Depth · one concept from first principles · step 2 of 4
At 7:00 a.m., some cars are parked in a parking lot, and more cars keep arriving at a steady rate every minute. If cars leave at an average of cars every minutes, the lot becomes empty at 8:12 a.m. If cars leave at an average of cars every minutes, the lot becomes empty at 9:32 a.m. How many cars were parked in the lot at 7:00 a.m.?
From 7:00 to 8:12 is 72 minutes; to 9:32 is 152 minutes. The leaving rates are 7/9 and 1/2 car per minute.
Rates can only be compared once they share a unit.
7.RP.A.1Analyze The UnitsLet a be the starting cars and r the arrivals per minute. When the lot is empty, everything that was there or arrived has left.
Both equations say: start + arrivals = departures.
7.EE.B.4Convert To AlgebraThe second case runs 80 minutes longer and 20 more cars leave; the start is the same, so those 20 cars arrived in those 80 minutes.
Subtracting one story from the other cancels a.
7.EE.B.4Identify SubproblemsThe extra cars that leave in the second case are exactly the cars that arrive in its extra minutes, so .
When the lot is empty, the cars that left are exactly the starting cars plus the cars that arrived.
The starting cars are the same in both cases, so taking the first equation from the second removes them and leaves .
Cars arrive at a steady rate, so the extra minutes bring in cars.
Put r = 1/4 into the first equation.
One car arrives every 4 minutes, so 18 cars came in during the first 72 minutes.
7.EE.B.3Convert To AlgebraTwo versions of the same story differ only in what changed, and that difference reveals the hidden rate.
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