What if the rug is 104.7 cm?paulb203 said:Homework Statement: A rectangular rug is 2 m long and 1m wide. Both measurements are given correct to the nearest 10cm. State the maximum possible width of the rug in cm.
Relevant Equations: No equations involved.
My answer was 104cm.
That's 105 cm to the nearest cm.paulb203 said:I was thinking that 105cm would be rounded up to 110cm.
The answer in the text book was 105cm.
In addition to what @PeroK has already said, I'd like to add this.paulb203 said:My answer was 104cm. I was thinking that 105cm would be rounded up to 110cm.
The answer in the text book was 105cm.
“What if the rug is 104.7 cm?”PeroK said:What if the rug is 104.7 cm?
That's 105 cm to the nearest cm.
PeroK said:PS in general "to the nearest ##x## units" means ##\pm \frac x 2## units.
“You can avoid ambiguity by giving the answer as a recurring decimal: maximum width = 1.04999...m.Steve4Physics said:In addition to what @PeroK has already said, I'd like to add this.
You can avoid ambiguity by giving the answer as a recurring decimal: maximum width = ##1.04 \dot 9##m.
However, since ##1.04 \dot 9## is considered equal to ##1.05##, giving the answer as 1.05m is considered correct (and is what is required here).
Of course if you were told that the width is 1.05m and asked to round, then 1.1m would (by convention) be correct.
Also, note that the original question should have given the length and width as 2.0m and 1.0m (not 2m and 1m).
The textbook answer is correct. I'm struggling a little to see your logic. Look at it this way. Suppose you have a space exactly ##104 cm## wide and you have an object (perhaps a rigid object is better than a rug) which is ##100 cm## to the nearest ##10 cm##. What you are saying is that it will definitely fit into that space? It cannot be ##104.1 cm## it cannot be ##104.2 cm## and it cannot be ##104.9 cm##. I don't get that. What if it is ##104.5 cm## wide? That's definitely closer to ##100 cm## than ##110 cm##.paulb203 said:“What if the rug is 104.7 cm?”
Ah, thanks, I didn’t think of decimals.
If it was 104.7cm and it had been rounded to the nearest 10cm would the rounded measurement still be be 100cm? There’s a zero in the TENS column; the number to the right is the DECIDER, which is less than 5, so the zero remains zero; is that right?
“That's 105 cm to the nearest cm.”
Not sure what you mean. The 105cm answer in the textbook is saying that 105cm is the maximum width of the rug.
Is the textbook answer wrong?
I assume that's someone's explanation for the answer of ##105 cm##?paulb203 said:“You can avoid ambiguity by giving the answer as a recurring decimal: maximum width = 1.04999...m.
However, since 1.049˙ is considered equal to 1.05, giving the answer as 1.05m is considered correct (and is what is required here).”
Yes, that's a reasonable informal proof.paulb203 said:Bit of a digression, but, have I remembered correctly why 0.999... is considered equal to 1?
0.333... x 3 = 1 because 0.333... is equal to 1/3 and 3 x 1/3 = 1
And 0.333... x 3 is also equal to 0.999... because if you have an infinite line of 3s, each one multiplied by 3, you get an infinite line of 9s.
Is that correct?
Yes.paulb203 said:And therefore 1.04999... is equal to 1.05 because if 0.999... is equal to 1, then, 0.00999... is equal to 0.01?
And we add the 0.01 to 0.04 to give 0.05 which we add to the 1.0 to give 1.05?
No, IS equal. You have proved it (1 = 1/3 x 3 = 0.333... x 3 = 0.999...).paulb203 said:*”...considered equal.”? Only considered? Not IS equal?
My wording was poor. ##1.04\dot9## and ##1.05## are equal. Similarly, ##0.\dot9## and ##1## are equal.paulb203 said:Thanks. I’m guessing that 1.04999... is considered* equal to 1.05 in the same way that 0.999... is considered equal to 1; is that correct?
Yes, I can't see any merit in overanalysing why the answer is given as 105 cm.PeroK said:This question, IMO, is a practical question about the width of a rug.
Yes, as I said abovehutchphd said:Does anyone else hate the statement of the question?
@paulb203 if you understand the ## 0.999... = 1 ## proof then I suspect the textbook you are using is at too low a level for you - it's time you moved on from measuring rugs.pbuk said:If you want better answers, use a better textbook.
I do. I hate all such questions. I think questions like this contribute to young students learning to hate science and math. Even though they really have nothing to do with science or math.hutchphd said:Does anyone else hate the statement of the question?
The windows in my flat are non-standard, apparently; and I had to get the blinds made to measure. They explicitly asked for the width of the alcove, which was 119cm. When the blinds arrived, they were 119.5cm and couldn't fit in the space. These things are important.gmax137 said:I do. I hate all such questions. I think questions like this contribute to young students learning to hate science and math. Even though they really have nothing to do with science or math.
There, now I feel a little better.
Would 104.7 cm be equal to 105.2 cm to the nearest 10cm? That is for the greatest upper bound and 104.2 cm for the least upper bound.PeroK said:What if the rug is 104.7 cm?
That's 105 cm to the nearest cm.
As you have written it the words "to the nearest 10 cm" apply to the measurement "105.2 cm". What is 105.2 cm to the nearest 10 cm? Is this equal to 104.7 cm?chwala said:Would 104.7 cm be equal to 105.2 cm to the nearest 10cm?
pbuk said:As you have written it the words "to the nearest 10 cm" apply to the measurement "105.2 cm". What is 105.2 cm to the nearest 10 cm? Is this equal to 104.7 cm?
