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Which of the following fractions has the greatest value?
We need to find which of these five fractions gives us the biggest number when we calculate it out. Think of it like this: if you had to choose between getting \(\frac{1}{4}\) of a pizza versus \(\frac{1}{2}\) of a pizza, you'd want \(\frac{1}{2}\) because it's bigger. Here we have more complex fractions, but the same principle applies.
Each fraction has a numerator (top number) and a denominator (bottom number) that contains powers of 2 and powers of 5. The bigger the numerator and the smaller the denominator, the larger our fraction will be.
Let's write out what we're comparing:
Process Skill: TRANSLATE - Converting the mathematical notation into plain language understanding
To compare these fractions fairly, let's first calculate what each denominator actually equals. This is like finding a common way to measure all our pizza slices.
Let's work out the denominators step by step:
Now our fractions look much cleaner:
Process Skill: SIMPLIFY - Breaking down complex expressions into manageable numbers
Now let's convert each fraction to a decimal so we can easily see which is biggest. This is like converting all our measurements to the same units.
Looking at these decimal values:
Option D gives us \(0.062\), which is larger than all the others.
Let's double-check our calculation for Option D and compare it with the closest competitor (Option E):
Option D: \(\frac{62}{1000} = 0.062\)
Option E: \(\frac{122}{2000} = \frac{122}{2000} = \frac{61}{1000} = 0.061\)
Indeed, \(0.062 > 0.061\), confirming that Option D is larger.
We can also verify this makes sense: Option D has a numerator of \(62\) and when we scale Option E to the same denominator \((1000)\), it becomes \(\frac{61}{1000}\). Since \(62 > 61\), Option D wins.
The fraction with the greatest value is Option D: \(\frac{62}{(2^3)(5^3)}\), which equals \(0.062\). This fraction is larger than all other options when converted to decimal form.
1. Misunderstanding the comparison task: Students may think they need to find a common denominator for all fractions (like adding fractions) rather than simply determining which has the greatest value. This leads to unnecessarily complex calculations and potential errors.
2. Attempting to use cross-multiplication incorrectly: When comparing multiple fractions, students might try to use cross-multiplication methods meant for comparing just two fractions, leading to confusion when dealing with five different options simultaneously.
3. Overlooking the decimal conversion approach: Students may get intimidated by the exponential notation and attempt complex algebraic manipulations instead of recognizing that converting to decimals provides the most straightforward comparison method.
1. Calculation errors in computing powers: Students frequently make mistakes when calculating exponentials, especially confusing \(2^3 = 8\) vs \(2^4 = 16\), or \(5^2 = 25\) vs \(5^3 = 125\). These errors cascade through the entire solution.
2. Arithmetic mistakes in multiplication: When computing denominators like \(2^3 \times 5^3 = 8 \times 125\), students often make multiplication errors (getting \(1125\) instead of \(1000\), for example), leading to incorrect decimal conversions.
3. Decimal conversion errors: Students may incorrectly convert fractions to decimals, such as computing \(\frac{28}{500}\) as \(0.56\) instead of \(0.056\), often dropping or misplacing decimal places.
1. Misreading decimal comparisons: Students may incorrectly order decimals like \(0.061\) and \(0.062\), thinking \(0.061\) is larger because \(61 > 62\), forgetting to consider the decimal place values properly.
2. Selecting based on numerator size alone: Seeing that option E has the largest numerator \((122)\), students might incorrectly assume it must be the largest fraction without considering the denominator's impact.