Good question on sets.

[Anand Iyer]

A survey shows that 63% of Americans like cheese whereas 76% like apples.  If x% of the Americans like both cheese and apples, find the value of x.  

Hint: x is a range, not one value.

Left me scratching my head for some time, even while I was staring at the answer:)

[Pradeep S]

Is it 39-63?

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[Anand Iyer]

Great!  Power to you...

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Cheers,

Anand

[Malabika Guha Mustafi]

@ prdp

Exactly

Maximum people is  63Percent(not more than that since 63 percent likes cheese).

Minimum people  is  39 .( if we consider the general overlapping set solution).

@ Anand, really good one. Thank you.

[Anand Iyer]

welcome @Malabika!

[Tejasvi Hegde]

Thanks @Anand for bringing up question, it helps!

[Sanjaykumar]

Thanks @Anand. 

[Venkatesan Shesha Srikanth]

Hi anand,

                Thank you so much for bringing up questions like this on the discussion forum.  

Regards,

Srikanth Venkatesan

Course Instructor

IITM Online Degree Project

IIT Madras.

   

On Saturday, November 7, 2020 at 2:01:04 PM UTC+5:30 Tejasvi Hegde wrote:

[Anand Iyer]

you're welcome!  I'll look out for such unusual but relevant questions and post here.  

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Cheers,

Anand

[Tejasvi Hegde]

This is how the derivation I found on net:

>Using set (found on net)

Let C and A denotes the percentage of people likes cheese and apples respectively. 

Given, n(C)=63,n(A)=76 and n(C∩A)=x.
We know that,
n(C∪A)=n(C)+n(A)−n(C∩A)
⇒n(C∪A)=63+76−x
⇒x=139−n(C∪A)

Now, n(C∪A)≤100

⇒139−n(C∪A)≥139−100=39

⇒x≥39

Also, n(C∩A)≤n(C) and n(C∩A)≤n(A)
⇒x≤63
∴39≤x≤63.

But this is how I derived using logical calculation:

A = 76% of people who like Apple 

C = 63% of people who like Cheese

Since its %, it cannot be more than 100%

Since A + C = 76% + 63% = 139%, Since total cannot be more than 100%, the minimum percentage of people who like both should be 139 - 100 = 39

It is possible that all people who like cheese may also like Apple (or other way), 

Since people who like Apple are more, all apple lovers cannot be fond of cheese, so, since cheese percentage of cheese lovers are lesser than apple lovers, there is a possibility that all cheese lovers may like Apple, hence max 63%

On Saturday, November 7, 2020 at 3:06:43 PM UTC+5:30 Malabika Guha Mustafi wrote:

Of 60 students in a class, anyone who has chosen to study maths elects to do physics as well. But no one does maths and chemistry, 16 do physics and chemistry. All the students do at least one of the three subjects and the number of people who do exactly one of the three is more than the number who do more than one of the three. What are the maximum and minimum number of people who could have done Chemistry only?  

[Malabika Guha Mustafi]

@ Tejasvi

I followed the combined logic. To get the minimum , computed as described in net . To get maximum, the same logic as you applied.

On Saturday, 7 November 2020 at 15:41:58 UTC+5:30 Malabika Guha Mustafi wrote:

In the question, please assume  " But no one does maths and chemistry" means not only the combination of maths and chemistry only. Few people may take three subjects.  

On Saturday, 7 November 2020 at 15:06:43 UTC+5:30 Malabika Guha Mustafi wrote:

Of 60 students in a class, anyone who has chosen to study maths elects to do physics as well. But no one does maths and chemistry, 16 do physics and chemistry. All the students do at least one of the three subjects and the number of people who do exactly one of the three is more than the number who do more than one of the three. What are the maximum and minimum number of people who could have done Chemistry only?  

[Vinayak Kumar]

I am getting 39 as answer but how 63 can anyone explain?

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[Tejasvi Hegde]

Please refer to my latest response

[Anand Iyer]

Unfortunately, that solution on the internet is really confusing and involves a lot of abstract stuff.  Here's something simpler I could think of.

Assume there're a totally 100 Americans.  Out of that 63 are fond of cheese, 76 are fond of apples.  

Now, there are two cases.

1.  All 100 are fond of either of these.  THis is the case where you get 39.

2.  There might be people out of the total 100, that is fond of neither.

So, there are 63 cheese guys, 76 apple guys, and some who're neither.  

We don't know for sure, how many, but sure it's more than zero, thus the total guys is somewhere upwards of 39.  Let's find the upper boundary of the range.

Since there're 76 apple guys (76 is the greater number out of the two), there should be at least 76 guys who like either.  This means, 24 guys like neither apple nor cheese.  Now, all 63 cheese guys also happen to like apple, which makes 63%.

I agree, the statement could sound confusing, but that's the best I could do.

Combining both 1 and 2, we arrive at the range 39 <= x <= 63

[Tejasvi Hegde]

@Malabika, May be you have to re-phrase the sentence ?

On Saturday, November 7, 2020 at 3:06:43 PM UTC+5:30 Malabika Guha Mustafi wrote:

Of 60 students in a class, anyone who has chosen to study maths elects to do physics as well. But no one does maths and chemistry, 16 do physics and chemistry. All the students do at least one of the three subjects and the number of people who do exactly one of the three is more than the number who do more than one of the three. What are the maximum and minimum number of people who could have done Chemistry only?  

[Anand Iyer]

Malabika, do you want to post your question as a separate post, so more people can see this?  It's a very nice one, and quite challenging by the way...

[Malabika Guha Mustafi]

@ Anand,  

ok.

[Malabika Guha Mustafi]

Today's problem:

70% of the people like Coffee, 80% of the people like Tea, 85% of the people like Milk; then at least what % of people like all three?  

[Malabika Guha Mustafi]

Solution : 35 percent