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State X has \(\mathrm{77}\) acres designated as wilderness, which is \(\mathrm{0.0003}\) percent of its total acreage. Which of the following expresses the total acreage of State X?
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
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 \(0.0003\%\).
\(\text{Total acreage} = 77 \div 0.0003\%\)
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:
\(\text{Total acreage} = 77 \div 0.000003\)
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
Looking at our result: \(\frac{(10^6)(77)}{3}\)
Comparing with the answer choices:
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.
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 \(\text{Total} = 77 \div 0.0003\%\).
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.
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."
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 \(\frac{(77 \times 10^6)}{3}\).
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.
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.