Rounding Errors to nearest 10cm

[paulb203]

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. I was thinking that 105cm would be rounded up to 110cm.
The answer in the text book was 105cm.

 

[PeroK]

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.

What if the rug is 104.7 cm?

I was thinking that 105cm would be rounded up to 110cm.
The answer in the text book was 105cm.

That's 105 cm to the nearest cm.

 

[PeroK]

PS in general "to the nearest $x$ units" means $\pm \frac x 2$ units.

 

[Steve4Physics]

My answer was 104cm. 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.

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).

 

[paulb203]

What if the rug is 104.7 cm?

That's 105 cm to the nearest cm.

“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?

 

[paulb203]

PS in general "to the nearest $x$ units" means $\pm \frac x 2$ units.

So with this example (to the nearest 10 units) they mean plus or minus 10/2, which means plus or minus 5?

Not sure what that means. How would the question be phrased if they were to explicitly state this thing about plus or minus x/2 units?

 

[paulb203]

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).

“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).”

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?

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?

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?

*”...considered equal.”? Only considered? Not IS equal?

 

[PeroK]

“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?

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 $110 cm$.

In order to guarantee that your object fits into the gap, the gap would need to be $105cm$. Not $104 cm$.

 

[PeroK]

“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).”

I assume that's someone's explanation for the answer of $105 cm$?

I don't know that recurring decimals are relevant.

A better way to look at it is that $105 cm$ is the only possible answer. $104.9 cm$ is no good, because it could be $104.99cm$ etc.

This question, IMO, is a practical question about the width of a rug.

 

[pbuk]

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, that's a reasonable informal proof.

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?

Yes.

*”...considered equal.”? Only considered? Not IS equal?

No, IS equal. You have proved it (1 = 1/3 x 3 = 0.333... x 3 = 0.999...).

 

[Steve4Physics]

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?

My wording was poor. $1.04\dot9$ and $1.05$ are equal. Similarly, $0.\dot9$ and $1$ are equal.

I'm reluctant to say more, as lots of good explanations are already available.

You will find plenty of information if you do a search such as "Is 0.9 recurring equal to 1?". You'll even find a Wikipedia article about it! https://en.wikipedia.org/wiki/0.999...

 

[pbuk]

This question, IMO, is a practical question about the width of a rug.

Yes, I can't see any merit in overanalysing why the answer is given as 105 cm.

A more advanced textbook would state that "the length of the rug lies in the range $[95, 105)$ cm" so if you want better answers, use a better textbook.

 

[hutchphd]

Does anyone else hate the statement of the question?

 

[pbuk]

Does anyone else hate the statement of the question?

Yes, as I said above

If you want better answers, use a better textbook.

@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.

We have a forum that muight help: https://www.physicsforums.com/forums/science-and-math-textbooks.21/

 

[gmax137]

Does anyone else hate the statement of the question?

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.

 

[PeroK]

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.

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.

 

[chwala]

What if the rug is 104.7 cm?

That's 105 cm to the nearest cm.

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.

 

[pbuk]

Would 104.7 cm be equal to 105.2 cm to the nearest 10cm?

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.

 

[chwala]

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.

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. The accuracy here is to 0.1 cm and not 1 cm because of the decimal. Unless, i am wrong, i stand corrected.

 

[pbuk]

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.

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?"

 

[chwala]

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?"

I gave my thoughts on post $19$.

When you have decimals involved I think we approximate the upper bound and lower bound differently as opposed to whole numbers...but I could be wrong.

 

[Mark44]

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?

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.

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.

When you have decimals involved I think we approximate the upper bound and lower bound differently as opposed to whole numbers...

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]

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.

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?Edit
I think I am confusing myself with the terms : Rounding a number and finding the upper bound of a number. I think it should be clear now.

 

[pbuk]

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?

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?

 

[Mark44]

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?

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.

 

[pbuk]

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.

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/6

 

[chwala]

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?

Noted boss. My interpretation or rather my english was the problem here. Cheers.