Ratio Equations

Relevant wiki: Setting Up Equations

Let $A$ and $B$ denote the number of candy bars Alice and Bob initially have, respectively.

From the first paragraph, $A = 6B \qquad (\bigstar)$ .

From the second paragraph, we know that Alice and Bob now have $A + 6$ and $B+6$ candy bars respectively. And we are given that Alice now has only 4 times as many candy bars as Bob does, so

$\dfrac{A+6}{B+6} = 4 \qquad (\bigstar\bigstar)$

So we have two equations, and we can solve them simultaneously by substituting $\bigstar$ into $\bigstar\bigstar$ to get

$\dfrac{6B + 6}{B+6} = 4 \qquad \Rightarrow \qquad 6B + 6 = 4(B+6) = 4B+ 24 \qquad \Rightarrow 2B = 18 \qquad \Rightarrow \qquad B = 9.$

And so $A = 6B = 6\times 9 =54$ .

Thus, from the second paragraph, after receiving an additional 6 extra candy bars, Alice and Bob now has $A+6 = 60$ and $B+6 = 15$ candy bars, respectively.

From the third paragraph, let $n$ denote the number of additional candy bars (on top of the 6 already received) for Alice to have only 2 times as many candy bars as Bob does, then setting up the equation gives

$\dfrac{60 + n}{15 + n} = 2 \qquad \Rightarrow \qquad 60 + n = 2(15+n) = 30 + 2n \qquad \Rightarrow \qquad n =\boxed{30} .$