Question

Timmy and Ben shared a bag of marbles. Timmy had 3/4 of what Ben had. When Ben gave 36 marbles to Timmy, Ben had 1/3 of what Timmy had. How many marbles were there in the bag?

Answer 1

Let $t$ be the inital number of Timmy's marbles, and $b$ be the inital number of Ben's marbles.

At the beginning, "Timmy had 3/4 of what Ben had", which means

$t = \cfrac{3}{4}b \iff 4t=3b$

(I wrote the equation in an equivalent form to avoid denominators)

Then, Ben gives 36 marbles to Timmy. So, now Ben has $b-36$ marbles, and Timmy has $t+36$ marbles. The new ratio is 1 to 3 in Timmy's favour, so we have

$b-36 = \cfrac{t+36}{3} \iff 3(b-36) = t+36 \iff 3b-108 = t+36$

(same as before)

So, we have the following linear system:

$\begin{cases}4t=3b\\3b-108=t+36\end{cases}$

From the first equation we know that $3b=4t$, so the second equation becomes

$4t-108=t+36 \iff 3t = 144 \iff t = \cfrac{144}{3} = 48$

Now that we know the value for $t$, we can simply plug it into the first equation to get

$3b = 4\times 48 = 192 \iff b=\cfrac{192}{3} = 64$