## What you need to know

When ordering numbers, they can either go in ascending order (smallest to largest) or descending order (largest to smallest). Understanding place value is important for ordering numbers. Click here to learn more.

Example: Write these numbers in ascending order of size.

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

The question asks for ascending order, so we’ll 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.

Looking at the list, we can see two numbers than are under 100, and one has “4” in the tens column whilst the other has “6”, so our list will begin: 46, 64. Then, 346 is the next smallest as it’s under 400. Finally, the last two are close but 460 is smaller, since 476 has a “7” in the tens column compared to a “6”. So, our final order is

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

Example: 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, that means that -21 is the smallest number in the list. So, the completed list is

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

Example: Write the following decimals in order from smallest to largest.

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

Make sure you understand decimal place by clicking here before approaching this question. When comparing digits after the decimal point, first compare the tenths, then the hundredths, then the thousandths, and so.

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.

Looking at the digits in the “tenths” column, we see that the remaining three, from smallest to largest, are: 0.31, 0.6, 0.998. So, the complete ordering is

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

Ordering Numbers of Different Types

To put numbers of different types (fractions, decimals, percentages) in order, we have to convert them all to be the same type. If you’re unfamiliar with converting between these three types of numbers, you should click here.

Example: 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, whereas fractions are less obvious if they don’t have matching denominators.

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. To do this, we’re going to treat the fraction like a division (numerator \div denominator), but first we’ll manipulate the fractions to make our lives easier.

\dfrac{2}{5}=\dfrac{2\times 2}{5\times 2}=\dfrac{4}{10}

We do this because dividing by 10 is fairly straightforward:

\dfrac{4}{10}=4\div 10=0.4

Now, for the second fraction:

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

This time we’ll be dividing by 100, which is also fairly straightforward:

\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 numbers listed below in descending order.

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

Descending order means “from largest to smallest”. None of the negative numbers will be largest, so let’s consider the positives: 1, 4, and 23. 23 is the only one in the tens, so that is the biggest, and then clearly 4 is bigger than 1, so our list begins

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

There are two negative numbers: -23.5 and -42. 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 completed ordering is

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. The first observation is that only one of these values is bigger than three: 3.5, so this will be the biggest. Now, we consider the other four.

They all begin with a 2, but 2.04 has a zero in the ‘tenths’ column, meaning it must be smaller than all the others that have some number bigger than 0.

Next, 2.5 and 2.58 are close, but 2.58 has that extra ‘8’ in the ‘hundredths’ column making it slightly bigger.

Lastly, 2.8 has an ‘8’ in the tenths column so must be second biggest. Thus, the completed list is

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

3) Write the following numbers 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. You could reasonably choose to turn them all to percentages, but we’re going to go for decimals again.

Firstly, 0.633 is already in decimal form.

Secondly, to turn the percentages into decimals, we divide by 100:

• 64\%=64\div 100=0.64.

• 64.4\%=64.4\div 100=0.644.

Lastly, to turn the fraction to a decimal we need to treat it like a division and divide 5 by 8. Use whichever method you prefer – here we’re going to do the bus stop method:

So, the fraction is equal to 0.625. So, putting these decimals in order, we get

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\%

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