AMC 10 · 2009 · #7
Grade 7 rate-ratioPick an answer.
The end price is pinned to the start price, and April's drop is the only unknown. So compute the price through March forward, then work backwards from the fixed end value to find the drop April must have made. Picking a convenient start value of 100 turns every percent into an easy multiplication.
Pick a starting price of 100
A convenient starting price makes every percent easy.
Percent answers stay the same no matter the starting amount, so pick the amount that makes the arithmetic easiest.
7.RP.A.3Solve An Easier Related ProblemApply January and February
The first two months land it below the start.
Up 20% then down 20% loses ground, because the drop is a bigger share of the raised price.
Up twenty percent then down twenty percent loses ground, because the drop is a bigger share of the raised price.
▸ Why?
A percent is always a count out of a hundred of whatever it is applied to, so the base matters.
▸ Why?
The same fraction of a larger amount is a larger amount, so the fall outweighs the rise.
Apply March
The third month pushes it back above.
Multiply by 1.25 to add a quarter of the current price on top.
7.RP.A.3Identify SubproblemsWork backwards to find April's drop
The final drop is measured against that raised price.
A percent drop is always measured against the price you are dropping from, which here is 120, not 100.
7.RP.A.3Work BackwardsRound to the nearest integer
Rounding gives 17, choice (E).
A decimal part above 0.5 rounds up to the next whole number.
5.NBT.A.4Work BackwardsA percent change always works on the current price, so undoing a rise needs a smaller percent than the rise itself.
- Pick a starting price of 100
- Apply January and February
- Apply March
- Work backwards to find April's drop
- Round to the nearest integer