- Translate the problem requirements: We need to find the difference between two fractions: \(\frac{0.99999999}{1.0001}\) minus \(\frac{0.99999991}{1.0003}\). The numbers are very close to 1, suggesting we should use approximation.
- Recognize the approximation opportunity: Since the numerators are very close to 1 and denominators are slightly above 1, we can approximate each fraction as close to 1 and focus on the small differences.
- Apply strategic approximation: Use the fact that when dividing a number close to 1 by a number slightly above 1, we can approximate the result using the relationship \(\frac{1-\mathrm{small\ amount}}{1+\mathrm{small\ amount}} \approx 1-2(\mathrm{small\ amount})\).
- Calculate the difference and match to answer choices: Compute the approximate difference and compare with the given options to identify the correct order of magnitude.
Execution of Strategic Approach
1. Translate the problem requirements
Let's start by understanding what we're being asked to find. We have two fractions that we need to subtract:
First fraction: \(0.99999999 \div 1.0001\)
Second fraction: \(0.99999991 \div 1.0003\)
We need to find: (First fraction) - (Second fraction)
Looking at these numbers, I notice something important: both numerators are extremely close to 1 (they're just tiny amounts less than 1), and both denominators are just slightly above 1. This tells me that both fractions will be very close to 1, and their difference will be a small number.
Process Skill: TRANSLATE - Converting the decimal expressions into a clear subtraction problem and recognizing the proximity to 1
2. Recognize the approximation opportunity
Since we're dealing with numbers so close to 1, let me rewrite them to see the pattern more clearly:
First numerator: \(0.99999999 = 1 - 0.00000001 = 1 - 10^{-8}\)
Second numerator: \(0.99999991 = 1 - 0.00000009 = 1 - 9 \times 10^{-8}\)
First denominator: \(1.0001 = 1 + 0.0001 = 1 + 10^{-4}\)
Second denominator: \(1.0003 = 1 + 0.0003 = 1 + 3 \times 10^{-4}\)
Now I can see that both fractions have the form "something very close to 1" divided by "something slightly bigger than 1". This is perfect for approximation techniques.
Process Skill: SIMPLIFY - Expressing complex decimals in terms of powers of 10 to reveal the underlying structure
3. Apply strategic approximation
When we have a fraction like \(\frac{1 - \mathrm{small\ amount}}{1 + \mathrm{small\ amount}}\), we can use this approximation:
In plain English: "A number slightly less than 1, divided by a number slightly more than 1, gives us a result that's about 1 minus twice the denominator's extra amount."
Let me apply this to each fraction:
For the first fraction \(\frac{1 - 10^{-8}}{1 + 10^{-4}}\):
Since \(10^{-4}\) is much larger than \(10^{-8}\), the main effect comes from dividing by \((1 + 10^{-4})\)
This gives us approximately: \(1 - 10^{-4}\)
For the second fraction \(\frac{1 - 9 \times 10^{-8}}{1 + 3 \times 10^{-4}}\):
Again, \(3 \times 10^{-4}\) dominates over \(9 \times 10^{-8}\)
This gives us approximately: \(1 - 3 \times 10^{-4}\)
Technical notation: For \(\frac{1-a}{1+b}\) where a,b are small, this \(\approx 1-b\) when \(b \gg a\)
4. Calculate the difference and match to answer choices
Now I can find the difference:
Difference = \([1 - 10^{-4}] - [1 - 3 \times 10^{-4}]\)
Difference = \(1 - 10^{-4} - 1 + 3 \times 10^{-4}\)
Difference = \(-10^{-4} + 3 \times 10^{-4}\)
Difference = \(3 \times 10^{-4} - 10^{-4}\)
Difference = \(2 \times 10^{-4}\)
Looking at the answer choices:
- \(10^{-8} = 0.00000001\)
- \(3 \times 10^{-8} = 0.00000003\)
- \(3 \times 10^{-4} = 0.0003\)
- \(2 \times 10^{-4} = 0.0002\)
- \(10^{-4} = 0.0001\)
Our calculated result of \(2 \times 10^{-4}\) matches choice (D) exactly.
Final Answer
The answer is (D) \(2 \times 10^{-4}\).
This makes sense because the key insight was recognizing that when we divide numbers very close to 1 by numbers slightly above 1, the small differences in the numerators become negligible compared to the differences in the denominators. The calculation reduced to finding the difference between the reciprocals of the denominators, giving us \(2 \times 10^{-4}\).
Common Faltering Points
Errors while devising the approach
- Missing the approximation opportunity: Students often get intimidated by the complex decimal numbers and attempt to perform exact arithmetic calculations instead of recognizing that the numbers are very close to 1, which makes this problem perfect for approximation techniques. This leads to extremely tedious calculations that are prone to errors.
- Misunderstanding the magnitude hierarchy: Students may fail to recognize that the denominators (\(10^{-4}\) and \(3 \times 10^{-4}\)) have much larger magnitudes than the numerator differences (\(10^{-8}\) and \(9 \times 10^{-8}\)). This hierarchy is crucial for determining which terms dominate the approximation.
- Converting decimals incorrectly: Students might make errors when expressing decimals like 0.99999999 as \((1 - 10^{-8})\) or 1.0001 as \((1 + 10^{-4})\), either miscounting decimal places or incorrectly determining the power of 10.
Errors while executing the approach
- Applying approximation formulas incorrectly: When using the approximation \(\frac{1-a}{1+b} \approx 1-b\) for small values, students might incorrectly include both terms, writing something like \(1-a-b\) instead of recognizing that when \(b \gg a\), the dominant effect comes from \(-b\) alone.
- Sign errors during subtraction: When computing \([1 - 10^{-4}] - [1 - 3 \times 10^{-4}]\), students frequently make sign errors, especially when distributing the negative sign. They might incorrectly get \(-2 \times 10^{-4}\) instead of \(+2 \times 10^{-4}\).
- Arithmetic mistakes with scientific notation: Students may struggle with operations like \(3 \times 10^{-4} - 10^{-4} = 2 \times 10^{-4}\), either by incorrectly handling the coefficients or getting confused about the powers of 10.
Errors while selecting the answer
- Magnitude confusion: Students might calculate the correct numerical result but select an answer choice with the wrong order of magnitude, such as choosing \(2 \times 10^{-8}\) instead of \(2 \times 10^{-4}\), especially if they made errors in the decimal-to-scientific notation conversion.
- Missing the negative sign consideration: If students obtained \(-2 \times 10^{-4}\) due to sign errors, they might look for this exact value among the choices and when not finding it, randomly guess instead of recognizing they need to check their sign.
