AMC 10 · 2013 · #10
Grade 6 rate-ratioA flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?
Pick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A bouquet has four kinds of flowers: pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of all the flowers are pink. Find what percent of all the flowers are carnations.
Givens: Each flower is exactly one of: pink rose, red rose, pink carnation, red carnation; Among the pink flowers, $\tfrac{1}{3}$ are roses; Among the red flowers, $\tfrac{3}{4}$ are carnations; $\tfrac{6}{10}$ of all the flowers are pink; Answer choices: (A) $15$, (B) $30$, (C) $40$, (D) $60$, (E) $70$
Unknowns: The percent of all flowers that are carnations
Understand
Restated: A bouquet has four kinds of flowers: pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of all the flowers are pink. Find what percent of all the flowers are carnations.
Givens: Each flower is exactly one of: pink rose, red rose, pink carnation, red carnation; Among the pink flowers, $\tfrac{1}{3}$ are roses; Among the red flowers, $\tfrac{3}{4}$ are carnations; $\tfrac{6}{10}$ of all the flowers are pink; Answer choices: (A) $15$, (B) $30$, (C) $40$, (D) $60$, (E) $70$
Plan
Primary tool: #9 Solve an Easier Related Problem
Secondary: #7 Identify Subproblems, #8 Analyze the Units
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.
Execute — Answer: E
6.RP.A.3 Step 1 Use 100 flowers as the total
- No real count is given, only fractions, and the question asks for a percent.
- Percent means 'out of 100', so pretend the bouquet has exactly $100$ flowers.
- With that choice, however many flowers turn out to be carnations is automatically the percent that are carnations.
- Picking $100$ does not change the answer because every condition is a fraction, which scales with the total.
💡 Since the answer is a percent, choosing 100 flowers makes the final count equal to the percent with no extra division.
6.RP.A.3 Step 2 Split by color: pink and red
- Six tenths of the flowers are pink, so out of $100$ flowers there are $\tfrac{6}{10}\cdot 100 = 60$ pink flowers.
- The rest are red, so there are $100 - 60 = 40$ red flowers.
- Every flower is now sorted into one of these two color groups.
💡 Pink and red are the only two colors, so once you know the pink count the red count is just what is left.
5.NF.B.6 Step 3 Count the pink carnations
- Among the $60$ pink flowers, one third are roses, so the other two thirds are carnations.
- Two thirds of $60$ is $\tfrac{2}{3}\cdot 60 = 40$, so there are $40$ pink carnations.
💡 If one third of a group are roses, the remaining two thirds must be the carnations.
5.NF.B.6 Step 4 Count the red carnations
- Among the $40$ red flowers, three fourths are carnations.
- Three fourths of $40$ is $\tfrac{3}{4}\cdot 40 = 30$, so there are $30$ red carnations.
💡 Taking three fourths of a group is the same as splitting it into four equal parts and keeping three of them.
6.RP.A.3 Step 5 Add the carnations and read off the percent
- Carnations come in two kinds: pink and red.
- Add the two counts: $40 + 30 = 70$ carnations.
- Out of the $100$ flowers we started with, $70$ are carnations, which is $70\%$.
- So the answer is $\textbf{(E)}\ 70$.
💡 Total carnations is just pink carnations plus red carnations, and out of 100 flowers that count is the percent.
6.RP.A.3 No real count is given, only fractions, and the question asks for a percent. Per 6.RP.A.3 Six tenths of the flowers are pink, so out of $100$ flowers there are $\tfrac{6} 5.NF.B.6 Among the $60$ pink flowers, one third are roses, so the other two thirds are ca 5.NF.B.6 Among the $40$ red flowers, three fourths are carnations. Three fourths of $40$ 6.RP.A.3 Carnations come in two kinds: pink and red. Add the two counts: $40 + 30 = 70$ c Review
Reasonableness: Check that all four groups add back to the whole bouquet: $20$ pink roses $+ 40$ pink carnations $+ 10$ red roses $+ 30$ red carnations $= 100$ flowers, exactly the total we chose. The red roses come from $40 - 30 = 10$, matching one fourth of the red flowers. Carnations ($70$) plus roses ($20 + 10 = 30$) also give $100$. Since more than half the flowers are pink and most pink flowers are carnations, a carnation percent well above $50$ is sensible, so $70\%$ fits.
Alternative: Work with fractions of the whole instead of $100$ flowers. Pink is $\tfrac{6}{10}$ and red is $\tfrac{4}{10}$. Pink carnations are $\tfrac{2}{3}\cdot\tfrac{6}{10} = \tfrac{4}{10}$ of all flowers, and red carnations are $\tfrac{3}{4}\cdot\tfrac{4}{10} = \tfrac{3}{10}$ of all flowers. Adding gives $\tfrac{4}{10} + \tfrac{3}{10} = \tfrac{7}{10} = 70\%$, the same answer.
CCSS standards used (min grade 6)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Treating the total as 100 flowers, converting 'six tenths are pink' into a count, and reading the final carnation count of 70 out of 100 as 70 percent.)5.NF.B.6Solve real-world problems involving multiplication of fractions and mixed numbers (Taking two thirds of the 60 pink flowers to get 40 pink carnations and three fourths of the 40 red flowers to get 30 red carnations.)
⭐ When 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.
⭐ When 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.
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