Each year for 4 years, a farmer increased the number of trees in a certain orchard by 1/4 of the...

1. Translate the problem requirements: Each year, the farmer increases trees by \(\frac{1}{4}\) of the previous year's amount, meaning each year the total becomes \(125\%\) of the previous year (original + \(\frac{1}{4}\) more). We need to find the starting amount given that after 4 years of this growth, there are 6250 trees.
2. Identify the growth pattern: Since adding \(\frac{1}{4}\) means multiplying by 1.25 each year, we can establish that after 4 years, the original amount has been multiplied by \((1.25)^4\).
3. Set up the relationship equation: If we call the original number of trees 'x', then \(x \times (1.25)^4 = 6250\), so we need to solve for x.
4. Calculate backwards to find the starting amount: Determine \((1.25)^4\) and divide 6250 by this value to find the original number of trees.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what's happening in plain English. The farmer starts with some number of trees. Each year, he plants more trees - specifically, he adds \(\frac{1}{4}\) of whatever he had the previous year.

For example, if he started with 100 trees:

• Year 1: He adds \(\frac{1}{4}\) of 100 = 25 more trees, so he has 100 + 25 = 125 trees
• Year 2: He adds \(\frac{1}{4}\) of 125 = 31.25 more trees, so he has 125 + 31.25 = 156.25 trees

Notice that adding \(\frac{1}{4}\) of the current amount is the same as multiplying by 1.25 (since \(100\% + 25\% = 125\% = 1.25\)).

We know that after 4 years of this pattern, he ends up with 6250 trees. We need to find how many he started with.

Process Skill: TRANSLATE - Converting the verbal description into mathematical understanding

2. Identify the growth pattern

Since each year the farmer adds \(\frac{1}{4}\) of the previous year's trees, the total becomes \(125\%\) of what it was before.

This means each year we multiply by 1.25:

• After 1 year: \(\text{Original} \times 1.25\)
• After 2 years: \(\text{Original} \times 1.25 \times 1.25 = \text{Original} \times (1.25)^2\)
• After 3 years: \(\text{Original} \times (1.25)^3\)
• After 4 years: \(\text{Original} \times (1.25)^4\)

So the growth pattern follows a consistent multiplication factor of 1.25 each year.

3. Set up the relationship equation

Let's call the original number of trees 'x'.

After 4 years of multiplying by 1.25 each year:
\(x \times (1.25)^4 = 6250\)

This equation captures the relationship between our unknown starting value and the known ending value.

4. Calculate backwards to find the starting amount

Now we need to solve: \(x \times (1.25)^4 = 6250\)

First, let's calculate \((1.25)^4\):

• \(1.25^2 = 1.5625\)
• \(1.25^4 = (1.25^2)^2 = (1.5625)^2 = 2.44140625\)

So our equation becomes: \(x \times 2.44140625 = 6250\)

To find x: \(x = 6250 \div 2.44140625 = 2560\)

Let's verify: \(2560 \times (1.25)^4 = 2560 \times 2.44140625 = 6250\) ✓

4. Final Answer

The farmer started with 2560 trees at the beginning of the 4-year period.

This matches answer choice D: 2560.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the growth pattern direction

Students often confuse whether they should work forward from the starting number or backward from the ending number. Since we know the final result (6250 trees) and need the initial amount, students might try to guess-and-check forward instead of setting up the equation to work backward systematically.

2. Incorrect interpretation of "increased by 1/4"

Students frequently misinterpret "increased by \(\frac{1}{4}\) of the number of trees" as simply adding \(\frac{1}{4}\) each year (additive growth) rather than understanding it as multiplicative growth where each year's total becomes 1.25 times the previous year's amount.

3. Setting up the wrong time relationship

Students may incorrectly count the number of growth periods. Since the farmer increases trees "each year for 4 years," there are exactly 4 multiplication periods, but some students might think there are only 3 growth periods or get confused about when the counting starts.

Errors while executing the approach

1. Calculation errors with (1.25)⁴

Students often make arithmetic mistakes when calculating \((1.25)^4\), especially when computing \(1.25^2 = 1.5625\) first, then squaring it again. The decimal calculations can be error-prone, leading to an incorrect final multiplier.

2. Division errors when solving for x

When solving \(x = 6250 \div 2.44140625\), students may make computational mistakes with the division, especially since they're dividing by a decimal with many places. This can lead them to select an incorrect answer choice.

Errors while selecting the answer

1. Verification confusion

Students might get the correct numerical answer but then second-guess themselves during verification. They may incorrectly check their work by applying the growth pattern in the wrong direction or making arithmetic errors during the check, leading them to doubt the right answer and select a different choice.