Can someone please explain the lets stop and think problem?

quietcamel

The explanation is hard for me to understand.

debbie

@quietcamel Thanks for writing, and I'm sorry it was a confusing explanation! I've updated the solution on the course website, and also putting it down here.

There are a lot of numbers and percentages in this problem, and keeping track of all of them will help you go a long way toward solving this problem. So, what do we know? There are $60$% of the total people who support A, and therefore $40$% of the total people who support B.

What if we suppose that there are $100$ people in total? We don't lose anything logically if we do so. Then we can get rid of the percents and use numbers, and so $60$ of the people support A, and forty of them support B.

If $30$% of the people who support A told the pollster, then $30$% of the red people on the left told the pollster, which means

$$\begin{aligned} 60 \times \frac{30}{100} &= 60 \times 0.30 \\\\ &= 18 \text{ people who support A told the pollster} \\ \end{aligned} $$

Similarly, for the people who support B (the blue people), we can do the same multiplication, except it's a little tricky, because we don't know what percent of the B supporters told the pollster. Since we don't know, let's call this percentage $x$%.

Now, for the final piece of information: $40$% of the people who told the pollster actually support candidate B. This is equal to a fraction, where the top is the number of B supporters who told (blue people with speech bubbles), and the bottom is the total number of people who told (red people with speech bubbles, plus blue people with speech bubbles).

It's easy to make a mistake and think that the bottom is the total people, or equal to the total blue people. Actually, the denominator equals the people who told the pollster, and they should have speech bubbles.

Now, let's just plug in the expressions we found for each of these quantities! The number of A supporters who told is $18,$ and the number of B supporters who told is $40 \times x,$ where $x$ equals the percentage as a decimal of B supporters who told the pollster.

$$\begin{aligned} 40 \text{ percent } &= \frac{40x}{18 + 40x} \\\\ \frac{40}{100} &= \frac{40x}{18 + 40x} \\\\ \frac{2}{5} &= \frac{40x}{18 + 40x} \\\\ \end{aligned} $$

Now, let's multiply both sides by $5$ and $18 + 40x,$ in order to cancel the denominators:

$$\begin{aligned} 40x \times 5 &= \left( 40x + 18 \right) 2 \\\\ 200x &= 80x + 36 \\\\ 200x - 80x &= 80x + 36 - 80x \\\\ 120x &= 36 \\\\ x &= \frac{36}{120} \\\\ &= \frac{3 \times 12}{10 \times 12} \\\\ &= \boxed{\frac{3}{10}} \\\\ \end{aligned} $$

Thus the fraction of B supporters who told the pollster is $\boxed{30 \text{ percent}}.$

quietcamel

Thank you so much, debbie!

debbie

@quietcamel You're welcome!