I encounter many students who struggle with the concept and practice of rounding.

The general procedure taught in school is that if the last digit is 0-4 we round down, and if it is 5-9 we round up, but what does “rounding up” and “rounding down” mean, and why is this important?

The word “rounding” can be quite confusing, since it sounds like you are doing something round, like a circle, but this is not what this term is concerned with. So let us dispel any notion that rounding has something to do with circles. Rather, the term “round” when used as a verb (to round) is concerned with find a round number, which is the closest number to another number, rather than as an adjective (round) which is used to describe circular objects.

Thus the term “rounding” means to find the closest number of a particular form. If we are asked to round to the nearest 10, then the outcome should be a number like 10, 20, 30, etc… and if we are asked to round to the nearest 2 decimal places, then we are expected to provide a number like 243.23 or 0.17, but not 243.2 or 243.22785.

In some cases we have to found to the nearest 5, and for this the outcome will be a number which is divisble by 5 such as 0, 5, 10, 15, 20, 25, ….

## Why do we round?

It is important in all mathematics to understand why we do things and what the different mathematical structures are for. Putting mathematics into context makes it more meaningful and less abstract.

Rounding is used for us to determine an approximate amount since this is often easier to communicate. For example, if I am describing to somehow how far it is for me to drive to work every day I will say “about 20km”, rather than saying “exactly 18.6km”. No-one expects me to tell them exactly how far it is to my work, and to get an idea of a close amount is sufficient. This is rounding to the nearest 5km, since I could have also said 15km or 25km, but neither of these are that good and are not that close. I could have said 18km or 19km and this would also have been informative, but when asked a request such as how far it is to drive to work people do not expect a detailed answer and only an approximation.

I am sure that almost every day you are engaged in conversations in which you are discussion measures and numbers in a way in you use the work “about” to approximate the number when you do not know the exact number or if it is not necessary to be so exact.

For example, if I ask you how much per week do you spend on airtime you will probably not say “exactly R20.45”, but will rather say “about R20”.

Rounding is the mathematical operation which allows us to find the number which is closest according fo a simple set of rules, and which is also appropriate to the context.

## Rounding to the nearest unit

The simplest case of rounding is where we have to round to the nearest unit.

As an example, we have a number 452.35 and we want the nearest whole number to this. This can be achieved by simply dropping the decimal fraction .35 leaving us with 453. But what if the the numebr is 452.93. This is clearly closer to 453 than to 452 and this it is better to round UP to 453.

The rule mentioned above will apply. If the first digit is 0-4 we round down and if it is 5-9 we round up. In this case we do not need to look at any other digits in the decimal fraction.

For example, all of the numbers 3.4, 3.49, 3.495, 3.4957 will ALL round down to 4, since the first digit of the decimal fraction is 4, and this indicates we should round down.

This example may be confusing to some, since if we take 3.49 to one decimal place this will become 3.5 and if we then round this it will become 4, but rounding up since the digit 5 will cause a round up. However, this means that when we round we do this in a single operation and not in multiple operations, and in all cases we only need to look at a single digit to make the decision.

## Rounding to Two Decimal Places

It is very common in examinations that answers are required to be entered to two decimal places if this is not specified elsewhere. This means that you need to know how to round to two decimal places since this may be used many times on every examination which you write. If you fail to round the answer you are likely to lose a mark, even if this was not specified in the question.

In this case and answer of 2.456 must be rounded UP to 2.46, since the last digit is 6 which causes you to round up.

So what about 2.45633?

In this case you must look at the first decimal digit AFTER the point you are to stop. In this case, since your rounding to TWO decimal places you must look at the digit after the second decimal position, which is the 6 after the .45 and is not the additional 33 at the end, which are not needed for determining how to round.

## Try these

Original number | Rounded number |
---|---|

2.45 | |

0.6 | |

0.798 | |

23.1999 | |

452.5 |

## Rounding to nearest 5

In most cases you are asked to round to a number which is a power of ten, such as to the nearest hundred (100), ten (10), unit (1), tenth (0.1 – one decimal place), or hundredth (0.01 – 2 decimal places).

If you are asked to round to a number other than a power of ten, then you cannot use the rule about that 0-4 rounds down and 5-9 rounds up.

For example consider the problem of rounding 372 to the nearest 5. For this you will need to find the lowest and hight numbers around 372 which are multiple of 5, and then to select which is the closest. You should be able to this by inspection, but to help you understand this better you can find these numbers youself. In this case the numbers are 370, which is the multiple of 5 which is lower than 372 and 375, which is the multipel of 5 larger than 372. It is clear that 372 is closer to 370, which is a difference of 2 (372-370), rather than 375 which is a difference of 3 (375-372).

## Try these

Original number | Rounded to nearest 5 |
---|---|

245 | |

343 | |

798 | |

2319 | |

452 |

## Rounding to nearest 10, 100, 1000

These are essentially the same process as rounding to the nearest unit. But the answer will be a multipe of 10, 100, or 1000.

For example, rounding 2459 to the nearest 10 is 2460, because the last digit is 9, which means we round up. Rounding this to the nearest 100 give 2500, since the last digit is now 5, being the digit after the 100 position. Rounding to the nearest 1000, then gives us 2000, since the last digit when considering 1000s is 4 which means round down.

## Try these

Original number | Rounded to nearest | Answer |
---|---|---|

2459 | 10 | |

2459 | 100 | |

2459 | 1000 | |

765 | 10 | |

765 | 100 | |

765 | 100 | |

245.595 | 1 | |

245.595 | 10 | |

245.595 | tenth / 1 dec place | |

245.595 | hundredth / 2 dec place |

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