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).

$$\begin{aligned} \frac{90}{90 + 110} &= 45\% \\\\ \frac{0.45n}{ 0.45n + 0.55n} &= 45\% \\\\ \frac{450}{ 450 + 55} &= 45\% \\\\ \end{aligned} $$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

$$ \frac{30}{100} \times \left( \text{ number of people who support A } \right) = 45. $$

Then, solving,

$$ \text{ number of people who support A } = \frac{45}{0.3} $$

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,

$$\begin{aligned} \text{ number of people who support A and tell } &= 30\% \times f \times n \\\\ \text{ number of people who support B } &= (1 - f) \times n \\\\ \text{ number of people who support B and tell } &= 50% \times (1 - f) \times n \\\\ \end{aligned} $$Since \(45%\) of the people who tell support A,

$$ \frac{\text{ number of people who support A and tell }}{ \text{ number of people who support A and tell } + \text{ number of people who support B and tell}} = 45 \text{ percent } = 0.45 $$

This gives us

$$ \frac{0.3 \times f \times n}{0.3 \times f \times n + 0.5 (1-f)n} = 0.45 $$

The \(n\)'s cancel out, leaving us with \(f\) as the only variable. Solving this for \(f\) gives us \( f\) = \(\boxed{\frac{15}{26}},\) which is the fraction of people who support A. That's the answer!

Happy Learning,

The Daily Challenge Team