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In a survey of 348 employees, 104 of them are uninsured, 54 work part time, and 12.5 percent of employees who are uninsured work part time. If a person is to be randomly selected from those surveyed, what is the probability that the person will neither work part time nor be uninsured?

A. \(\frac{7}{12}\) B. \(\frac{8}{41}\) C. \(\frac{91}{348}\) D. \(\frac{1}{8}\) E. \(\frac{41}{91}\)

In a survey of 348 employees, 104 of them are uninsured, 54 work part time, and 12.5 percent of employees who are uninsured work part time. If a person is to be randomly selected from those surveyed, what is the probability that the person will neither work part time nor be uninsured?

A. \(\frac{7}{12}\) B. \(\frac{8}{41}\) C. \(\frac{91}{348}\) D. \(\frac{1}{8}\) E. \(\frac{41}{91}\)

Here, we must determine the probability that a employee randomly selected from a pool of 348 employees is not uninsured, AND does not work part time. This survey divides employees into four groups: employees who are uninsured but are not part time, employees who work part time but who are not uninsured, part time employees who are also uninsured, and employees who neither are uninsured nor work part time. Assign variables accordingly:

We need to find the fraction \(\frac{z}{348}\).

The question tells us that the total number of uninsured employees is 104, so \(w + x = 104\). Similarly, the total number of those who work part time is 54, so \(x + y = 54\). We also know that \(12.5\) percent, or \(\frac{1}{8}\) of the employees who are uninsured also work part time: \(\frac{1}{8}(w + x) = x\). Finally, recall that all four quantities sum to 348, so \(w + x + y + z = 348\) and \(z = 348 - w - x - y\).

Use substitution to solve for each variable. First, solve \(\frac{1}{8}(w + x) = x\) for \(w\). Multiply both sides of \(\frac{1}{8}(w + x) = x\) by 8: \(w + x = 8x\). Combine like terms: \(w = 7x\). Now substitute into \(w + x = 104\) to get \(7x + x = 104\). Combine like terms and divide: \(8x = 104\), \(x = 13\).

Plug back into \(w = 7x\): \(w = 7(13) = 91\).

Next, substitute the value for \(x\) into \(x + y = 54\) to get \(13 + y = 54\). Subtract: \(y = 41\).

Finally, substitute and solve for \(z\): \(z = 348 - w - x - y = 348 - 91 - 13 - 41 = 203\).

Therefore, \(\frac{z}{348} = \frac{203}{348}\). This is not one of the answer choices, so we need to reduce the fraction. Factor a 29 out of the top and the bottom: \(\frac{7}{12}\).

I think this is a high-quality question. How do we quickly recognise that 203/348 will be divisible by 29.I wasted so much time on this and so i skipped this question with a guess.

P(Part Time or Uninsured)={P(Part Time)+P(Uninsured)-P(Part Time and Uninsured)}(Possible)/348(Total) ={104+54-(12.5 % of 104 i.e. 13)}/348 =(158-13)/348=145/348

Thus, P(Neither Part Time Nor Uninsured)=1-P(Part Time or Uninsured) =1-(145/348) =203/348=7/12

Once you reach 203/348 and since it is not in the options. Start factoring 348 . You will get 2*2*3*29 . That is how you will know that both numerator and denominator are divisible by 29. Now you can easily arrive at 7/12.

You don't have to know that 203 is not prime or that 29 goes into it.

203/348 is over 50%, there's only one answer choice that was over 50%.

In fact you don't even need to know anything. 104 uninsured plus 54 part time is still less than half of the total population even before subtracting for the overlap.

Plus the ven diagram isn't necessary. 25% of 104 is 26. 12.5% is half of 25% subtract 13 part time workers from the 104.

Once you reach 203/348, you can use POE to eliminate wrong choices. B, C, D and E are all values below 0.5 while 203/348 is above 0.5. No need to even factor out 29. Therefore, A must be correct.

12.5 percent of employees who are uninsured work part time ---- how come this got translated as \(\frac{1}{8}\)(w+x) = x .... because i think it actually means \(\frac{1}{8}\)(w+x) = x+ y.....

12.5 percent of employees who are uninsured work part time ---- how come this got translated as \(\frac{1}{8}\)(w+x) = x .... because i think it actually means \(\frac{1}{8}\)(w+x) = x+ y.....

Please explain me why I am wrong

\(12.5\) percent, or \(\frac{1}{8}\) of the employees who are uninsured also work part time.

Segment of those who are uninsured is represented by (w+x). Segment of those who are uninsured and also work part time, so both, is represented by x.

I think this is a high-quality question. How do we quickly recognise that 203/348 will be divisible by 29.I wasted so much time on this and so i skipped this question with a guess.

Same here. The question is really really tricky.. I believe this is definitely a 750 level question. We need be more calm and go through all the options before marking- that's what I have learnt from the question and hope the same technique will help to improve further.
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Bunuel, maybe a stupid question... how did you know to factor a 29?

Here's how i did it.

I got till 203/348 using the matrix method.

Once i got to 203/348, since it was not in the options, i checked by using the options present. Eliminated C outright. Checked with D, the simplest fraction. Dividing 348/8 doesn't give an integer. Eliminated. Checked with A. Dividing 348/12 gives 29. Integer. So need to check the numerator also. Divided 203/29. Got 7. Hence A.

I am not sure if this is the fastest method though.

Anyone with a simpler/faster method? Please do reply.