Jared and Willie are always trading baseball cards during their 6th grade math class. When their teacher decided she wasn’t having it any more and confiscated the cards, the other students wailed, in chorus, “That’s not fair; you should give them back.” The teacher, Mrs. Bibble, said, “Well, maybe, but you have to do some thinking first. Try this scenario. On Monday these two come in with their cards. On Tuesday, Jared gives Willie as many cards as Willie already has. On Wednesday, Willie then gives Jared back as many as Jared has left. On Thursday, Jared gives Willie back as many cards as Willie has left. Now here it is, Friday, and Willie has 80 cards altogether, but poor Jared has none. If you can tell me how many cards each of them had on Monday, then I will return the cards. But no more trading in class.” Can you help the boys get their cards back?
This can be done by simply guessing numbers, or by programming a spreadsheet. Or by algebra:
| Willie | Jared | |||
| M | w | j | ||
| T | w + w = 2w | j – w | ||
| W | 2w – (j – w) = 3w – j |
2(j – w) = 2j – 2w | ||
| T | 2(3w – j) | 2j – 2w – (3w – j) = 3j – 5w |
||
| F | 6w – 2j = 80 | 3j – 5w = 0 | → |
w + j = 80 → j = 80 – w 3j = 5w → 3(80 – w) = 5w 240 – 3w = 5w 240 = 8w w = 30 j = 50. |