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