In a certain orchard, 4/5 of the trees are apple trees and 1/10 of the trees are pear trees. If...

1. Translate the problem requirements: We need to understand what "\(\frac{4}{5}\) of trees are apple trees" and "\(\frac{1}{10}\) of trees are pear trees" means in terms of the total number of trees, then express apple trees in terms of the given \(\mathrm{x}\) pear trees.
2. Establish the total using pear trees as reference: Since we know there are \(\mathrm{x}\) pear trees and this represents \(\frac{1}{10}\) of all trees, we can determine the total number of trees.
3. Calculate apple trees using the total: Once we know the total number of trees, we can find the number of apple trees since they represent \(\frac{4}{5}\) of all trees.
4. Express result in terms of x: Simplify our expression to match one of the given answer choices.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is telling us in everyday language:

• We have an orchard with different types of trees
• Out of every 5 trees in the orchard, 4 are apple trees (that's what "\(\frac{4}{5}\) of the trees are apple trees" means)
• Out of every 10 trees in the orchard, 1 is a pear tree (that's what "\(\frac{1}{10}\) of the trees are pear trees" means)
• We're told there are exactly \(\mathrm{x}\) pear trees
• We need to find how many apple trees there are

Think of it like this: if you walked through the orchard and counted trees, for every pear tree you found, there would be a specific number of apple trees. Our job is to figure out that relationship.

Process Skill: TRANSLATE - Converting the fraction language into a clear understanding of tree ratios

2. Establish the total using pear trees as reference

Since we know there are \(\mathrm{x}\) pear trees, and these represent \(\frac{1}{10}\) of all trees, we can figure out the total number of trees.

Here's the reasoning: If 1 out of every 10 trees is a pear tree, and we have \(\mathrm{x}\) pear trees, then we must have 10 times as many total trees.

For example, if there were 3 pear trees (\(\mathrm{x} = 3\)), and pear trees make up \(\frac{1}{10}\) of all trees, then:

• 3 pear trees = \(\frac{1}{10}\) of total trees
• Total trees = \(3 \times 10 = 30\) trees

In general terms:

• \(\mathrm{x}\) pear trees = \(\frac{1}{10}\) of total trees
• Total trees = \(\mathrm{x} \times 10 = 10\mathrm{x}\) trees

3. Calculate apple trees using the total

Now that we know there are \(10\mathrm{x}\) total trees, we can find the number of apple trees.

We're told that \(\frac{4}{5}\) of all trees are apple trees. This means:

• Apple trees = \(\frac{4}{5}\) of total trees
• Apple trees = \(\frac{4}{5} \times 10\mathrm{x}\)
• Apple trees = \(\frac{4 \times 10\mathrm{x}}{5}\)
• Apple trees = \(\frac{40\mathrm{x}}{5}\)
• Apple trees = \(8\mathrm{x}\)

Let's verify with our example: if there were 30 total trees, then apple trees = \(\frac{4}{5} \times 30 = 24\) apple trees. And indeed, \(8 \times 3 = 24\), so our formula works!

4. Express result in terms of x

We found that the number of apple trees = \(8\mathrm{x}\).

Let's double-check this makes sense:

• Pear trees: \(\mathrm{x}\) trees (given)
• Total trees: \(10\mathrm{x}\) trees
• Apple trees: \(8\mathrm{x}\) trees
• Remaining trees: \(10\mathrm{x} - \mathrm{x} - 8\mathrm{x} = \mathrm{x}\) trees (these would be other types)

This means the fractions work out correctly:

• Pear trees: \(\frac{\mathrm{x}}{10\mathrm{x}} = \frac{1}{10}\) ✓
• Apple trees: \(\frac{8\mathrm{x}}{10\mathrm{x}} = \frac{8}{10} = \frac{4}{5}\) ✓

Final Answer

The number of apple trees is \(8\mathrm{x}\).

Looking at our answer choices:

1. \(\frac{7\mathrm{x}}{10}\)
2. \(\frac{9\mathrm{x}}{10}\)
3. \(3\mathrm{x}\)
4. \(4\mathrm{x}\)
5. \(8\mathrm{x}\) ✓

The answer is E.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the relationship between fractions and actual quantities

Students often struggle to connect that if pear trees represent \(\frac{1}{10}\) of total trees and there are \(\mathrm{x}\) pear trees, then the total must be \(10\mathrm{x}\). They may try to work directly with the fractions \(\frac{4}{5}\) and \(\frac{1}{10}\) without establishing the total number of trees first, leading to confusion about how to express apple trees in terms of \(\mathrm{x}\).

2. Confusing which quantity to use as the reference point

Since the problem gives the number of pear trees as \(\mathrm{x}\), students might mistakenly try to express apple trees directly as a multiple of pear trees (like "apple trees = 8 × pear trees") without recognizing they need to go through the total number of trees as an intermediate step.

Errors while executing the approach

1. Arithmetic errors when multiplying fractions

When calculating \(\frac{4}{5} \times 10\mathrm{x}\), students frequently make errors such as getting \(4\mathrm{x}\) instead of \(8\mathrm{x}\), or incorrectly computing \(\frac{4 \times 10\mathrm{x}}{5} = \frac{40\mathrm{x}}{5}\). The fraction multiplication and simplification step is a common source of computational mistakes.

2. Incorrectly setting up the proportion

Students may set up the relationship backwards, thinking "if \(\frac{1}{10}\) of trees are pear trees and there are \(\mathrm{x}\) pear trees, then total = \(\frac{\mathrm{x}}{10}\)" instead of correctly reasoning that "total = \(10\mathrm{x}\)." This fundamental error in proportion setup leads to completely wrong calculations.

Errors while selecting the answer

1. Selecting fractional answers when the result should be a whole multiple

After correctly calculating that there are \(8\mathrm{x}\) apple trees, some students might doubt their answer because it seems "too simple" and instead select one of the fractional options like \(\frac{7\mathrm{x}}{10}\) or \(\frac{9\mathrm{x}}{10}\), thinking these look more sophisticated or reasonable for a proportion problem.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient total number of trees
Since we need fractions like \(\frac{4}{5}\) and \(\frac{1}{10}\) to work out to whole numbers, let's choose the total number of trees to be the LCM of the denominators (5 and 10), which is 10.

Step 2: Calculate the number of each type of tree
With 10 total trees:
• Apple trees = \(\frac{4}{5} \times 10 = 8\) trees
• Pear trees = \(\frac{1}{10} \times 10 = 1\) tree
So we have \(\mathrm{x} = 1\) pear tree.

Step 3: Find the relationship between apple trees and x
We have 8 apple trees when \(\mathrm{x} = 1\) pear tree.
Therefore, the number of apple trees = \(8\mathrm{x}\).

Step 4: Verify with the answer choices
When \(\mathrm{x} = 1\):

• Choice A: \(\frac{7\mathrm{x}}{10} = \frac{7(1)}{10} = 0.7\) ✗
• Choice B: \(\frac{9\mathrm{x}}{10} = \frac{9(1)}{10} = 0.9\) ✗
• Choice C: \(3\mathrm{x} = 3(1) = 3\) ✗
• Choice D: \(4\mathrm{x} = 4(1) = 4\) ✗
• Choice E: \(8\mathrm{x} = 8(1) = 8\) ✓

The answer is E.

Why this smart number approach works:
By choosing 10 as our total (the LCM of denominators 5 and 10), we ensure that all fractional parts result in whole numbers, making calculations simple and avoiding messy arithmetic. The relationship \(8\mathrm{x}\) holds regardless of the actual number of trees in the orchard.