450 / 600 = 0.75
150 / 600 = 0.25
This when multiplied by 100 gives 75 and 25 respectively.
So 450 is 75% of 600.
So 150 is 25% of 600.
32 / 88 = 0.3636....
56 / 88 = 0.6363....
So the percentages are 36% and 63% to the nearest whole number.
300 / 480 = 0.625
180 / 480 = 0.375
So the percentages are 63% and 38% using the convention of rounding
upwards if there is a five after the digit being rounded even though
the number '0.5' is half way inbetween 0 and 1.
Notice that they do not quite add up to 100%, but this is purely due
to the rounding, apart from that they would add up to 100% in problems
where you have worked out percentages of a total amount, and have included
all non overlapping components exactly once which add to form the total.
Notes on conventional rounding of numbers:
Let us suppose you are rounding to the nearest whole number a number
which has digits after the decimal point.
The following convention usually applies to numbers between 0 and 1:
0.1 rounds down to 0
0.2 rounds down to 0
0.3 rounds down to 0
0.4 rounds down to 0
0.5 is a borderline case we usually round this up to 1
0.6 rounds up to 1
0.7 rounds up to 1
0.8 rounds up to 1
0.9 rounds up to 1
Notice that 0.0 can sometimes be used to mean that the accuracy is to
one decimal place. Obviously to the nearest whole number it is 0, and
indeed it equals 0.
Similarly notice that 1.0 can be used to indicate that the accuacy is to
one decimal place. Obviously to the nearest whole number it is 1, and
it of course equals 1.
Often you will have to give an answer to a certain number of decimal places.
Example 1: Round 1.476 to 2 decimal places.
Method: Since the second digit after the decimal point is 7 we look at the next
number. It is greater than 5. Therefore rounding up is appropriate.
Example 2: Round 4.685 to 2 decimal places.
Method: Since the second digit after the decimal point is 8 we look to the next
digit. It is 5 and by convention an upward rounding occurs.
Example 3: Round 3.595 to 2 decimal places.
Method: Since the second digit is a 9 care has to be taken because an upward
round of this number will cause an overflow carry since a "10" will result
meaning that the one higher place value digit to the left must go up by one.
As it happens the next digit to the right is 5 so upward rounding occurs by
convention. Therefore a 10 results and "59" becomes "60".
A few for you to try:
Q1: Round to the nearest whole number 4.7
Q2: Round to the nearest whole number 7.5
Q3: Round to the nearest whole number 3.2
Q4: Round to 2 decimal places 8.469
Q5: Round to 2 decimal places 2.755
Q6: Round to 2 decimal places 3.933
Q7: Round to 2 decimal places 9.695
Q8: Round to 2 decimal places 1.005
Q9: Round to 2 decimal places 0.999
Q10: Round to 2 decimal places -4.867
Note that if you have something like 0.49999999... (recurring) then it is considered the same as 0.5 so
it rounds up to 1 not down to zero, so be careful about that exception. Most calculators will spot the
series of nines and round to 0.5 for you if that is the result of a division or similar so you do not usually
need to worry. Of course if you just had 0.49 and a finite series of nines then it rounds downward when
rounded to the nearest whole number. A number with many nines at the end that terminates is rare in practice,
but with calculators that do not automatically round you could get the recurring 9 happen by something like
the process that I have described here:
(This will not work on all calculators. Some will correct the rounding error automatically and others are caught out.)
A method to get 0.49999.... is to start with 0.4 in the calculator and then add (1/30) three times.
On my calculator it rounds up to 0.5 upon the third addition of 0.033333333.... but has retained a small
discrepancy. You can then subtract the 0.5 and then obtain "-1 E-14" in other words -0.00000000000001
a very small negative number: minus a hundred trillionth.
If you do 1 divide 6 then times 3 and subtract 0.5 then the answer is positive because the rounding at the
stage of 1 divided by 6 is an upward round - the recurring digit is 6, so it rounds up to 7. Since 7 times 3
is 21 the last digit is 1, which gives the positive extra 1 at the end of the 12 zeros after the five. Then subtract
the 0.5 and you get plus a hundred trillionth ("1 E-14") 0.00000000000001
The number of digits will vary according to what calculator is used.
Obviously the correct answer to 0.4 + (1/30) + (1/30) + (1/30) - 0.5 is exactly zero.
Also the answer to ((1/6) * 3) - 0.5 is also exactly zero.
The calculator answer is an example of an inevitable rounding error caused by the limits of accuracy,
some calculators will not be caught out by that trick depending on how they have been programmed to work.
Last edited by SteveB (2013-11-19 07:52:01)