Rounding in everyday life: up, down or to the nearest.
	Procedures for rounding numbers
	Significant figures: When 12.00 is not the same as 12.0 or 12.
	Items to promote class discussion (lesson from Teaching Section)

Rounding in everyday life

a) Rounding up
If you needed to buy 123 cm of fabric, you would find that, at most shops, they cut fabric in multiples of 20 cm. So you cannot buy 123 cm of fabric: a decision needs to be made between 120 cm and 140 cm. Usually, people buy the larger amount and have a small amount of wastage rather than risk not having enough. In this case, we have rounded up to the nearest 20 cm. The first row of the following table shows the length required, and the second row shows the length purchased, after rounding up to the nearest 20 cm.

b) Rounding down
If you buy groceries at the supermarket in Australia, the cost of the good may be $24.63 but it is not possible to pay this amount of money, due to the abolition of one and two cent coins. Rounding can be done in various ways; the one which seems 'fairest' to the customer is when the amount is rounded down to the nearest 5c. The first row of the following table shows the total amount of the bill, and the second row shows the amount to be paid, after rounding down to the nearest 5c.

c) Rounding to the nearest 5c
If you visit a small shop it is more likely that your total will be adjusted by rounding to the nearest 5c. In this case the shop-owner is not prepared to make a loss on (almost) every sale of a few cents as these will accumulate in the long term to considerable amounts. The following table shows the same first row (the total of the bill) as above, but the amounts to be paid are different as this time the amounts are rounded to the nearest 5c.

*** Logically, because $550 is exactly halfway between $500 and $600, $550 rounded to the nearest $100 could be either $500 or $600. By convention, we say it is $600.

Top

Procedures for rounding numbers

Here are some examples of rounding to the nearest hundred, ten and tenth.

a) Rounding to the nearest hundred
Consider rounding 936 to the nearest hundred. If we draw a numberline in hundreds,(e.g. 8 hundred, 9 hundred, 10 hundred, 11 hundred) then 936 is closer to 9 hundred than any other hundred. In fact, any number between 850 and 950 will round to 9 hundred (see lower diagram). How to treat the numbers which are exactly equal distance from the hundred markers (e.g. 850 and 950) needs to be decided. Usually, these are rounded up, so 850 is rounded to 9 hundred, and 950 is rounded to 10 hundred (i.e. one thousand).

While this process can be taught as an algorithm, (e.g. identify the digit after the relevant place value column; if it is 5 or above, then round up, otherwise round down) it is important to understand the process involved.

b) Rounding to the nearest ten
If we repeat the above ideas for rounding 936 to the nearest ten, we need a numberline with marks on the tens, e.g. 92 tens, 93 tens, 94 tens and 95 tens. Then 936 will be seen to nearest to 94 tens (usually called 940). The first diagram shows this, while the second shows that any number from 935 to 945 is rounded to 940 (when rounding to the nearest ten). Remember that the numbers which are exactly in the middle need special consideration; we usually round these up, but this has just been agreed by people (there is no logical reason).

Instead of rounding to the nearest ten, we could try some other rounding. To round 936 up to the nearest ten also gives 940, and to round 936 down to the nearest ten gives 930.

c) Rounding to the nearest tenth
This time we need a numberline with tenths marked on it. For example, 27.53 can be thought of as 2753 hundredths, or 275 tenths and 3 hundredths (see number expanders). The numberline marked in tenths (273 tenths, 274 tenths, 275 tenths, 276 tenths) is usually marked as below. Then 275.3 tenths is closer to 275 tenths than 276 tenths, so 27.53 is closer to 27.5 than 27.6. As the second diagram shows, any number between 27.45 and 27.55 is rounded to 27.5 when rounding to the nearest tenth.

Bank interest
Interest of 3% for 1 year on $12000 is $360 per year, or $6.923076923 per week, or $0.986301369 per day. If the bank calculated the interest daily, rounding to the nearest cent, they would pay 99 cents a day or $361.35 over the year. If, on the other hand, they round down to the nearest cent, they would pay 98 cents daily or $357.70 over a year, saving themselves $2.30. Rounding is certainly an important issue when money is concerned!

Top

Significant Figures - When 12.00 is not the same as 12.0 or 12

How measurements are made
Consider the pencil shown in the diagram. We can try to measure its length using a measuring device placed below the pencil. The length that we decide upon depends on the divisions on the measuring device. For example, if our measuring device only has cm marked on it, the pencil can only be judged to be close to 4 cm, or maybe nearly 4 1/2 cm but not 4.0 cm or 40 mm as these imply we have made finer judgements than is possible with this measuring device.

By measuring the same pencil with the second measuring device, which is marked in tenths of centimetres (i.e. millimetres), we can make a more accurate decision of 4.3 cm or 43 mm. Again, we could not write 4.30cm or 43.0 mm as this implies we have made judgements about tenths of millimetres.

Under a microscope, using a measuring device marked in tenths of millimetres, we can further judge that the length is 4.32cm.

So the measuring device that we use to measure an object affects the judgement we make and therefore the way we record the length. The following table summarises these ideas.

So, is 12 equal to 12.00 and 12.00?
There are two answers to this simple question:

• As exact mathematical numbers, the answer is YES!
• As measurements, the answer is NO!

A measurement of 12 indicates a real size of between 11.5 and 12.5
A measurement of 12.0 indicates a real size of between 11.95 and 12.05
A measurement of 12.00 indicates a real size of between 11.995 and 12.005

Top