Rounding

Rounding can be done is several different ways and to different precisions. Consider:

a) Rounding to a specified power of ten ( or 'column' in the decimal system ):

558 rounded to the nearest 10 = 560
75995 rounded to the nearest 1000 = 76000
6888342 rounded to the nearest 10000 = 6890000

b) Rounding to the nearest 25, 50, 250 or 500:

4079 rounded to the nearest 250 = 4000
389 rounded to the nearest 50 = 400
19773 rounded to the nearest 500 = 20000

c) Rounding a whole number to significant figures:

581720 rounded to 4 significant figures = 581700
34396 rounded to 3 significant figures = 34400
23315 rounded to 4 significant figures = 23320
58115 rounded to 2 significant figures? 58000
4555035 rounded to 2 significant figures? 4600000

d) Rounding a decimal number to significant figures:

0.00345 rounded to 1 significant figure = 0.003
0.00027575 rounded to 2 significant figures = 0.00028
0.00831251 rounded to 3 significant figures? 0.00831
0.0004268095 rounded to 4 significant figures? 0.0004268

e) Rounding a decimal number to decimal places:

52.299 rounded to 1 decimal place? 52.3 9.942698 rounded to 2 decimal places? 9.94 677.90 rounded to 1 decimal place? 677.9 5.0289 rounded to 3 decimal places? 5.029 3.50053 rounded to 4 decimal places = 3.5005
5.09694 rounded to 2 decimal places = 5.10

The rules

The basics
As an example consider £12, £15 and £19. If we round to the nearest ten pound, £12 is clearly nearer to £10 than £20.
The answer then is a digit "1" , and a zero. However £19 is nearer to £20, so we 'round up' the 1 to a 2.
The problem comes when we have £15, as that is equally near to £10 as it is to £20. However the rule is that 5 "rounds up:"
"For 5 and above, give it a shove." This is the golden rule when a decision is split in the middle. It applies to all rounding.
Rounding to the nearest 10, or 100 etc.
means what is closest to the value to be rounded.
For example, £13 rounded to the nearest £10 is £10. £17 rounded to the nearest £10 is £20.
Basically the answer can only be in multiples of the 'nearest' amount. So 2xxx can only be either 2000 or 3000.
£15 rounded to the nearest £10 is £20.
£150 rounded to the nearest £100 is £2000.
Rounding to 25, 50, 250, 500 etc.
Rounding to the nearest 25 or 50 is similar to tens, hundreds etc.
For example, 'to the nearest 25' means the answer can only be a multiple of 25:
£57 rounded to the nearest 25 is £50, because that is the nearest multiple of 25.
£27 rounded to the nearest 25 is £25.
£79 rounded to the nearest 25 is £75.
£776 rounded to the nearest £250 would be £750.
Significant figures.
Reading a number left to right, the significant figures *start* with the first non-zero digit.
For whole numbers, this is simply the same as the initial digits, as whole numbers can't start with a zero:
3456 to 1 sig fig is 3000.
723 to 2 sig figs is 720.
794 to 1 sig fig is 800 ( you always check the digit to the right. 9 rounds up. )
For decimal numbers, you have to ignore the initial zeros:
0.00425 to 1 sig fig is 0.004.
0.05062 to 3 sig figs is 0.0506. Note the sandwiched zero counts as a sig fig, as we have already hit a non-zero digit, 5.
Decimal places.
Only makes sense for decimals. Literally to the number of places to the right of the decimal point:
0.0537 to 2 decimal places is 0.05.
0.39493 to 1 decimal place is 0.4 ( always look to the one digit to the right; 9 is higher than 4 so round up. )

Estimation

Estimation is more a 'real-world' skill but the rules can be counter-intuitive. Indeed, the 'rules' they expect you to follow for full marks according to the learning scheme is often not what people do in the real world. The general rule is that estimation = round to 1 significant figure. Often, the numbers are chosen so as not to be confusing. Example:

"Estimate 1943 + 203" -> this would become 2000 + 200 = 2200. So far so good. But consider this:

"Estimate £1501 + $149"

In the "real world," anyone with any sense would see this as £1500 + £150, which would be £1650. But that's not corret for GCSE!

Working strictly to 1 sig fig, it becomes 2000 + 100 = 2100.

As stated, normally the numbers won't be chosen to be deliberately misleading, but you never know.

The exceptions

These 'other cases' are a big problem, because the guidance and expectations varies between exam boards! The scenarios are:

- squaring numbers e.g. 16.7²
- taking square roots e.g. √83
- where there is a fractional / decimal amount in the denominator e.g. 10 / 0.5

In general however, the exam boards prioritize "completable math" over rigid adherence to the 1 sig fig rule when it results in a nightmare calculation.

Squares
Example: 16.7²) - you usually stick to the 1 sig fig rule : 16.7 becomes 20² = 400.
Note: Even though 17 or 16 might seem "closer," 20 is the expected estimation for a non-calculator paper because 20 x 20 is an "instant" calculation.

Be aware however if they trick you with an 'easy' known square, say 15.1 , most exam boards would accept 15 x 15 = 225.

Square Roots
Example: √67 - This is the biggest exception - you do not round to 1 sig fig. Instead, you round to the nearest square number.
For √67, rounding to 1 sig fig would give √70 which obviously will be a difficult decimal. Instead, choose 64, as this is the nearest perfect square number. THe answer then is √64 = 8.

Do not round to 17 and square it! This gives 289, but 17 squared is not considered "immediate maths" which is what estimation is all about.

Fractional denominators
Denominators near 0.5 (123 + 61)/0.51 - When a number is very close to a "simple" fraction like 0.25, 0.5, 0.75 , most boards expect you to round to that specific decimal: Round the denominator to the nearest "friendly" decimal or 1 sig fig.

So for ( 123 + 61 ) / 0.51 , this becomes (100 + 60) / 0.5 = 160/0.5 = 160 x 2 = 320.

Nearly always you don't get "horrible" numbers. One such scenario might be:

( 51 + 9.1 )
0.59 .

It's tempting to "see" this as 60 / 0.6 , which would be 100. But that might lose marks! You have to show a different process:

51 indeed becomes 50, but 9.1 to 1 sig fig is 9, so we get 59 / 0.6 . At this point, you would be allowed to 'nudge' it further, so rounding 59 to 1 sig fig becomes 60, and 60 / 0.6 is 100.

However the most important rule is that for fractions/decimals, you round the denominator to a simple fraction (like 0.5, 0.25, 2, 10) and adjust the calculation accordingly.