Splitting the Grant Fairly
A youth club receives a $1,200 grant to split between three activities: art, sport and music. Rules say no activity may receive more than half the grant, and each must receive at least $200. The committee proposes art $700, sport $300, music $200. A member objects that art is over the cap. They suggest art $600, sport $400, music $200. A second member wants music raised to $300 and asks where the money should come from without breaking any rule.
1. Test both proposals against the two rules. Which one passes, and why does the other fail?
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Rule check: cap is half of $1,200 = $600. Floor is $200 each. Total must equal $1,200.
Proposal 1: art $700 is over the $600 cap, so it fails immediately, even though it adds to $1,200 and every share is at least $200.
Proposal 2: art $600 (at the cap, allowed), sport $400, music $200 (at the floor). Total $1,200. All shares within limits. It passes.
So the first member is right that art broke the cap, and their replacement fixes it. The key is that 'no more than half' allows exactly $600 - the cap is a ceiling, not a forbidden value.
2. The second member wants music at $300. Show one valid split that does this, and explain why the money cannot come from art alone.
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Why not from art alone: art is already at the $600 cap. Lowering art to $500 frees $100, which would raise music from $200 to $300. But then sport stays at $400 and the total is $500 + $400 + $300 = $1,200 - that actually works. So art can fund it; the cap only limits increases, not decreases.
A valid split: art $500, sport $400, music $300. Check: total $1,200; each at least $200; none over $600. Passes.
Another valid split: art $600, sport $300, music $300. Also passes.
Both work because sport and art together can absorb the shift. The only hard limits are the $600 ceiling and $200 floor, and both splits respect them.