Thread: Year 10 - Unfair Question

Hi all, Just asking for an opinion on a recent practice paper.
The question (on a multiple choice paper) was a string is 16cm long to the nearest CM. What is the longest possible length it can be?
and the options where (15.5, 16.4, 16.5) and afew stupid others that I can't renember.
I put down 16.4, as anything under 16.5 rounds down(As we had been told for many years).
Apparently though, the answers IS 16.5 because

0.9(rec) = X
9.9(rec) = 10X <times both by 10>
9.0 = 9X <minus 1X>
1 = X <divide both by 9>

Therfore

16.49(rec) = 16.5
and 14.49(rec) rounds down.

Even though I can't argue the theory, I still think its is a very unfair question.
Any thoughts?

Use whatever rules they gave you (the rules can vary somewhat) for determining "accuracy". (It is not a matter merely of rounding.)

A string is 16cm long to the nearest CM. What is the longest possible length it can be?

Let's say the length of the string is 16.x centimetres, where x is a digit (from 0 to 9 inclusive).

If x = 0, 1, 2, 3, 4, then the length of the string is 16 cm, to the nearest cm.
If x = 5, 6, 7, 8, 9, then the length of the string is 17 cm, to the nearest cm.

Now... You may conclude that the longest possible length of the string is 16.4 cm. Perhaps, but a string measuring 16.45 cm has its length as 16 cm to the nearest cm, as well. So does a string measuring 16.46 cm, 16.47 cm, 16.48 cm, 16.49 cm, 16.499 cm, 16.4999 cm, and so on... We could write this in inequality form as 15.5 <= Possible length of string < 16.5.

16.4999... has its limit as 16.5 as the number of 9s after the decimal point tend to infinity. So, a very controversial question there... Theoretically speaking, 16.5 cm is the maximum possible length... Practically, you can't really have 16.499999999 cm of string. And to make things worse, 16.5 cm is 17 cm rounded to the nearest cm.

But, note that... The answer cannot be 16.4 cm, because you can always have 16.41 cm of string. Also, 16.5 cm is the minimum possible length of string if a string is 17 cm long, to the nearest cm.

I hope that helps.

ILoveMaths07.

as 16.5 rounds up, it should not be the answer. even if 16.5 is written as 16.49999999...

it's a neat "thinking" question but their answer is just wrong. they would have had to have an answer that did not continue with '9's forever which would have been an easy pick for you.

They should've asked what the minimum possible length of the string is!

If I had to answer that question, I know I wouldn't have chosen 16.4 cm. Because that is incorrect. The most 'sensible' answer is 16.5 cm , and you know WHY. :P Even in inequality form, you write it like this: 15.5 <= Possible length of string < 16.5, NOT 15.5 <= Possible length of string <= 16.4. Notice the upper limit. It's 16.5, not 16.4.

But what were the other options?

Remember the rule for rounding decimals is the same as the rule for whole numbers.
Step 1: Decide on the place the number is to be rounded to.
Step 2: Look at the first digit to the right of that place.
Step 3: If the digit is equal to or more than 5, round up
Step 4: If the digit is less than 5, round down.
Step 5: We may drop the zeros.

In your example, the you want to have a number 16.49 and want to round to one decimal point (the tenths position). The digit to the right is a '9'. Since it is equal or more than 5, you round the '4' up to a '5'. Therefore your answer is 16.5.

Check out Rounding Decimals (with worked solutions & videos) for examples and more information on math-related topics.

Originally Posted by rhx123

Even though I can't argue the theory, I still think its is a very unfair question.

I suppose it is a poor question (with an arguably incorrect solution), but if nothing else, you should be able to recognize that even by your reasoning the string could have been something like 16.45 cm long, so 16.4 cm could not possibly be the answer.

And anyway, 16.5 is no nearer to 17 than it is to 16. It is just by convention that we round it up. But it is not uncommon to round a .5 to the nearest even number, as a means of eliminating bias.

Though I think it would be more appropriate to say that there is no string of maximum length. For any valid string you give me, I can give you a longer string that is still within range. 16.5 is, however, the least upper bound (supremum) of possible string lengths. This is what the question should probably have asked for.