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The easy way to answer this is to plug in numbers. Because you are given "12" and you are given "twice as many", the answer has to be a variable of 12 or 6. If you start plugging in numbers, you will find that 36 is the answer...
Ben and Jerry start with 36 cards.
Ben gives 12 of his cards to Jerry. Ben now has 24 and Jerry now has 48. 24 is one half of 48.
For the actual math, you have to use variables. Please stick with it, this gets a little ugly if the child you are working with is as young as mine is.
Ben at the start is equal to B0
Jerry at the start is equal to J0
At the start B0=J0
Now, Ben gives 12 of his cards to Jerry. The new value for Ben is B1 and B1 = B0-12.
The new value for Jerry is J1 and J1 = J0+12.
Also, we know that the new Jerry is twice the amount of the new Ben...J1 =B1 x 2.
So, the below are known:
1: B0=J0
2: B1 = B0-12
3: J1 = J0+12
4: J1 = B1 x 2
So, using the last known, start to substitute the other values to get everything equal to the same variable. In this case, I'm going to solve it for B0.
J1 = B1 x 2
J1 = (B0-12)x2 - Substitute B1 for B0-12, see line 2
J0+12=(B0-12)x2 - Substitute J1 for J0+12, see line 3
B0+12=(B0-12)x2 - Substitute J0 for B0, see line 1
1/2B0+6 = B0-12 - Divide the whole equation by 2
+6=B0-12-1/2B0 - Subtract 1/2B0 from both sides
+18=B0-1/2B0 - Add 12 to both sides
+18=1/2B0 - Simplify equation
36=B0 - Multiply both sides by 2
There is probably an easier way to do the above, but that's how I worked it out on paper.
Well, well, well, Ben must have had 24 cards to start with. Jerry was probably sitting pretty with 12 cards, but after Ben's generous donation, Ben ended up with 12 cards, doubling Jerry's stash. Math can be a real game-changer, can't it?
Let's denote the initial number of cards Ben and Jerry had as x. After Ben gives 12 cards to Jerry, Ben has x-12 cards and Jerry has x+12 cards. Given that Ben then has twice as many cards as Jerry, we can set up the equation x-12 = 2(x+12). Solving this equation, we find that x = 36. Therefore, Ben had 36 cards at the start.
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