BMAT question

As.1997

A survey of primary school children showed that between 55% and 65% owned a laptop computer, between 70% and 80% owned a mobile phone and fewer than 5% owned neither of these items. What percentage of the children owned both a laptop computer and a mobile phone?

A between 15% and 30%
B between 25% and 45%
C between 25% and 50%
D between 30% and 45%
E between 30% and 50%

Answer is C.
Could someone please explain how this is the correct answer?

Reply 1

mqb2766

21

Original post by As.1997

A survey of primary school children showed that between 55% and 65% owned a laptop computer, between 70% and 80% owned a mobile phone and fewer than 5% owned neither of these items. What percentage of the children owned both a laptop computer and a mobile phone?

A between 15% and 30%
B between 25% and 45%
C between 25% and 50%
D between 30% and 45%
E between 30% and 50%

Answer is C.
Could someone please explain how this is the correct answer?

Its just a variant of a venn diagram with some uncertainty in the values.

Lower bound: 55% laptop & 70% mobile. Adding gives 125%.
Upper bound: 65% laptop & 80% mobile. Additing gives 145%

In each case the total is between 95% & 100%

Lower bound 125-100 = 25%
Upperbound 145-95 = 50%

Reply 2

As.1997

OP

16

Original post by mqb2766

Its just a variant of a venn diagram with some uncertainty in the values.

Lower bound: 55% laptop & 70% mobile. Adding gives 125%.
Upper bound: 65% laptop & 80% mobile. Additing gives 145%

In each case the total is between 95% & 100%

Lower bound 125-100 = 25%
Upperbound 145-95 = 50%

Thank you for your response. I am unsure as to why you add 55% and 70% to get 125%. Perhaps if you explain how you would do this question using a Venn diagram, I may understand why you have done this.

Reply 3

mqb2766

21

Original post by As.1997

Thank you for your response. I am unsure as to why you add 55% and 70% to get 125%. Perhaps if you explain how you would do this question using a Venn diagram, I may understand why you have done this.

The venn diagram is for visualising. If you add laptops and phones, you double count the overlap. This causes the percentage to be greater than 100 in this case (assuming everyone had a phone or laptop). The amount over 100 gives you the overlap or joint probability or percentage.

Edit. If you add the two complete (overlapping) circles in
https://www.bbc.com/bitesize/guides/z8nfrdm/revision/2
you double count the "oval" in the center.

(edited 5 years ago)

Reply 4

As.1997

OP

16

Original post by mqb2766

The venn diagram is for visualising. If you add laptops and phones, you double count the overlap. This causes the percentage to be greater than 100 in this case (assuming everyone had a phone or laptop). The amount over 100 gives you the overlap or joint probability or percentage.

Edit. If you add the two complete (overlapping) circles in
https://www.bbc.com/bitesize/guides/z8nfrdm/revision/2
you double count the "oval" in the center.

Thank you again. I understand how you got the values 125% and 145%. However, I'm not sure about the "fewer than 5% owned neither of these items." So, what I do get is this would mean that either there are between 95%-100% that own both items. But, I don't really get how this affects the lower and upper bound.

fewer than 5% owned neither of these items

Reply 5

mqb2766

21

Original post by As.1997

Thank you again. I understand how you got the values 125% and 145%. However, I'm not sure about the "fewer than 5% owned neither of these items." So, what I do get is this would mean that either there are between 95%-100% that own both items. But, I don't really get how this affects the lower and upper bound.

fewer than 5% owned neither of these items

You're trying to either minimize the value of the overlap or maximize it.
When the overlap is 125%, you're trying to find the minimum overlap. So somewhere between 95% and 100% reprsents the union of the two circles. The overlap is the difference between 125 and this union. The difference is minimized when the union is 100%. if the union was 95%, the overlap would be 30% and not a minimum.
Similar for the other case, but you're trying to find a maximum for the overlap.

Original post by As.1997

So, what I do get is this would mean that either there are between 95%-100% that own both items. But, I don't really get how this affects the lower and upper bound.

No, this just means that there are >95% of the pupils who own either ONLY a laptop, or ONLY a phone, or both.

(edited 5 years ago)

Reply 7

As.1997

OP

16

Thanks guys :smile:

-- using all your info I have come up with a summary: -- could you correct me if any of it is wrong or misunderstood?

So alone there are 55%-65% who own a laptop computer and similarly alone there are 70%-80% who own a mobile phone. The final rule is that there are 95%-100% who own either a laptop computer only, a phone only or both. The question asks, what percentage of children own both a laptop computer and a mobile phone. Since the answer options are given as a range we have to find the lowest possible (lower bound) and the highest possible (upper bound) outcome whereby the children own both a laptop computer and a mobile phone.
Lower bound:
55% for laptop computer only and 70% for a mobile phone only --> (I tried to picture it as 2 separate circles)
If we put these together the bit that overlaps will count twice.
So 55% + 70% = 125%. This is 25% over 100%.
Going back to the < 5% own neither which in other words mean 95%-100% own either a LC, MP or both. Since we want to find the lowest value possible. We would take it to be 100% rather than any other value between 95%-100%.
This would give 125%-100%= 25%.

Upper bound:
65% own a LC and 80% a MP.
65%+80%= 145%.
To find the highest possible value we would choose 95% rather than any other value between 95%-100%.
Therefore, 145%-95%= 50%.

Therefore, the percentage of people who own both a LC and MP would be between 25% and 50% which is option C.

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