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Rounding numbers

rounding numbers rule

Have you ever wondered what to do when you just have too many digits to do a quick calculation? When you don’t need an exact, but an approximate result? Do your decimal numbers have an infinite repeating decimal expansion – like when you try to divide number 10 by 3 and the threes just go on forever? Well, there’s a light at the end of the tunnel – a great method called rounding.

Rounding is a method in which use to reduce the number of digits in order to simplify the calculation or if we do not need that much digits, while trying to keep the value of a number approximately the same as the initial value. The choice of values you want to round a number to depends on the smallest order of magnitude you want to work with – in front of the decimal point (units, tens, hundreds, thousands…) or behind it (tenths, hundredths, etc.). The difference between the exact value and the rounded (or approximate) value is called the round-off error.

rounding numbers rule

The sign we use to note that two values are approximately the same (which means “close enough for our needs”) is “≈”. So if we want to say the number 47 is approximately number 50, we write 47 ≈ 50.


Rounding methods

There are five common methods of rounding numbers which you can use, depending on your current needs. They are:

  • rounding up,
  • rounding down,
  • rounding towards zero,
  • rounding away from zero and
  • rounding to the nearest acceptable value.

To round up is increasing the value of a number to the nearest acceptable larger value. So, if we round up to the nearest ten, any number larger than number  30 and smaller than number 40 will be rounded to number 40.

round up round down

To round down means the exact opposite – we decrease the value of a number to the nearest acceptable smaller value. So, if we round down to the nearest ten, any number larger than number 30 and smaller than number 40 will be rounded to number 30.

To round towards zero means to change the value of a number to the nearest acceptable value that is closer to zero than the number we are currently changing.

To round away from zero is to change the value of a number to the nearest acceptable value that is further away from zero than the number we are changing.

To round the number to the nearest acceptable value is to change the value of a number to the nearest acceptable value, and it doesn’t matter if it is larger or smaller than the number we are changing.

  • If the difference between the number we are rounding and the nearest acceptable smaller value is lesser than the difference between that number and the nearest acceptable larger value – we round down.
  • If it’s the other way around – we round the number up. This last method is guided by a special rule that we use in case the number is equally distant from the nearest larger and the nearest smaller acceptable value (if we are rounding to the nearest ten, an example for such a case would be the number 35 – which is exactly five units larger than the number 30 and exactly five units smaller than the number 40).

Most commonly we use the “round half up” rule, which means that we take the digit we want to round and add the value it represents. For example, if we want to round the number 45 to the nearest ten, we add to it 5 units and, by doing so, round it up to 50. And if we want to round the number 4500 to the nearest thousand, we add 5 hundreds and get the number 5000 as the result.

Examples

Natural numbers

Rounding natural numbers is pretty straightforward so we’ll just go through a couple of examples. Let’s say we use the number 114 and we want to get rid of the units by rounding them to the appropriate multiple of ten (ten being the lowest order of magnitude we want to work with).
If we want to round the number 114 up, we will round it to the number 120. (114 ≈ 120)

round positive whole numbers

If we want to round the number 114 down, we will round it to the number 110. (114 ≈ 110)
If we want to round the number 114 towards zero, we will round it to the number 110. (114 ≈ 110)
If we want to round the number 114 away from zero, we will round it to the number 120. (114 ≈ 120)
If we want to round the number 114 to the nearest acceptable value (in this case, the nearest multiple of number 10), we will have to do a bit of addition and subtraction to decide where we want to go.

114 - 110 = 4

120 - 114 = 6

Since number 4 is less than nuber 6, the nearest multiple of 10 to the number 114 is the number 110, which means that we will round it down to 110. (114 ≈ 110)

 

Integers

The method for rounding integers is a bit different from the method we use to round natural numbers – because what we consider to be up and down when it comes to positive numbers is basically reversed when we are talking about negative numbers. The larger number is the one that is closer to zero, while the smaller number is the one further away from zero.
To make it clearer, let’s look at some more examples. This time we’ll use the number -114 and round it to a multiple of ten.
If we want to round the number -114 up, we will round it to the number -110. (-114 ≈ -110)
If we want to round the number -114 down, we will round it to the number -120. (-114 ≈ -120)
If we want to round the number -114 towards zero, we will round it to the number -110. (-114 ≈ -110)
If we want to round the number -114 away from zero, we will round it to the number -120. (114 ≈ -120)

If we want to round the number -114 to the nearest acceptable value (in this case, the nearest multiple of 10), we will have to do a bit of addition and subtraction again. Since it doesn’t change the distance between numbers, we will use their positive counterparts (or absolute values) to make it a bit easier for us.

114 - 110 = 4

120 - 114 = 6

Since number 4 is less than number 6, the nearest multiple of 10 to the number -114 is the number -110, which means that we will round it up to -110. (-114 ≈ -110)

Decimal numbers

Decimal numbers, positive or negative alike, are rounded as  integers – the only difference being that decimals can be rounded to the nearest unit or tenth, hundredths, thousandths, ten thousandths, hundred thousandths, millionths, ten millionths, hundred millionths, billionths etc. – depending on the desired level of precision.

Other than that, the same rules apply.

rounding decimal numbers

For example, let’s say we want to round the decimal number 0.77 to the appropriate tenth.
If we want to round the number 0.77 up, we will round it to the number 0.8. (0.77 ≈ 0.8)
If we want to round the number 0.77 down, we will round it to the number 0.7. (0.77 ≈ 0.7)
If we want to round the number 0.77 away from zero, we will round it to the number 0.8. (0.77 ≈ 0.8)
If we want to round the number 0.77 to the nearest acceptable value (in this case, the nearest tenth), we have to do a bit of subtraction to decide the direction in which to go in again.

0.77 - 0.7 = 0.07

0.8 - 0.77 = 0.03

Since number 0.03 is less than 0.07, the nearest tenth to the number 0.077 is the number 0.08, which means that we will round it up to 0.08. (-114 ≈ -110)

Fractions

Rounding fractions follows the same principals as rounding decimals. You can convert them to decimal numbers before rounding or you can round them as fractions. In case you want to round them as fractions, you can round them to another fraction or to an integer. In any case, the rules are the same – even if the numbers seem different. Take the fraction you want to round and position it in relation to the fraction you have to round to – just like you would do with decimals or integers. After that, all you have to do is to use the desired method and that is it.

rounding fractions

Whether you want to round an natural number, integer, decimal number or fraction, just follow the previous rules and you’ll never have to worry about it again. If you want to practice a bit more or test your knowledge, please feel free to use our worksheets.

Rounding worksheets

  Rounding to ones (84.6 KiB, 847 hits)

  Rounding to tens (84.3 KiB, 788 hits)

  Rounding to hundreds (84.9 KiB, 631 hits)

  Rounding to thousands (84.5 KiB, 611 hits)

  Rounding to ten thousands (84.8 KiB, 561 hits)

  Rounding to hundred thousands (84.4 KiB, 594 hits)

  Rounding to millions (84.9 KiB, 563 hits)

  Rounding to ten millions (85.8 KiB, 640 hits)

  Rounding to hundred millions (90.4 KiB, 599 hits)

  Rounding to billions (90.1 KiB, 559 hits)

  Rounding to tenths (88.2 KiB, 1,950 hits)

  Rounding to hundredths (87.8 KiB, 511 hits)

  Rounding to thousandths (88.2 KiB, 615 hits)

  Rounding to ten thousandths (89.2 KiB, 681 hits)

  Rounding to hundred thousandths (88.9 KiB, 488 hits)

  Rounding to millionths (91.0 KiB, 622 hits)

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