The student's reasoning for estimating the sum using cluster estimation is not accurate because they are rounding the numbers in a way that does not maintain the overall magnitude of the original numbers, especially with regard to the decimals. Let's break down the estimation step by step:
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Grouping Numbers: The student correctly observes that all the numbers begin with either a 7 or an 8, so it makes sense to group them for estimation. However, the rounding should reflect a more precise understanding of the numbers rather than simply rounding to the nearest whole number without considering the decimals.
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Rounding: The student rounds each number as follows:
- \(7.95\) is rounded up to \(8\)
- \(8.11\) is rounded up to \(8\)
- \(78.5\) is rounded up to \(80\)
- \(8.05\) is rounded up to \(8\)
- \(79.4\) is rounded up to \(80\)
- \(0.815\) is rounded up to \(0.8\)
While these rounds simplify the calculation, they do not accurately capture the values of the original numbers. For example, \(78.5\) should be more appropriately seen as approximately \(78\) rather than rounding to \(80\).
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Accuracy of the Estimate: By rounding \(78.5\) to \(80\) and \(79.4\) to \(80\), the student is effectively overestimating the contributions of these numbers. This results in an inflated total since the actual values are lower than what was estimated.
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Result of the Estimate: The accurate estimate should ideally take into account a more reasonable rounding where \(78.5\) might be rounded to \(79\) and \(79.4\) to \(79\), so an accurate adjustment could yield:
- \(7.95\) to approximately \(8\)
- \(8.11\) to approximately \(8\)
- \(78.5\) to approximately \(79\)
- \(8.05\) to approximately \(8\)
- \(79.4\) to approximately \(79\)
- \(0.815\) to approximately \(0.8\)
Therefore, the estimated sum would have been \(8 + 8 + 79 + 8 + 79 + 0.8 = 182.8\), which is closer to the actual sum.
In conclusion, the student's estimation method was not accurate because it rounded some numbers too aggressively, leading to an overall inflated estimate. A more precise rounding leads to a more accurate total.