1550.39 – Trading Cards

At Whitney M. Young High School, Jared and Willie traded "Garbage-Pail Kids" cards all week. Finally, on Friday, the teacher confiscated them.

"No fair," the class complained. "Return the cards."

"OK," Mrs. Gay replied, "But this is a math class. On Monday these guys came with their cards. On Tuesday, Jared gave Willie as many cards as he already had. On Wednesday, Willie gave Jared back as many as Jared had left. On Thursday, Jared, not to be out done by Willie's generosity, gave Willie back as many cards as Willie had left. Here it is Friday. Poor Jared is broke and Willie has 80 cards. If you can tell me how many each had on Monday, I'll give the cards back."

Solution

This can be attacked by judicious guessing or by programming a spread sheet or by algebra. Here is an outline of the algebra route. A table summarizes the week's transactions:

day ∣Willie ∣JaredM ∣W∣JT ∣W+W=2W∣J−WW ∣2W−(J−W)=3W−J∣2(J−W)Th ∣2(2W−J)=6W−2J∣2J−2W−(3W−J)=3J−5WF ∣80∣0 \begin{aligned} \text{day } &|& \text{Willie } &|& \text{Jared} \\ \text{M } &|& W &|& J \\ \text{T } &|& W+W = 2W &|& J-W \\ \text{W } &|& 2W - (J-W) = 3W - J &|& 2(J - W) \\ \text{Th } &|& 2(2W - J) = 6W - 2J &|& 2J - 2W - (3W - J) = 3J - 5W \\ \text{F } &|& 80 &|& 0 \end{aligned}

This leads to two equations:

6W−2J=80,3J−5W=0. \begin{aligned} 6W - 2J &=& 80, \\ 3J - 5W &=& 0. \end{aligned}

From the first equation we learn that J=3W−40J = 3W - 40. Plugging this into the second equation:

3(3W−40)−5W=0,9W−5W=120,4W=120. \begin{aligned} 3 (3W - 40) - 5W &=& 0, \\ 9W - 5W &=& 120, \\ 4W &=& 120. \end{aligned}

So W=30W = 30 and J=50.J = 50.