Here we are trying to find the number of people who support A. Not all of the people who support A actually admitted that they did. Only 30% of them told.
First, we can pretend that 45 people told the pollster they support A, and 55 people told the pollster that they support B. This satisfies the condition that 45% of the people who responded actually support A. It doesn't matter if you use 90 and 110 instead, or 450 and 550 instead, or even 0.45n and 0.55n (where n is the number of people who told).
90 / (90 + 110) = 45%
0.45n / (0.45n + 0.55n) = 45%
450 / (450 + 55) = 45%
This works because we want to know the fraction of the total people who support A, not the number of people who told A.
Now let's work backward, knowing that 45 people told that they support A, and this is 30% of the total people who support A (the number who told plus the number who didn't tell.)
The purple section refers to the 45 people, so
30% × (# people who support A) = 45.
Then, solving,
#. of people who support A = 45 ÷ .3
Another way to do this is to say:
n = total number of people
f = fraction of people who support A
1 - f = fraction of people who don't support A
Then,
#. people who support A and tell = 30% × f × n
#. people who support B = (1 - f) × n
#. people who support B and tell = 50% × (1 - f) × n
Since 45% of the people who tell support A,
#. people who support A and tell ÷ ( # people who support A and tell + # people who support B and tell) = 45% = 0.45
This gives us
= 0.45
The n's cancel out, leaving us with f as the only variable. Solving this for f gives us f = 15/26, which is the fraction of people who support A. That's the answer!
Happy Learning,
The Daily Challenge Team