At a certain hospital, 75% of the interns receive fewer than

Thanks!

Bunuel, this tabular method for solving Venn Diagram problems: is it explained anywhere with other examples?

You can just use logic or a Venn diagram, whatever works for you. I will give the logical solution below. Try to figure out the Venn diagram for it. Get back if it gets tricky.
Look at the given data line by line.

At a certain hospital, 75% of the interns receive fewer than 6 hours of sleep and report feeling tired during their shifts.

n(FEW and TIRED) = 75%

At the same time, 70% of the interns who receive 6 or more hours of sleep report no feelings of tiredness.
.... We will come back...

If 80% of the interns receive fewer than 6 hours of sleep,

n(FEW) = 80%
Therefore, n(Not FEW) = 20%
Also since n(FEW and TIRED) = 75%, n(FEW and Not TIRED) = 5%

Now we go back to the second sentence.
We know that n(Not FEW) = 20%. 70% of these are not tired.
So n(Not TIRED and Not FEW) = 70% of 20% = 14%

what percent of the interns report no feelings of tiredness during their shifts?

n(Not TIRED) = n(FEW and Not TIRED) + n(Not TIRED and Not FEW) = 5% + 14% = 19%

The notation is used only to explain it to you. You obviously don't need to do that. Just remember that when you get some data '80% people get less sleep,' it also implies that 20% get enough or more.

5% of 80 =4 but why you guys took it as 5% am I mistaking something

A very important thing in percentages is to note what is the % of i.e. when I say 5%, it is 5% of what?

5% of total interns or 5% of interns who receive less than 6 hrs of sleep or 5% of interns who report tiredness etc

The question says:
75% of the interns receive fewer than 6 hours of sleep and report feeling tired during their shifts (75% of total interns)
If 80% of the interns receive fewer than 6 hours of sleep (again, 80% of the total interns)

So out of 80% of total interns i.e. if total interns are 100, 80 of them receive fewer than 6 hrs of sleep. Out of 100, 75 receive fewer than 6 hrs of sleep and report tiredness. Therefore, 5 receive fewer than 6 hrs of sleep but report no tiredness.

Guys,
As per the solution : interns with LT6 hours of sleep and tired = 75% of total interns(75 assuming a smart number)
However I calculated it as : 0.75x0.80N. i.e. 75% of 80% of interns with LT6 hours of sleep.(which is obviously wrong).
I am not going very far with such mistakes.
Next comes a sillier question: how to avoid such mistakes.
Regards,
shivdeep

Needless to say, Bunuel's method is perfect

There is no need to calculate 75% of 80%

As the answer is expected in percentage, better take the Total interns = 100 so that the %age factor gets computed along

How this queston can be done with venn diagram. Can anyone suggest. I have tried the following combinations for the circles:

a. Tired one circle & Few than 6 hrs another circle.

Here How do I represent the balance more or equal to 6 hrs ( by another circle??)

2. Non tired one & more than 6 hrs another circle.

Still not gettign there.

Hi ankushbagwale,

Presenting the Venn diagram approach for the question. Please refer the diagram below:

Given
Number of interns who receive fewer than 6 hours of sleep and report feeling tired = \(b\) = 75% of total interns.

Number of interns who receive 6 or more hours of sleep and report no feeling of tiredness = \(d\) = 70% of interns who receive more than 6 hours of sleep

Number of interns who receive less than 6 hours of sleep = \(a + b = 80\)% of total interns




We are asked to find the percent of interns who report no feeling of tiredness.

Number of interns who report no feeling of tiredness = \(a + d\).

Hence we need to find \(\frac{a +d}{a + b + c+ d}\).

Approach
As we are not given any info about the number of interns, to find the ratio of of interns who report no feeling of tiredness to the total interns we need to express them in the same terms. With this understanding let's proceed to the working out.

Working Out
Let's assume the total number of interns to be \(x\). i.e. \(a + b + c + d = x\)

\(b = 75\)% of total interns \(= 0.75x\)

\(a + b = 80\)% of total interns \(= 0.8x\) which would give us \(a = 0.05x\)

It also tells us that \(c + d = 0.2x\).

\(d = 70\)% of interns who receive more than 6 hours of sleep

Interns who received more than 6 hours of sleep \(= c +d\)

So, we can write \(d = 70\)% of \((c + d) = 70\)% of \(0.2x = 0.14x\)

Hence \(a + d = 0.05x + 0.14x = 0.19x\)

So \(\frac{a +d}{a + b + c+ d} = \frac{0.19x}{x} = 19\)%

Answer: Option C

Hope it's clear

Regards
Harsh

Thanks Harsh for your prompt response. You have demonstrated how venn diagram can be utilized effectively. Although I could grasp your approach, I wonder how you came up with > 6 hrs of sleep & not tired being shown by the empty spaces.

However, I have tried to to visualize in the following way. can you pl. comment on the same? I would like to move from this three circle format to two. Any suggestions.

Also, the double matrix method seems quite neat & crystal clear.

We can let the total number of interns surveyed = 100. Thus, 80 interns receive fewer than 6 hours of sleep, and 20 interns receive at least 6 hours of sleep. Furthermore, 75 interns receive fewer than 6 hours of sleep and report feeling tired during their shifts. In other words, 5 interns who receive fewer than 6 hours of sleep report no feelings of tiredness during their shifts. Moreover, 0.7 x 20 = 14 interns who receive at least 6 hours of sleep report no feelings of tiredness during their shifts. Therefore, a total of 5 + 14 = 19 interns, or 19% of all interns, report no feelings of tiredness during their shifts.

Answer: C

I have the same question here, how can you know if 75% if of the total and 70% given is of the intern who sleep > 6 hrs?

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