## What you need to know

## Ordering Numbers

When ordering numbers, they can either go in ascending order (smallest to largest) or descending order (largest to smallest). Understanding place value and fractions, decimals and percentages is important for ordering numbers.

### Example 1: Ascending Order

Write these numbers in ascending order of size.

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

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

Now, the biggest numbers here are hundreds. To compare sizes, you should first compare the hundreds, then the tens, then the ones. If there were numbers in then we would start by comparing thousands, and so on. The final answer is.

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

### Example 2: Descending Order

Write these numbers in descending order of size.

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

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, so, the completed list is

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

### Example 3: Ascending Order

Write the following decimals in order from smallest to largest.

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

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

### 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}

You can choose which format to express you numbers in, but in general decimals is a good choice. This is because decimals are very easy to compare side-by-side.

0.45 is already in decimal form.

To convert a percentage to a decimal, we divide the value by 100:

44.5\%=44.5\div 100=0.445

Now, we have two fractions to convert to decimals.

\dfrac{2}{5}=0.4

Now, for the second fraction, we make this our of 100 in order to convert to a decimal:

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

\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

### Example Questions

1) Write the following numbers in descending order:

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

**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

2) Write the following numbers in ascending order:

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

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

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

3) Order the following numbers from smallest to largest,

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

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

4) Write the following in order from smallest to largest,

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

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\%

5) Write the following numbers in ascending order, given that x is a positive integer greater than 2.

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

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

### Worksheets and Exam Questions

#### (NEW) Ordering Numbers Exam Style Questions

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