AMC 10 · 2009 · #5
Grade 7 rate-ratioPick an answer.
AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The sentence hands you the finished value on the left and asks for the starting number on the right, so the natural move is to compute the left side, then undo the 'one-third more' step to walk back to the original number.
Compute twenty percent less than 60
Twenty percent of 60 is 12, and 'less than' means subtract it: 60 - 12 = 48.
'Percent less' means slice that share off the whole, not add it on.
7.RP.A.3Identify SubproblemsWrite 'one-third more than n'
Call the number n. 'One-third more than n' is n plus a third of n, which is four-thirds of n, and that equals 48.
'One-third more' scales the number up to 4/3 of itself, not just to a third of it.
One third more scales the number up to four thirds of itself, not just to a third of it.
▸ Why?
The bigger amount is the original plus the added share, so both pieces have to be counted.
▸ Why?
The added share is measured against the original, so the whole thing scales by one common factor.
Undo the scaling to find n
Four-thirds of n is 48, so undo it: multiply 48 by three-fourths to get n = 36, choice (D).
To find the original from a scaled-up value, divide by the same factor you multiplied by.
6.EE.B.7Work Backwards'One-third more' means times four-thirds, so to go back you divide by four-thirds.
- Compute twenty percent less than 60
- Write 'one-third more than n'
- Undo the scaling to find n