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12 Jul 2019, 00:52
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
64% (02:49) correct
36%
(02:35)
wrong
based on 230
sessions
History
Date
Time
Result
Not Attempted Yet
Attachment:
712.png [ 7.28 KiB | Viewed 4162 times ]
In the diagram, each of \(a, b, c, ... ,g\) represents one of the numbers \(1, 2, 3,...\) and \(7\), with no repetition. These values correspond to the number of elements in each part of the sets. If sets \(A, B\) and \(C\) have the same number of elements and the sum of the numbers of elements of \(A, B\) and \(C\) is as small as possible, find the value of \(b+d+f\).
\(A. 5\)
\(B. 7\)
\(C. 9\)
\(D. 11\)
\(E. 13\)
14 Jul 2019, 18:09
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=>
The number of elements in set \(A\) is \(a + b + c + d\). The number in set \(B\) is \(b + c + e + f,\) and the number of elements in set \(C\) is \(c + d + f + g.\) The sum of the sizes of these three sets is then \(a + 2b + 3c + 2d + e + 2f + g.\)
To minimise this total, \(c\) must have the smallest value as its coefficient is the largest. Thus, \(c = 1.\) The next smallest values must be given to \(b, d\), and \(f\) as they have coefficients of \(2\). Thus, \(b,d\) and \(f\) are chosen from \(2, 3\) and \(4.\)
It follows that \(b + d + f = 2 + 3 + 4 = 9.\)
Therefore, the answer is C.
Answer: C
The number of elements in set \(A\) is \(a + b + c + d\). The number in set \(B\) is \(b + c + e + f,\) and the number of elements in set \(C\) is \(c + d + f + g.\) The sum of the sizes of these three sets is then \(a + 2b + 3c + 2d + e + 2f + g.\)
To minimise this total, \(c\) must have the smallest value as its coefficient is the largest. Thus, \(c = 1.\) The next smallest values must be given to \(b, d\), and \(f\) as they have coefficients of \(2\). Thus, \(b,d\) and \(f\) are chosen from \(2, 3\) and \(4.\)
It follows that \(b + d + f = 2 + 3 + 4 = 9.\)
Therefore, the answer is C.
Answer: C
General Discussion
14 Mar 2026, 16:19
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In the diagram, each of \(a, b, c, ... ,g\) represents one of the numbers \(1, 2, 3,...\) and \(7\), with no repetition. These values correspond to the number of elements in each part of the sets. If sets \(A, B\) and \(C\) have the same number of elements and the sum of the numbers of elements of \(A, B\) and \(C\) is as small as possible, find the value of \(b+d+f\).
c is included in all three sets.
b, d, and f, are each included in two of the sets.
a, e, and g are each included in only one set.
So, the sum of the elements in the sets will be the following:
3c +
2b + 2d + 2f +
1a + 1g + 1e
Thus, c has a triple impact on the sum, and b, d, and f each have a double impact on the sum.
So, to minimize the sum, make c the smallest possible number of the 7 and b, d, and f the three next smallest.
1 is the smallest.
2, 3, and 4 are the next three smallest numbers.
So, b + d + f = 2 + 3 + 4 = 9
\(A. 5\)
\(B. 7\)
\(C. 9\)
\(D. 11\)
\(E. 13\)
Correct answer: C
c is included in all three sets.
b, d, and f, are each included in two of the sets.
a, e, and g are each included in only one set.
So, the sum of the elements in the sets will be the following:
3c +
2b + 2d + 2f +
1a + 1g + 1e
Thus, c has a triple impact on the sum, and b, d, and f each have a double impact on the sum.
So, to minimize the sum, make c the smallest possible number of the 7 and b, d, and f the three next smallest.
1 is the smallest.
2, 3, and 4 are the next three smallest numbers.
So, b + d + f = 2 + 3 + 4 = 9
\(A. 5\)
\(B. 7\)
\(C. 9\)
\(D. 11\)
\(E. 13\)
Correct answer: C
