Intersection of percentages

Question

I have a problem with the following details:

50% total = comfortable
75% total = without
25% total = with
70% of *with* are comfortable

What percentage of without are comfortable? Please help I'm lost.

Answer 1

HINT

Assume $1000$ people

• $50\%$ total = comfortable: $500$

• $75\%$ total = without: $750$

• $25\%$ total = with: $250$

• $70\%$ of with are comfortable: $175$

therefore $500-175=325$ without are comfortable.

Answer 2

Make up the joint probability table: $$\begin{array}{c|c|c} &\text{comfortable}&\text{uncomfortable}&\text{total}\\ \hline \text{with}&17.5\%&&25\%\\ \text{without}&32.5\%&&75\%\\ \hline\ \text{total}&50\%&&100\%\\ \end{array}$$ How do you find the numbers:

1) marginal probabilities (percentages) $25\%, 75\%, 50\%$ are given.

2) $70\%$ of $25\%$ is $25\%\cdot 0.7=17.5\%$.

3) $50\%-17.5\%=32.5\%$ are comfortable and without of total.

4) percent of without that are comfortable is: $\frac{32.5\%}{75\%}\cdot 100\%=\frac{130}{3}\%\approx 43.33\%$ (final answer).

5) finding the rest numbers is an exercise for you.

Answer 3

The facts should suffice as they are outlined, without the need for assumption; and may be reorganized into one coherent statement as follows:

Given a population where 50 percent is "comfortable", 75 percent is "without", 25 percent is "with", and the percent "with" intersects the percent "comfortable" at a rate of 70 percent; solve for the rate at which the percent "without" intersects the percent "comfortable".

It is understood that some percent (x) of "without" (75%) is proportional to some percent (y) of "comfortable" (50%).

Solve for x where x of 75% is equal to y of 50%.

x * 75% = y * 50%
x = y * 50% / 75%

It is also understood that 70% of "with" (25%) is proportional to the remainder (100% - y) of "comfortable" (50%).

Solve for y, where 70% of 25% is equal to y from 100% then of 50%.

70% * 25% = (100% - y) * 50%
100% - y = 70% * 25% / 50%
-y = (70% * 25% / 50%) - 100%
y = 100% - (70% * 25% / 50%)
y = 100% - 35%
y = 65%

Recall that x = y * 50% / 75% and solve for x.

x = 65% * 50% / 75%
x = 43.333...%

An alternative solution is to calculate the percent of the population that is "with" and "comfortable".

a = 70% * 25% = 17.50%

Calculate the remainder of "comfortable".

b = 50% - 17.50% = 32.50%

Divide the remainder into the percent that is "without" (75).

x = 32.5% / 75%
x = 43.333...%