State X has 77 acres designated as wilderness, which is 0.0003 percent of its total acreage. Which of the following...

1. Translate the problem requirements: We know that 77 acres represents \(0.0003\%\) of the total acreage, and we need to find an expression for the total acreage of State X.
2. Set up the percentage relationship: Use the basic percentage formula where 77 acres = \(0.0003\%\) of total acres to create an equation.
3. Convert percentage to decimal form: Transform \(0.0003\%\) into its decimal equivalent to make the calculation clearer.
4. Solve for total acreage: Rearrange the equation to isolate the total acreage and express it in a form that matches the answer choices.
5. Match with answer choices: Compare our result with the given options to identify the correct expression.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have and what we need to find. We're told that State X has 77 acres of wilderness land. This wilderness area represents a tiny fraction of the state's total land - specifically, it's \(0.0003\) percent of all the state's acreage.

Think of it this way: if you had a huge property and someone told you that your small garden was only \(0.0003\%\) of your entire property, you'd realize your total property must be much, much larger than that garden.

What we need to find: An expression for the total acreage of State X
What we know: \(77 \text{ acres} = 0.0003\% \text{ of the total acreage}\)
Process Skill: TRANSLATE - Converting the word problem into a clear mathematical relationship

2. Set up the percentage relationship

Now let's express this relationship in simple terms. When we say "77 acres is \(0.0003\) percent of the total," we can write this as:

\(77 \text{ acres} = 0.0003\% \times (\text{Total acreage})\)

In everyday language: "The wilderness area equals a tiny percentage times the whole state."

To find the total acreage, we need to rearrange this relationship. If 77 equals \(0.0003\%\) of the total, then the total equals 77 divided by \(0.0003\%\).

\(\text{Total acreage} = 77 \div 0.0003\%\)

3. Convert percentage to decimal form

Before we can do the division, we need to convert the percentage to a decimal. Remember that percent means "per hundred," so:

\(0.0003\% = 0.0003 \div 100 = 0.000003\)

Think of it this way: \(0.0003\%\) is already a very small number, and dividing by 100 makes it even smaller - it becomes \(0.000003\).

Now our equation becomes:
\(\text{Total acreage} = 77 \div 0.000003\)

4. Solve for total acreage

Now we need to calculate \(77 \div 0.000003\). When dividing by a very small decimal, it's easier to work with scientific notation.

\(0.000003 = 3 \times 10^{-6}\)

So: \(\text{Total acreage} = 77 \div (3 \times 10^{-6})\)

When dividing by a number in scientific notation, we can rewrite this as:
\(\text{Total acreage} = 77 \times (1 \div (3 \times 10^{-6}))\)
\(\text{Total acreage} = 77 \times (10^6 \div 3)\)
\(\text{Total acreage} = (77 \times 10^6) \div 3\)

This can be written as: \(\frac{(10^6)(77)}{3}\)
Process Skill: MANIPULATE - Converting complex division into a more manageable form using scientific notation

5. Match with answer choices

Looking at our result: \(\frac{(10^6)(77)}{3}\)

Comparing with the answer choices:

1. \(3(10^4)(77)\) - This multiplies by 3 instead of dividing, and uses \(10^4\)
2. \(3(10^6)(77)\) - This multiplies by 3 instead of dividing
3. \(\frac{(10^2)(77)}{3}\) - This has the right structure but wrong power of 10
4. \(\frac{(10^4)(77)}{3}\) - This has the right structure but wrong power of 10
5. \(\frac{(10^6)(77)}{3}\) - This exactly matches our result!

4. Final Answer

The correct answer is E. \(\frac{(10^6)(77)}{3}\)

This makes intuitive sense: since 77 acres represents such a tiny percentage (\(0.0003\%\)) of the total, the total acreage must be enormous - millions of times larger than 77, which is exactly what our expression shows with the \(10^6\) factor.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the percentage relationship

Students often confuse which quantity is the part and which is the whole. They might incorrectly think that the total acreage is \(0.0003\%\) of the 77 acres, rather than understanding that 77 acres represents \(0.0003\%\) of the total acreage. This fundamental misunderstanding leads to setting up the equation backwards: \(\text{Total} = 77 \times 0.0003\%\) instead of \(\text{Total} = 77 \div 0.0003\%\).

2. Confusion about what the question is asking for

The question asks for an expression that represents the total acreage, not the numerical value. Students might attempt to calculate the exact numerical answer and then try to match it with the choices, rather than recognizing they need to find the algebraic expression that equals the total acreage.

Errors while executing the approach

1. Incorrect percentage to decimal conversion

When converting \(0.0003\%\) to decimal form, students frequently forget to divide by 100. They might use \(0.0003\) instead of \(0.000003\), leading to an answer that's off by a factor of 100. This error occurs because students sometimes treat the % symbol as just notation rather than recognizing it means "divided by 100."

2. Errors in scientific notation manipulation

When working with \(0.000003 = 3 \times 10^{-6}\), students often make mistakes with the negative exponent. They might incorrectly convert this to \(3 \times 10^6\) or struggle with the division rule for exponents, writing \(77 \div (3 \times 10^{-6})\) as \((77 \times 3 \times 10^{-6})\) instead of correctly getting \(\frac{(77 \times 10^6)}{3}\).

3. Arithmetic errors with powers of 10

Students may incorrectly determine the power of 10 needed. Since \(0.000003\) has 6 decimal places, they might count incorrectly and use \(10^4\) or \(10^5\) instead of \(10^6\), leading them to select answer choices C or D instead of the correct answer E.

Errors while selecting the answer

1. Choosing expressions that multiply by 3 instead of dividing by 3

After correctly determining they need \(10^6\) and 77 in their expression, students might select answer choice B: \(3(10^6)(77)\) because it contains the right components but in the wrong mathematical relationship. They fail to notice that their derived expression requires division by 3, not multiplication by 3.