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If a is the sum of the reciprocals of the consecutive integers from 11 to 20, and b is the estimated sum of the same numbers, so each even integer is rounded up to the nearest 10 and each odd integer is rounded down to the nearest 10, then a - b is between.
A. -1/4 and 1/4
B. 1/4 and 3/4
C. 3/4 and 5/4
D. 5/4 and 3/2
E. 7/4 and 11/2
A. -1/4 and 1/4
B. 1/4 and 3/4
C. 3/4 and 5/4
D. 5/4 and 3/2
E. 7/4 and 11/2
02 Nov 2021, 00:17
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02 Nov 2021, 01:05
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Range for a: If all the numbers were the smallest number \(a = \frac{10}{20}\)
If all the numbers were the largest number \(a = \frac{10}{11}\)
So the range will be: \( \frac{10}{20} < a < \frac{10}{11} \)
Range for b: I'm going to assume that what they mean is that 11 becomes 10, 12 becomes 20, 13 becomes 10, 14 becomes 20 etc, and then we take their reciprocals.
As there are 5 odd and 5 even numbers between 11 and 20 inclusive, we will get \(\frac{5}{10}\) for the odd numbers and \(\frac{5}{20}\) for the even numbers. When added together that gives us \(\frac{15}{20}\)
a-b
Minimum value of the range: \(\frac{10}{20} - \frac{15}{20} = - \frac{5}{20} \) or simplified \(-\frac{1}{4}\)
Maximum value of the range: \( \frac{10}{11} - \frac{15}{20}\)
\(= \frac{200 - 165 }{ 220}\)
\(= \frac{35}{220}\)
\(= \frac{7}{44}\)
which is somewhere between \(\frac{1}{7}\) and \(\frac{1}{6}\)
The only answer choice which covers the range of \(-\frac{1}{4} < a-b < \frac{1}{7}\) is answer choice A.
Answer A
If all the numbers were the largest number \(a = \frac{10}{11}\)
So the range will be: \( \frac{10}{20} < a < \frac{10}{11} \)
Range for b: I'm going to assume that what they mean is that 11 becomes 10, 12 becomes 20, 13 becomes 10, 14 becomes 20 etc, and then we take their reciprocals.
As there are 5 odd and 5 even numbers between 11 and 20 inclusive, we will get \(\frac{5}{10}\) for the odd numbers and \(\frac{5}{20}\) for the even numbers. When added together that gives us \(\frac{15}{20}\)
a-b
Minimum value of the range: \(\frac{10}{20} - \frac{15}{20} = - \frac{5}{20} \) or simplified \(-\frac{1}{4}\)
Maximum value of the range: \( \frac{10}{11} - \frac{15}{20}\)
\(= \frac{200 - 165 }{ 220}\)
\(= \frac{35}{220}\)
\(= \frac{7}{44}\)
which is somewhere between \(\frac{1}{7}\) and \(\frac{1}{6}\)
The only answer choice which covers the range of \(-\frac{1}{4} < a-b < \frac{1}{7}\) is answer choice A.
Answer A
