Upper and Lower Bounds

Question
The length of a side of a square is 6.81 cm, correct to 3 significant figures.
(a) Work out the lower bound for the perimeter of the square.

(b) Give the perimeter of the square to an appropriate degree of accuracy.
You must show working to explain how you obtained your answer.

Ans a)

3 significant figures, hence subtract 0.005 from 6.81 to get 6.805cm (lower bound)
Lower bound for the perimeter is 6.805X 4= 27.22cm.

b).
Upper bound for the perimeter is 6.815X4= 27.26cm.( 6.81+0.005= 6.815)

Appropriate degree of accuracy =27cm ( as i was told) , did not understand why. Can someone shed some light on this.

Hello nazz

Quote:

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Originally Posted by nazz

Question
The length of a side of a square is 6.81 cm, correct to 3 significant figures.
(a) Work out the lower bound for the perimeter of the square.

(b) Give the perimeter of the square to an appropriate degree of accuracy.
You must show working to explain how you obtained your answer.

Ans a)

3 significant figures, hence subtract 0.005 from 6.81 to get 6.805cm (lower bound)
Lower bound for the perimeter is 6.805X 4= 27.22cm.

b).
Upper bound for the perimeter is 6.815X4= 27.26cm.( 6.81+0.005= 6.815)

Appropriate degree of accuracy =27cm ( as i was told) , did not understand why. Can someone shed some light on this.

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I'm not quite sure which bit you don't understand. So I'll just explain the last part. If you need any more, please let us know.

The true value of the perimeter lies between 27.22 cm and 27.26 cm. So we obviously can't be sure what the value of the second decimal place is: it's somewhere between 2 and 6, inclusive.

Can we instead give the value of the perimeter correct to 1 d.p.?

Well, correct to 1 d.p:

27.22 is 27.2;

and

27.26 is 27.3.

So the perimeter might be either 27.2 or 27.3, correct to 1 d.p.

So, no, we can't give the value to 1 d.p. and be sure of getting it right.

What about giving the value to the nearest whole number?

When we round 27.22 and 27.26 to the nearest whole number, each one is 27. So we can be sure that, correct to the nearest whole number, the perimeter is 27 cm.

Is that OK now?

Grandad

Thanks grandad for the prompt reply, but please have a look at the attached document. if i would follow your method then how did the author get 67cm3 as an answer.

The original data, r= 3.5 cm and h= 5.2 cm are both given to two significant figures. After doing the calculation, V= 66.7 cubic cm. the author rounded that to two significant figures also- sinc .7 is larger than .5, it is closer to 67 than 66 so 66.7 is rounded to 67.

To granddad
Really sorry if i am getting back to you on this. For the first question perimeter of square , the data was the following:

Upper bound perimeter=27.26cm
Lower bound perimeter=27.22cm

Give the perimeter of the square to an appropriate degree of accuracy

As you said
The true value of the perimeter lies between 27.22 cm and 27.26 cm. So we obviously can't be sure what the value of the second decimal place is: it's somewhere between 2 and 6, inclusive.

Question; so how come the final answer is 27cm as 27 does not lie between 27.22 and 27.26. Shouldn’t the final answer lie between the upper and lower bounds which is the best estimate.

Do you have any explanation notes on this topic?

Hello nazz

Quote:

---

Originally Posted by nazz

To granddad
Really sorry if i am getting back to you on this. For the first question perimeter of square , the data was the following:

Upper bound perimeter=27.26cm
Lower bound perimeter=27.22cm

Give the perimeter of the square to an appropriate degree of accuracy

As you said
The true value of the perimeter lies between 27.22 cm and 27.26 cm. So we obviously can't be sure what the value of the second decimal place is: it's somewhere between 2 and 6, inclusive.

Question; so how come the final answer is 27cm as 27 does not lie between 27.22 and 27.26. Shouldn’t the final answer lie between the upper and lower bounds which is the best estimate.

Do you have any explanation notes on this topic?

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I agree that 27 does not lie between 27.22 and 27.26, but that's not the point. You really only have a limited number of choices when it comes to giving values that aren't exact. That is, you have to give them to:

• a certain number of decimal places, or
• to the nearest whole number, or the nearest ten, hundred, ... etc, or
• to so many significant figures.

There's no other option.

Now as we've seen, we can't give the perimeter to 1 decimal place and be sure of getting it right - because it could be 27.2 or 27.3. So the only other option is to 'zoom out' a bit and give the answer to the nearest whole number - even though we know this is actually going to be smaller than the true value.

Another example is to consider the population of a town. If a friend asks you what is the population of the town you live in, you may know (if you work for the Borough Council Statistics Department) that on a certain day there were 83,405 citizens in that town. But you'd be a bit of an anorak to tell this to your friend. You'd probably say "about 80 thousand". And you'd be right! To the nearest ten thousand, that's correct.

Or, if you wanted to be a bit more accurate, you'd say "about 83 thousand". And again, you'd be right - correct to the nearest thousand.

Both of these answers are less than the true figure, but each is right as far as it goes. Well, it's just like that with the perimeter. If someone presses you for an answer you might say "It's between 27.2 and 27.3". But if they weren't too bothered about the last millimetre, they'd be quite happy if you said "About 27, to the nearest whole number".

One of the important lessons you need to learn is that real life is like that: there aren't necessarily any completely right answers - it's all a bit untidy round the edges!

Does that make it clearer?

Grandad