- Part (a): Represent each fraction as \(1 - \epsilon_i\). The sum is approximately \(4 - 0.32 = 3.68\), which is closest to \(\boxed{4}\).
- Part (b): Approximate each fraction as \(\frac{30}{41} \approx 0.75\), \(\frac{244}{1000} \approx 0.25\), and \(\frac{2}{2000} \approx 0\). The sum is approximately \(\boxed{1}\), which matches option \(\boxed{D}\).
Explanation
- Approximating the First Sum
We analyze the sum \(S_a = \frac{8}{9} + \frac{11}{12} + \frac{12}{13} + \frac{20}{21}\) by observing that each fraction is slightly less than 1. Specifically, each term can be written as \(1 - \epsilon_i\), where \(\epsilon_i\) represents the small difference between the fraction and 1:
\(\frac{8}{9} = 1 - \frac{1}{9} \approx 1 - 0.111\)
\(\frac{11}{12} = 1 - \frac{1}{12} \approx 1 - 0.083\)
\(\frac{12}{13} = 1 - \frac{1}{13} \approx 1 - 0.077\)
\(\frac{20}{21} = 1 - \frac{1}{21} \approx 1 - 0.048\)
Summing these, we get \(S_a = 4 - (\frac{1}{9} + \frac{1}{12} + \frac{1}{13} + \frac{1}{21}) \approx 4 - 0.319 = 3.681\). Comparing \(3.681\) to the options \(\{1, 2, 3, 4\}\), the value is clearly closest to \(4\).
- Approximating the Second Sum
We analyze the sum \(S_b = \frac{30}{41} + \frac{244}{1000} + \frac{2}{2000}\) by estimating the magnitude of each fraction:
\(\frac{30}{41} \approx \frac{30}{40} = 0.75 = \frac{3}{4}\)
\(\frac{244}{1000} = 0.244 \approx 0.25 = \frac{1}{4}\)
\(\frac{2}{2000} = 0.001 \approx 0\)
Summing these estimates gives \(\frac{3}{4} + \frac{1}{4} + 0 = 1\). Comparing this result to the given options \(\{\frac{3}{8}, \frac{3}{4}, 1, 2\}\), the best approximation is \(1\), which corresponds to option D.
- Conclusion
In conclusion, the best approximations are 4 for the first sum and 1 for the second sum. Thus, for part (a) the result is 4, and for part (b) the correct option is D.
Examples
Estimating sums is a vital skill in daily life, such as when shopping. If you are at a grocery store and have items costing \$8.90, \$11.10, \$12.10, and \$20.20, you can quickly estimate that your total will be close to \$40 rather than \$30 by observing that each item is slightly less than a certain dollar amount. This mental math helps you keep track of your budget without needing a calculator.