- Translate the problem requirements: We know \(\mathrm{15\%}\) of plants developed fungus AND died. We also know that of the plants that developed fungus, \(\mathrm{40\%}\) did NOT die (meaning \(\mathrm{60\%}\) of fungus plants died). We need to find what percent of all plants developed fungus.
- Set up the relationship between fungus plants and deaths: If \(\mathrm{60\%}\) of fungus plants died, and this represents \(\mathrm{15\%}\) of all plants, we can work backwards to find the total percentage with fungus.
- Calculate using proportional reasoning: Since \(\mathrm{60\%}\) of fungus plants = \(\mathrm{15\%}\) of all plants, we can find what \(\mathrm{100\%}\) of fungus plants represents as a percentage of all plants.
- Verify the logic with the survival rate: Check that our answer makes sense by confirming the \(\mathrm{40\%}\) survival rate among fungus plants.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know in plain English:
- \(\mathrm{15\%}\) of ALL plants in the nursery developed fungus AND died
- \(\mathrm{40\%}\) of the plants that developed fungus did NOT die (which means \(\mathrm{60\%}\) of fungus plants DID die)
- We need to find: What percent of ALL plants developed fungus?
Think of it this way: imagine we have a group of plants that developed fungus. Some of these fungus plants died (\(\mathrm{60\%}\) of them), and some survived (\(\mathrm{40\%}\) of them). The plants that died represent \(\mathrm{15\%}\) of the entire nursery.
Process Skill: TRANSLATE - Converting the problem language into clear mathematical relationships
2. Set up the relationship between fungus plants and deaths
Let's use concrete reasoning. If we call the percentage of plants that developed fungus "F%", then:
- F% of all plants developed fungus
- \(\mathrm{60\%}\) of those fungus plants died
- So \(\mathrm{60\%}\) of F% = \(\mathrm{15\%}\) of all plants
In everyday terms: "\(\mathrm{60\%}\) of the fungus group equals \(\mathrm{15\%}\) of the whole nursery."
Mathematically: \(\mathrm{0.60 \times F\% = 15\%}\)
3. Calculate using proportional reasoning
Now we solve for F%:
- \(\mathrm{0.60 \times F\% = 15\%}\)
- \(\mathrm{F\% = 15\% ÷ 0.60}\)
- \(\mathrm{F\% = 15\% ÷ \left(\frac{60}{100}\right)}\)
- \(\mathrm{F\% = 15\% \times \left(\frac{100}{60}\right)}\)
- \(\mathrm{F\% = 15\% \times \left(\frac{5}{3}\right)}\)
- \(\mathrm{F\% = \frac{15 \times 5}{3} = \frac{75}{3} = 25\%}\)
So \(\mathrm{25\%}\) of all plants in the nursery developed fungus.
4. Verify the logic with the survival rate
Let's check our answer makes sense:
- \(\mathrm{25\%}\) of plants developed fungus
- \(\mathrm{60\%}\) of these fungus plants died: \(\mathrm{60\% \text{ of } 25\% = 0.60 \times 25\% = 15\%}\) ✓
- \(\mathrm{40\%}\) of these fungus plants survived: \(\mathrm{40\% \text{ of } 25\% = 0.40 \times 25\% = 10\%}\)
- Total check: \(\mathrm{15\% + 10\% = 25\%}\) had fungus ✓
This confirms our logic is correct.
Final Answer
\(\mathrm{25\%}\) of the plants in the nursery developed fungus.
The answer is A. \(\mathrm{25\%}\)
Common Faltering Points
Errors while devising the approach
- Misinterpreting "\(\mathrm{40\%}\) of plants that developed fungus did not die": Students often confuse this statement and think it means \(\mathrm{40\%}\) of ALL plants in the nursery didn't die from fungus, rather than understanding it refers to \(\mathrm{40\%}\) of only those plants that actually developed fungus. This leads to setting up incorrect relationships.
- Confusing the subset relationships: Students may struggle to recognize that the plants that died (\(\mathrm{15\%}\) of all plants) are a subset of the plants that developed fungus, not a separate group. They might try to add percentages incorrectly instead of recognizing the nested relationship.
- Setting up the wrong equation: Instead of recognizing that "\(\mathrm{60\%}\) of fungus plants = \(\mathrm{15\%}\) of all plants," students might incorrectly set up equations like "\(\mathrm{40\% + 15\% = \text{total with fungus}}\)" or other combinations that don't reflect the actual relationships.
Errors while executing the approach
- Arithmetic errors in division: When solving \(\mathrm{15\% ÷ 0.60}\), students commonly make calculation mistakes, especially when converting between decimals and fractions. They might incorrectly calculate \(\mathrm{15 ÷ 0.6}\) or struggle with the fraction conversion \(\mathrm{15 \times \left(\frac{5}{3}\right)}\).
- Using wrong survival/death percentages: Students might use \(\mathrm{40\%}\) instead of \(\mathrm{60\%}\) as the death rate among fungus plants, forgetting that if \(\mathrm{40\%}\) survived, then \(\mathrm{60\%}\) died. This leads to the wrong equation: \(\mathrm{0.40 \times F\% = 15\%}\).
Errors while selecting the answer
- Selecting a partial result instead of the final answer: Students might calculate intermediate values like \(\mathrm{10\%}\) (the plants that had fungus and survived) or \(\mathrm{15\%}\) (the plants that died) and mistakenly select one of these as their final answer instead of the total \(\mathrm{25\%}\) that developed fungus.
Alternate Solutions
Smart Numbers Approach
Step 1: Choose a smart number for total plants
Let's say there are 100 plants in the nursery (this makes percentage calculations straightforward).
Step 2: Identify what we know
• \(\mathrm{15\%}\) of all plants (15 plants) developed fungus AND died
• \(\mathrm{40\%}\) of plants that developed fungus did NOT die
• This means \(\mathrm{60\%}\) of plants that developed fungus DID die
Step 3: Set up the relationship
If \(\mathrm{60\%}\) of fungus plants died, and this equals 15 plants:
\(\mathrm{60\%}\) of fungus plants = 15 plants
\(\mathrm{0.6 \times (\text{number of fungus plants}) = 15}\)
Step 4: Solve for total fungus plants
Number of fungus plants = \(\mathrm{15 ÷ 0.6 = 25}\) plants
Step 5: Convert to percentage
Percentage of plants with fungus = \(\mathrm{\frac{25}{100} = 25\%}\)
Step 6: Verify our answer
• Total fungus plants: 25
• Fungus plants that died: \(\mathrm{60\% \text{ of } 25 = 15}\) plants ✓
• Fungus plants that survived: \(\mathrm{40\% \text{ of } 25 = 10}\) plants
• This matches our given information that \(\mathrm{15\%}\) of all plants died from fungus
