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30 Dec 2023, 23:02
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D
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Difficulty:
55%
(hard)
Question Stats:
69% (02:29) correct
31%
(02:14)
wrong
based on 26
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Lisa wrote down 10 consecutive integers on a blackboard. However, Bart, being mischievous, erased one of them. If the sum of the remaining nine numbers is 146, what is the value of the number that Bart erased?
A. 7
B. 9
C. 12
D. 19
E. 21
A. 7
B. 9
C. 12
D. 19
E. 21
30 Dec 2023, 23:02
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Official Solution:
Lisa wrote down 10 consecutive integers on a blackboard. However, Bart, being mischievous, erased one of them. If the sum of the remaining nine numbers is 146, what is the value of the number that Bart erased?
A. 7
B. 9
C. 12
D. 19
E. 21
Assume the smallest of the 10 consecutive integers is \(x\). Then their sum is:
\(x + (x + 1) + ... + (x + 9) = 10x + (1 + 2 + ... 9) = 10x + \frac{1 + 9}{2}*9 = 10x + 45\).
Now, let's assume Bart erased the number \(k\). Then, we'd have:
\(10x + 45 = 146 + k\);
\(10x = 101 + k\).
Since \(x\) is an integer, the left-hand side of the equation is a multiple of 10. Therefore, the right-hand side must also be a multiple of 10, which means the units digit of \(k\) must be 9. This narrows our options down to B (9) and D (19). However, \(k\) cannot be 9 because in that case \(x\) would be 11, and the erased number cannot be smaller than the smallest number in the set. Therefore, Bart must have erased the number 19.
Answer: D
Lisa wrote down 10 consecutive integers on a blackboard. However, Bart, being mischievous, erased one of them. If the sum of the remaining nine numbers is 146, what is the value of the number that Bart erased?
A. 7
B. 9
C. 12
D. 19
E. 21
Assume the smallest of the 10 consecutive integers is \(x\). Then their sum is:
\(x + (x + 1) + ... + (x + 9) = 10x + (1 + 2 + ... 9) = 10x + \frac{1 + 9}{2}*9 = 10x + 45\).
Now, let's assume Bart erased the number \(k\). Then, we'd have:
\(10x + 45 = 146 + k\);
\(10x = 101 + k\).
Since \(x\) is an integer, the left-hand side of the equation is a multiple of 10. Therefore, the right-hand side must also be a multiple of 10, which means the units digit of \(k\) must be 9. This narrows our options down to B (9) and D (19). However, \(k\) cannot be 9 because in that case \(x\) would be 11, and the erased number cannot be smaller than the smallest number in the set. Therefore, Bart must have erased the number 19.
Answer: D
21 May 2024, 19:10
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Hello, I could not understand the part where 9 is eliminated; pls explain, thanks
