largest, hence the correct order is,

 

2.04,\,\,\,2.5,\,\,\,2.58,\,\,\,2.8,\,\,\,3.5

Ordering decimals requires comparing digits in the same columns, starting with the digits in the place value column that is furthest to the left, hence the correct order is,

 4.092,\,\,\,\,\,4.87,\,\,\,\,\,5.01,\,\,\,\,\,5.12,\,\,\,\,\,   5.23

To compare the sizes of these numbers, we need to have them all in the same form.

In order to turn a percentage into a decimal, we divide by 100, hence,

 

64\%=64\div 100=0.64

and,

64.4\%=64.4\div 100=0.644

 

We can also convert a fraction to a decimal,

 

\dfrac{5}{8} = 0.625

Placing the values in order then,

 

0.625,\,\,\,0.633,\,\,\,0.64,\,\,\,0.644

Finally, putting them in order in the original forms, we get

\dfrac{5}{8},\,\,\,0.633,\,\,\,64\%,\,\,\,64.4\%

A simple way to compare the sizes of these numbers is to substitute a value in for x that is greater than 2.

We will substitute in x=3 for each of the terms,

 

\dfrac{1}{x}=\dfrac{1}{3}

 

x^2=3^2=9

 

x=3

 

(x+1)=4

 

2x=6

 

Now it is straightforward to place them in order,

 

\dfrac{1}{3} ,\,\,\,\,\,3,\,\,\,\,\,\,4,\,\,\,\,\,6,\,\,\,\,\,9

 

Hence we can place the original terms in the same corresponding order,

 

\dfrac{1}{x} ,\,\,\,\,\,x,\,\,\,\,\,\,(x+1),\,\,\,\,\,2x,\,\,\,\,\,x^2

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