AMC 10 · 2013 · #10
Grade 6 rate-ratioPick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem never gives a real count of flowers, only fractions, and it asks for a percent. That makes Tool #9 (Solve an Easier Related Problem) the natural start: replace the abstract 'all the flowers' with a concrete, convenient total of 100 flowers. Percent means 'per hundred', so with 100 flowers the final carnation count is already the answer in percent (Tool #8, Analyze the Units). From there the bouquet splits cleanly into two color groups, and each group into roses and carnations, so Tool #7 (Identify Subproblems) lets us count pink carnations and red carnations separately and then add them.
Use 100 flowers as the total
Only fractions are given and the answer is a percent, so pretend the bouquet holds exactly 100 flowers.
Since the answer is a percent, choosing 100 flowers makes the final count equal to the percent with no extra division.
6.RP.A.3Solve An Easier Related ProblemSplit by color: pink and red
Six tenths of the 100 flowers are pink, so 60 are pink and the remaining 40 are red.
Pink and red are the only two colors, so once you know the pink count the red count is just what is left.
6.RP.A.3Analyze The UnitsCount the pink carnations
One third of the 60 pink flowers are roses, so the other two thirds, 40, are pink carnations.
If one third of a group are roses, the remaining two thirds must be the carnations.
If one third of a group are roses, the remaining two thirds must be the carnations.
▸ Why?
The group is exactly the roses plus the carnations, with nothing else in it.
▸ Why?
A third is one of three equal shares, so the other two shares are what is left.
Count the red carnations
Three fourths of the 40 red flowers are carnations, giving 30 red carnations.
Taking three fourths of a group is the same as splitting it into four equal parts and keeping three of them.
5.NF.B.6Identify SubproblemsAdd the carnations and read off the percent
Pink plus red carnations is 40 + 30 = 70 of the 100 flowers, so 70% are carnations — choice (E).
Total carnations is just pink carnations plus red carnations, and out of 100 flowers that count is the percent.
6.RP.A.3Identify SubproblemsWhen a problem gives only fractions and asks for a percent, pretend there are 100 things: count each group, add up the ones you want, and that count is the percent.
- Use 100 flowers as the total
- Split by color: pink and red
- Count the pink carnations
- Count the red carnations
- Add the carnations and read off the percent