2480.51 – Trading Marbles

Anthony had twice as many marbles as Russell but after he gave Russell 35 marbles, then Russell had three times as many as Anthony. How many marbles did each have to begin with?

Solution

One can start with guess and try: pick some number for Anthony's original number of marbles, say 100. Then Russell starts with 50. Anthony gives away 35, leaving him 65, and Russell now has 85. 85 is supposed to be 3 times 65, but it isn't, so that's no good. Pick another number – smaller or larger than 100?

But for students who are just getting started in algebra, this can be a good teaching problem. There are two equations: at the beginning of the story, $A = 2R$ and at the end, $(R + 35) = 3 (A - 35)$ .

We see (or can be helped to see) that Russell's final number, $R + 35, = 3A - 105$ , or $6 R - 105$ (remember that $A = 2R$ ). That is, $R + 35 = 6R - 105$ . If we trade things around a bit, we see that $5R = 140$ .

Therefore $R = 28$ and $A = 56$ .

Students might want to speculate as to why Anthony is giving away marbles in the first place, especially since he is left with very few.

Run the actual numbers through the original problem to see that they work.