I still had trouble reasoning through how to mesh a ratio with real numbers in this case, but I think I finally see what they did, so I thought I'd share. (I think visualizing the "after" diagram was the part that tripped me up, but once I could see the "after" Roses bar, the rest fell into place.) So here’s how I’d expand on their answer.

There's a reason this problem is an optional challenge problem

Julie

BEFORE:SarahP wrote:There were twice as many carnations as roses in a flower shop. After selling 50 carnations and 10 roses, there were 3 times as many roses as carnations left in the shop. How many roses were there in the shop at first?

/----------R----------/

/----------C----------/----------C----------/

AFTER:

/-R-/-R-/-R-/

/-C-/

We don't know the value of R or C, but we know the /R/ and /C/ bars are equal in each pair of diagrams.

TO GO BACK TO THE "BEFORE" AND INCORPORATE INFORMATION FROM THE "AFTER":

If we add back the amount sold, we go back to the original ratio (2:1):

/-R-/-R-/-R-/ + 10 SOLD = before

/-C-/ + 50 SOLD = before

so this...

/----------R----------/

/----------C----------/----------C----------/

becomes this...

/-R-/-R-/-R-/-10-/

/-C-/---------------50 SOLD--------------/ (twice as long)

or we could look at 2:1 like this...

/-R-/-R-/-R-/-10-/ (roses)

/-R-/-R-/-R-/-10-/-R-/-R-/-R-/-10-/ (carnations)

MAKE USE OF CONCRETE NUMBERS:

We could compare the two "after" carnation diagrams, making use of all concrete numbers given (50 and the 10),

and I start to ignore one bar in each, because I know they are equal, and focus on the red info:

/-C-/--------------50 SOLD--------------/ (carnations)

/-R-/-R-/-R-/--10--/-R-/-R-/-R-/-10-/ (carnations)

Then, rearranging a bit:

/-C-/--------------50 SOLD--------------/ (carnations)

/-R-/-R-/-R-/-R-/-R-/-R-/-10-/-10-/ (carnations)

COMPARING NUMBERS TO BARS:

Okay at this point, we can see that 5 of the unknown bars are equal to 30, right?

/-C-/-------30 SOLD-------/-20 SOLD-/ (carnations)

/-R-/-R-/-R-/-R-/-R-/-R-/-10-/-10-/ (carnations)

So one unknown bar is 6 (because 5x6=30 in the purple area),

which can be plugged back into every /R/ and /C/ in the "after" diagrams.

/-R-/-R-/-R-/-10-/ (roses) = 28

/-R-/-R-/-R-/-10--/-R-/-R-/-R-/-10-/ (carnations) = 56

OR

/-R-/-R-/-R-/-10-/ = 28

/-C-/---------------50 SOLD---------------/ = 56