We don't care at all about what Lyle received. We only want to know how much Bob received, and we can work backwards from the $32 that Chloe received.

Let's say that x dollars remained after Lyle got his share. Bob, then, received $4, leaving us with x - 4. He received one-third of this, leaving two-thirds for Chloe. So Chloe received \(\frac{2}{3}(x - 4)\).

This is equal to $32, so:

\(\frac{2}{3}(x - 4) = 32\)

Multiply both sides by \(\frac{3}{2}\):

\(x-4 = 48\)

Add 4 to both sides:

\(x = 52\)

So there were 52 dollars left after Lyle got his share. Bob received $4, and then one-third of the remaining $48. One-third of 48 is 16, so Bob received 4 + 16 = 20 dollars in all.

Notice that we need not consider Lyle's portion in the solution. We can just let K = the money REMAINING after Lyle has received his portion and go from there.
Our equation will use the fact that, once we remove Bob's portion, we have $32 for Chloe.
So, we get K - Bob's $ = 32

Bob received 4 dollars plus one-third of what remained
Once Bob receives $4, the amount remaining is K-4 dollars. So, Bob gets a 1/3 of that as well.
1/3 of K-4 is (K-4)/3
So ALTOGETHER, Bob receives 4 + (K-4)/3

So, our equation becomes: K - [4 + (K-4)/3 ] = 32
Simplify to get: K - 4 - (K-4)/3 = 32
Multiply both sides by 3 to get: 3K - 12 - K + 4 = 96
Simplify: 2K - 8 = 96
Solve: K = 52

Plug this K-value into K - Bob's $ = 32 to get: 52 - Bob's $ = 32
So, Bob's $ = 20

Answer: B

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Another approach:

This time, let K = the money REMAINING after Ann has received her portion AND after Bob has taken $4.
At this point, Bob receives 1/3 of K, and Chloe gets the rest.
This means that Chloe receives 2/3 of K

Since Chloe receives $32, we can say that: (2/3)K = 32
Multiply both sides by 3/2 to get: K = 48

Since Bob receives 1/3 of K plus $4, we can see that the amount Bob gets = (1/3)(48) + 4 = $20

Answer: B

Cheers,
Brent
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I looked at Magoosh's and Brent's solution, which makes sense and is relatively quick. When I tried it I went about it a long way. Specifically, I thought that:

Lyle + Bob + Chloe = Total

I could write Lyle and Bob in terms of Total with the given info and write in 32 for Chloe and then solve for Total. With that I could find Bob.

That was the idea. You can see in the attached calculations I got the wrong answer. I don't know why. Yes, in the future I will do the problem the quick way, as shown by others, but if someone could tell me where I went wrong in my long calculations I would appreciate it (or maybe their is a conceptual flaw in my thinking about it)

T - (0.5T + 2) - 4 simplifies to be T - 0.5T - 2 - 4

Cheers,
Brent
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Thank you very much Brent.

Here is the corrected version if anyone's interested. (Of course, as I said before, it's too long. Use the other method)

Lets, we (L, B, and C) have total 100$.

Now L received 4 + \(\frac{1}{2}\) of 96 ; (96, because after receiving 4 from 100, remaining is 96);

After that B received 4 + \(\frac{1}{3}\) of 48 ; (48, because after receiving \(\frac{1}{2 }\)of 96 we have remaining 48 only)

Now, we have remain only 32 which is C received. (32, because \(\frac{48}{3}\) = 16; 48 - 16 = 32 remaining lastly )

So, from B, ( 4 + \(\frac{1}{3}\) 48 ), we get, 4 + 16 = 20.
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Quick and easy method:
Since Bob received $4 plus 1/3 of the remainder, the total between Bob and Chloe can be represented as $32=2/3(T-4).
Simplify.
96=2(T-4), 48=T-4, T=52
Then, subtract Chloe's sum from the total between Bob and Chloe.
52-32=$20
B