- For sum \(S_a = \frac{8}{9} + \frac{11}{12} + \frac{12}{13} + \frac{20}{21}\), each term is close to \(1\), so the total is approximately \(4\).
- For sum \(S_b = \frac{30}{41} + \frac{244}{1000} + \frac{2}{2000}\), we approximate \(\frac{30}{41} \approx 0.732\), \(\frac{244}{1000} = 0.244\), and \(\frac{2}{2000} = 0.001\), which sums to \(\approx 0.977\), closest to \(1\).
- The correct option for the first sum's approximation is \(\boxed{4}\), which matches option C.
Explanation
- Analysis of Sums and Approximations
We analyze the two given expressions to find the best approximations from the provided sets.
The first sum is \(S_a = \frac{8}{9} + \frac{11}{12} + \frac{12}{13} + \frac{20}{21}\).
Each term in the sum is slightly less than \(1\):
- \(\frac{8}{9} \approx 0.889\)
- \(\frac{11}{12} \approx 0.917\)
- \(\frac{12}{13} \approx 0.923\)
- \(\frac{20}{21} \approx 0.952\)
Summing these values, we get \(S_a \approx 3.681\). Among the options \((1, 2, 3, 4)\), the closest integer is \(4\).
The second sum is \(S_b = \frac{30}{41} + \frac{244}{1000} + \frac{2}{2000}\).
- \(\frac{30}{41} \approx 0.732\)
- \(\frac{244}{1000} = 0.244\)
- \(\frac{2}{2000} = 0.001\)
Summing these values, we get \(S_b \approx 0.977\). Among the options \((\frac{3}{8}, \frac{3}{4}, 1, 2)\), the closest value is \(1\).
- Determining the Correct Option
Given the results:
- The best approximation for \(S_a\) is \(4\).
- The best approximation for \(S_b\) is \(1\).
The question asks us to choose the correct answer from the list A, B, C, D:
Looking at the structure, this appears to be asking for the approximation of the first sum \(S_a \approx 4\), which corresponds to option C.
Examples
In real life, estimating sums of fractions is crucial for quick mental calculations, such as checking a grocery bill. For instance, if items cost \(\frac{8}{9}, \frac{11}{12}, \frac{12}{13},\) and \(\frac{20}{21}\) dollars, a quick estimation of nearly \(4\) dollars helps confirm if your budget of \(\$5\) is sufficient without calculating the exact decimals. This skill saves time and helps verify the reasonableness of calculated totals.