A farmer's potato yield this year averaged 30 tonnes per hectare. This was 25% greater than last year's average yield...

1. Translate the problem requirements: We need to clarify three different yields: this year's actual (\(30\) tonnes), last year's actual (unknown), and this year's projected (unknown). The question asks for the percent increase from last year's actual to this year's projected.
2. Work backwards from known actual yield: Use this year's actual yield of \(30\) tonnes as our starting point to find last year's yield, since we know it was \(25\%\) greater than last year's.
3. Scale up to find projected yield: Use the relationship that this year's actual was only \(\frac{1}{3}\) of the projected yield to find this year's projected yield.
4. Calculate the percent increase: Compare last year's actual yield with this year's projected yield to determine the percent increase.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in plain English:

• This year's actual yield: \(30\) tonnes per hectare (this is our concrete starting point)
• Last year's actual yield: We don't know this yet, but we know this year was \(25\%\) better
• This year's projected yield: We don't know this yet, but we know the actual yield was only \(\frac{1}{3}\) of what was projected
• What we need to find: How much greater (as a percentage) was this year's projected yield compared to last year's actual yield

Think of it like this: We have three different numbers representing yields, and we need to compare two of them. We'll use the one we know (\(30\) tonnes) as our bridge to find the other two.

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

2. Work backwards from known actual yield

Since this year's actual yield (\(30\) tonnes) was \(25\%\) greater than last year's, let's work backwards.

In everyday terms: If this year is \(25\%\) more than last year, then this year represents \(125\%\) of last year's yield (\(100\% + 25\% = 125\%\)).

So if \(30\) tonnes = \(125\%\) of last year's yield, then:
Last year's yield = \(30 \div 1.25 = 24\) tonnes per hectare

Let's verify: \(24 + (25\% \text{ of } 24) = 24 + 6 = 30\) ✓

Technical notation: If \(L\) = last year's yield, then \(1.25L = 30\), so \(L = 24\)

3. Scale up to find projected yield

Now we know that this year's actual yield (\(30\) tonnes) was only \(\frac{1}{3}\) of what was projected for this year.

In simple terms: If \(30\) tonnes is \(\frac{1}{3}\) of the projection, then the full projection must be \(3\) times larger.

This year's projected yield = \(30 \times 3 = 90\) tonnes per hectare

Let's verify: \(\frac{1}{3} \text{ of } 90 = 30\) ✓

Technical notation: If \(P\) = projected yield, then \(\frac{P}{3} = 30\), so \(P = 90\)

4. Calculate the percent increase

Now we can compare the two yields we need:
• Last year's actual yield: \(24\) tonnes
• This year's projected yield: \(90\) tonnes

To find the percent increase, we use the formula:
Percent increase = \(\frac{\text{New value} - \text{Old value}}{\text{Old value}} \times 100\%\)

Percent increase = \(\frac{90 - 24}{24} \times 100\%\)
Percent increase = \(\frac{66}{24} \times 100\%\)
Percent increase = \(2.75 \times 100\% = 275\%\)

In plain English: The projected yield for this year was \(275\%\) greater than last year's actual yield. This means it was nearly \(4\) times as large (\(90\) is almost \(4\) times \(24\)).

Final Answer

The projected average yield for this year was \(275\%\) greater than last year's average yield.

This matches answer choice A. \(275\%\)

Verification: \(24 + (275\% \text{ of } 24) = 24 + 66 = 90\) ✓

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the relationship between actual and projected yields
Students often confuse which direction the comparison goes. The problem states that this year's actual yield (\(30\) tonnes) was "only \(\frac{1}{3}\) of the projected average yield." Students may incorrectly think that the projected yield was \(\frac{1}{3}\) of the actual yield, leading them to calculate \(30 \div 3 = 10\) tonnes instead of \(30 \times 3 = 90\) tonnes for the projected yield.

2. Confusion about which yields to compare in the final step
The question asks for the percent increase from "last year's average yield" to "this year's projected yield." However, students often get lost in the multiple yield values (last year's actual, this year's actual, this year's projected) and may incorrectly compare this year's actual yield to last year's actual yield, or compare this year's projected to this year's actual instead of the correct comparison.

3. Setting up the percent increase formula incorrectly
Students may confuse "what percent greater" with "what percent of." The question asks how much greater the projected yield was, which requires the formula \(\frac{\text{New} - \text{Old}}{\text{Old}} \times 100\%\). Some students might use \(\frac{\text{New}}{\text{Old}} \times 100\%\) instead, which would give them \(375\%\) rather than \(275\%\).

Errors while executing the approach

1. Arithmetic errors when working backwards from percentages
When calculating last year's yield from "this year was \(25\%\) greater," students may incorrectly subtract \(25\%\) from \(30\) (getting \(30 - 7.5 = 22.5\)) instead of recognizing that \(30\) represents \(125\%\) of last year's yield and dividing by \(1.25\) to get \(24\) tonnes.

2. Calculation errors in the final percentage computation
The final calculation requires computing \(\frac{90-24}{24} \times 100\% = \frac{66}{24} \times 100\%\). Students often make arithmetic mistakes here, such as incorrectly calculating \(66 \div 24 = 2.5\) instead of \(2.75\), leading to \(250\%\) instead of \(275\%\). Division of fractions like \(\frac{66}{24}\) can be particularly error-prone under time pressure.

Errors while selecting the answer

1. Selecting \(375\%\) instead of \(275\%\)
If students used the incorrect formula \(\frac{\text{New}}{\text{Old}} \times 100\%\) instead of \(\frac{\text{New}-\text{Old}}{\text{Old}} \times 100\%\), they would calculate \(\frac{90}{24} \times 100\% = 375\%\). While this isn't among the answer choices, students might then select the closest option or double-check their work but still make the same conceptual error about what "percent greater" means.

Alternate Solutions

Smart Numbers Approach

For this problem, we can use smart numbers by recognizing that we need values that work well with the given relationships: \(25\%\) increase and the fraction \(\frac{1}{3}\).

Step 1: Choose a smart number for last year's yield
Let's set last year's average yield = \(24\) tonnes per hectare
(We choose \(24\) because it works cleanly with \(25\%\) calculations: \(25\%\) of \(24 = 6\))

Step 2: Calculate this year's actual yield
This year's actual yield = Last year's yield + \(25\%\) increase
This year's actual = \(24 + (25\% \times 24) = 24 + 6 = 30\) tonnes per hectare
✓ This matches the given information

Step 3: Calculate this year's projected yield
We know: This year's actual yield = \(\frac{1}{3} \times\) This year's projected yield
So: \(30 = \frac{1}{3} \times\) Projected yield
Projected yield = \(30 \times 3 = 90\) tonnes per hectare

Step 4: Calculate the percent increase from last year's actual to this year's projected
Increase = \(90 - 24 = 66\) tonnes per hectare
Percent increase = \(\frac{66}{24} \times 100\% = 2.75 \times 100\% = 275\%\)

Verification:
• Last year's actual: \(24\) tonnes
• This year's actual: \(30\) tonnes (\(25\%\) greater than \(24\) ✓)
• This year's projected: \(90\) tonnes (\(30\) is \(\frac{1}{3}\) of \(90\) ✓)
• Percent increase: \(\frac{90-24}{24} \times 100\% = 275\%\) ✓

The answer is A. \(275\%\)