Or did you mean "would 104.7 cm to the nearest 10 cm be equal to 105.2 cm to the nearest 10 cm"? What do you think?
What is your objective in posing the question:
- Learning about how rounding of measurements works in a particular high school curriculum?
- then follow the materials for that curriculum.
- Learning about how imprecision and uncertainty is dealt with in science or engineering?
- then forget about "to the nearest" anything, that is not how we measure things in real science or engineering.
yes this is what i meant. The accuracy here is to 0.1 cm and not 1 cm because of the decimal. Unless, i am wrong, i stand corrected.Or did you mean "would 104.7 cm to the nearest 10 cm be equal to 105.2 cm to the nearest 10 cm"?
So what is 104.7 cm to the nearest 10 cm? And what is 105.2 cm to the nearest 10 cm? Are they equal?Or did you mean "would 104.7 cm to the nearest 10 cm be equal to 105.2 cm to the nearest 10 cm"? yes this is what i meant.
I gave my thoughts on post ##19##.pbuk said:So what is 104.7 cm to the nearest 10 cm? And what is 105.2 cm to the nearest 10 cm? Are they equal?
Edit: this is not a useful question though: if you can measure something in millimetres why would you want to round the measurement to the nearest 10 cm? A more sensible question might be "are 104.7 cm and 105.2 cm equal to within 10 cm?"
pbuk said:So what is 104.7 cm to the nearest 10 cm? And what is 105.2 cm to the nearest 10 cm? Are they equal?
The precision of the numbers that pbuk gave is to 0.1 cm, but that's of little relevance to rounding them to the nearest 10 cm.chwala said:yes this is what i meant. The accuracy here is to 0.1 cm and not 1 cm because of the decimal. Unless, i am wrong, i stand corrected.
No. All that matters for the given rounding level is whether the whole number part is less than 5 or more than 5. There are different rules that apply if the whole number part is right in the middle; i.e., at 5 in this case. Rounding to the nearest 10 cm, 104, 104.7, 104.782, and 104.9128 would all round to 100 cm.chwala said:When you have decimals involved I think we approximate the upper bound and lower bound differently as opposed to whole numbers...
Then what would the answer to the question:Mark44 said:The precision of the numbers that pbuk gave is to 0.1 cm, but that's of little relevance to rounding them to the nearest 10 cm.
To the nearest 10 cm, 104.7 cm would round to 100 cm and 105.2 cm would round to 110 cm.No. All that matters for the given rounding level is whether the whole number part is less than 5 or more than 5. There are different rules that apply if the whole number part is right in the middle; i.e., at 5 in this case. Rounding to the nearest 10 cm, 104, 104.7, 104.782, and 104.9128 would all round to 100 cm.
The answer would be "the question is not phrased correctly".chwala said:Then what would the answer to the question:
Find the upper bound and lower bound of ##46##cm and ##46.7##cm to the nearest cm?
If I'm understanding @pbuk's questions correctly, here are what I think are equivalent questions.pbuk said:We could have a question: "find the upper bound and lower bound of 46 cm measured to the nearest cm": can you answer this?
We could have a question: "find the upper bound and lower bound of 46.7 cm measured to the nearest 0.1 cm": can you answer this?
We could even have a question: "find the upper bound and lower bound of 46 cm measured to the nearest 0.1 cm": can you answer this?
Yes, those are equivalent, and better worded than mine, however the wording I have used is similar to that used in the English GCSE syllabus (which I believe @chwala may be using materials from): see e.g. https://www.bbc.co.uk/bitesize/guides/zscq6yc/revision/6Mark44 said:If I'm understanding @pbuk's questions correctly, here are what I think are equivalent questions.
- Find the upper and lower bounds of measurements, rounded to the nearest 1 cm, that round to 46 cm.
- Find the upper and lower bounds of measurements, rounded to the nearest 0.1 cm, that round to 46.7 cm.
- Find the upper and lower bounds of measurements, rounded to the nearest 0.1 cm, that round to 46 cm.
Noted boss. My interpretation or rather my english was the problem here. Cheers.pbuk said:The answer would be "the question is not phrased correctly".
We could have a question: "find the upper bound and lower bound of 46 cm measured to the nearest cm": can you answer this?
We could have a question: "find the upper bound and lower bound of 46.7 cm measured to the nearest 0.1 cm": can you answer this?
We could even have a question: "find the upper bound and lower bound of 46 cm measured to the nearest 0.1 cm": can you answer this?
But we could not have a question: "find the upper bound and lower bound of 46.7 cm measured to the nearest cm": can you see why?
A rounding error occurs when a number is approximated to a nearby value, often to make calculations simpler or to fit a certain level of precision. This can result in a small difference between the original value and the rounded value.
Rounding errors occur because the original measurement might not be an exact multiple of 10cm. When you round to the nearest 10cm, you adjust the value to the closest multiple of 10cm, which introduces a small discrepancy from the original measurement.
To round a measurement to the nearest 10cm, look at the digit in the units place. If it is 5 or more, round up to the next multiple of 10cm. If it is less than 5, round down to the nearest multiple of 10cm. For example, 34cm rounds to 30cm, and 36cm rounds to 40cm.
The maximum rounding error when rounding to the nearest 10cm is 5cm. This occurs because the largest difference between the original measurement and the nearest multiple of 10cm is half of 10cm.
Rounding errors can affect scientific measurements by introducing small inaccuracies. In some cases, these inaccuracies can accumulate, leading to significant errors in calculations, data analysis, and experimental results. Therefore, it's important to consider the impact of rounding and to use appropriate levels of precision in scientific work.