AMC 10 · 2008 · #10
Grade 6 rate-ratioPick an answer.
Two different units are hiding in this problem, and noticing the clash is the whole game — so tool #8 (Analyze the Units) leads. The solo speeds are given as times to finish, which measures work in chimneys: 1/9 and 1/10 of a chimney per hour, numbers that stay the same no matter how big the chimney is. The chatter penalty is given in bricks: 10 bricks per hour, a flat count that also ignores the chimney's size. Those two scales only line up through the unknown N, and that single point of contact is what pins N down. Tool #16 (Change Focus) is how the contact gets made: instead of tracking the work they did, track the work they lost, because the loss is the one quantity that can be measured in both units. Tool #7 (Identify Subproblems) keeps the fraction bookkeeping in order, and tool #6 (Guess and Check) closes the argument by feeding the number back into the original story — showing N not only is forced, but actually works.
Read each time as a rate
Each solo time becomes a rate.
Finishing a job in 9 hours and doing 1/9 of it each hour are two ways of saying one thing.
6.RP.A.2Analyze The UnitsHow much silence would have built
The sum of the rates would have built more than one job.
Two people laying bricks at once cover the sum of their hourly shares, so five hours of that sum is what silence would have produced.
5.NF.A.1Identify SubproblemsThe overshoot is exactly what talking cost
That overshoot is exactly what the slowdown cost.
Stop counting the bricks they laid and count the bricks they should have laid but didn't — the gap is the thing both units can name.
Stop counting the bricks they laid and count the bricks they should have laid: the gap is what talking cost.
▸ Why?
Two rates acting on the same job differ by a fixed amount, and that difference is the loss per hour.
▸ Why?
At a steady rate the amount done is the rate multiplied by the time, so the loss is one multiplication.
Price the same loss in bricks
The same loss in absolute units is 50.
One quantity measured two ways must come out the same, and that single equality is what the whole problem rests on.
6.RP.A.3Analyze The UnitsScale the piece to the whole
Scaling the piece to the whole gives 900.
If one slice out of eighteen equal slices weighs 50, the whole thing weighs eighteen fifties.
6.EE.B.7Identify SubproblemsCheck that 900 really works
Checking it forwards confirms 900, choice (E).
Forcing a number and confirming a number are two different jobs, so do both.
6.EE.B.5Guess And CheckMeasure the work they lost twice — once as a fraction of the chimney, once as a count of bricks — and setting those two measurements equal tells you how big the chimney is.
- Read each time as a rate
- How much silence would have built
- The overshoot is exactly what talking cost
- Price the same loss in bricks
- Scale the piece to the whole
- Check that 900 really works