If a certain garden plot contains 25 tomato plants, the yield per plant will be 10 tomatoes. If the plot...

1. Translate the problem requirements: Clarify the yield relationship - with 25 plants, each produces 10 tomatoes; for every additional plant added, each plant's yield decreases by 1 tomato. We need to find the total number of plants when the total yield was 174.
2. Set up the yield equation: Express total yield as (number of plants) × (yield per plant) using the given relationships.
3. Solve the quadratic equation: Transform the yield equation into standard quadratic form and solve for the number of additional plants.
4. Calculate total plants and verify: Add the additional plants to the base 25 plants and confirm the result produces 174 total tomatoes.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in this garden in plain English:

• We start with a baseline: 25 tomato plants, each producing 10 tomatoes
• When we add more plants, something interesting happens - each plant produces fewer tomatoes
• Specifically, for every 1 additional plant we add beyond 25, every single plant (including the original 25) produces 1 fewer tomato
• Last year, there were more than 25 plants and the total harvest was 174 tomatoes

Let's call the number of additional plants \(\mathrm{x}\). So if we have \(\mathrm{x}\) additional plants:

• Total plants = \(25 + \mathrm{x}\)
• Yield per plant = \(10 - \mathrm{x}\) (since each additional plant reduces everyone's yield by 1)
• Total yield = (Total plants) × (Yield per plant) = \((25 + \mathrm{x}) \times (10 - \mathrm{x})\)

Process Skill: TRANSLATE - Converting the word problem into mathematical relationships

2. Set up the yield equation

Now we can write our equation using the relationships we identified:

Total yield = (Number of plants) × (Yield per plant)
\(174 = (25 + \mathrm{x}) \times (10 - \mathrm{x})\)

Let's expand this step by step:
\(174 = (25 + \mathrm{x}) \times (10 - \mathrm{x})\)
\(174 = 25 \times 10 - 25 \times \mathrm{x} + \mathrm{x} \times 10 - \mathrm{x} \times \mathrm{x}\)
\(174 = 250 - 25\mathrm{x} + 10\mathrm{x} - \mathrm{x}^2\)
\(174 = 250 - 15\mathrm{x} - \mathrm{x}^2\)

3. Solve the quadratic equation

Let's rearrange this into standard quadratic form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\):

\(174 = 250 - 15\mathrm{x} - \mathrm{x}^2\)
\(174 - 250 = -15\mathrm{x} - \mathrm{x}^2\)
\(-76 = -15\mathrm{x} - \mathrm{x}^2\)
Multiplying by -1: \(76 = 15\mathrm{x} + \mathrm{x}^2\)
Rearranging: \(\mathrm{x}^2 + 15\mathrm{x} - 76 = 0\)

Now we need to solve this quadratic equation. Let's try factoring:
We need two numbers that multiply to -76 and add to 15.
Thinking through factor pairs of 76: 1×76, 2×38, 4×19...
Trying 19 and -4: 19 × (-4) = -76 and 19 + (-4) = 15 ✓

So: \(\mathrm{x}^2 + 15\mathrm{x} - 76 = (\mathrm{x} + 19)(\mathrm{x} - 4) = 0\)

This gives us \(\mathrm{x} = -19\) or \(\mathrm{x} = 4\)

Since \(\mathrm{x}\) represents additional plants, \(\mathrm{x} = -19\) doesn't make physical sense.
Therefore, \(\mathrm{x} = 4\) additional plants.

Process Skill: APPLY CONSTRAINTS - Rejecting the negative solution based on real-world meaning

4. Calculate total plants and verify

Total plants = \(25 + \mathrm{x} = 25 + 4 = 29\) plants

Let's verify this works:

• Number of plants: 29
• Yield per plant: \(10 - 4 = 6\) tomatoes per plant
• Total yield: \(29 \times 6 = 174\) tomatoes ✓

This matches our required total of 174 tomatoes, and \(29 > 25\) as required.

Final Answer

The plot contained 29 plants. This corresponds to answer choice A.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the yield reduction mechanism

Students often misinterpret how the yield reduction works. They may think that only the additional plants have reduced yield, or that the yield reduces by a different amount. The key insight is that for every 1 additional plant added beyond 25, ALL plants (including the original 25) produce 1 fewer tomato each.

2. Incorrectly defining the variable

Students might define \(\mathrm{x}\) as the total number of plants instead of the number of additional plants beyond 25. This leads to incorrect equations like "Total plants = \(\mathrm{x}\)" and "Yield per plant = \(10 - (\mathrm{x}-25)\)" which complicates the problem unnecessarily.

3. Missing the constraint about plant count

The problem states that "last year the plot contained more than 25 plants" but students often ignore this constraint when setting up their approach, which becomes crucial later when selecting the final answer.

Errors while executing the approach

1. Algebraic expansion errors

When expanding \((25 + \mathrm{x})(10 - \mathrm{x})\), students commonly make sign errors or miss terms. Common mistakes include writing "\(25 \times 10 + 25 \times \mathrm{x} + \mathrm{x} \times 10 - \mathrm{x}^2\)" (wrong sign on the \(25\mathrm{x}\) term) or forgetting the \(\mathrm{x}^2\) term entirely.

2. Factoring the quadratic incorrectly

For \(\mathrm{x}^2 + 15\mathrm{x} - 76 = 0\), students often struggle to find the correct factor pair. They may try combinations that don't work (like 38 and 2) or make arithmetic errors when checking if the factors multiply to -76 and add to 15.

3. Rearranging the equation incorrectly

When moving from \(174 = 250 - 15\mathrm{x} - \mathrm{x}^2\) to standard form, students frequently make sign errors, especially when multiplying by -1 or moving terms across the equals sign.

Errors while selecting the answer

1. Selecting x instead of total plants

After finding \(\mathrm{x} = 4\), students might mistakenly select 4 as their final answer, forgetting that \(\mathrm{x}\) represents additional plants and they need to calculate total plants = \(25 + 4 = 29\).

2. Choosing the negative solution

When solving the quadratic and getting \(\mathrm{x} = -19\) or \(\mathrm{x} = 4\), students might not properly apply the real-world constraint that the number of additional plants cannot be negative, leading them to consider both solutions as valid.