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26 Feb 2019, 11:29
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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:
75%
(hard)
Question Stats:
60% (02:55) correct
40%
(02:56)
wrong
based on 168
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\({{71} \over {11}} = A + {1 \over {B + {{\left( {C + 1} \right)}^{ - 1}}}}\)
In the equality above, \(A,B,C\) are positive integers. What is the value of \(2A+3B-4C\) ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
In the equality above, \(A,B,C\) are positive integers. What is the value of \(2A+3B-4C\) ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
27 Feb 2019, 07:43
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fskilnik
\(A,B,C\,\, \ge 1\,\,\,{\rm{ints}}\,\,\,\left( * \right)\)
\({{71} \over {11}} = A + {1 \over {B + {{\left( {C + 1} \right)}^{ - 1}}}}\)
\(? = 2A + 3B - 4C\)
\(\left. \matrix{\\
B + {1 \over {C + 1}}\,\,\mathop > \limits^{\left( * \right)} \,\,1\,\,\,\,\, \Rightarrow \,\,\,\,\,0 < \,{1 \over {B + {{\left( {C + 1} \right)}^{ - 1}}}} < 1\,\,\,\, \hfill \cr \\
{{71} \over {11}} = {{66 + 5} \over {11}} = 6 + {5 \over {11}} \hfill \cr} \right\}\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,A = 6\)
\(\left. \matrix{\\
{5 \over {11}} = {1 \over {B + {{\left( {C + 1} \right)}^{ - 1}}}}\,\,\,\,\, \Rightarrow \,\,\,\,\,B + {1 \over {C + 1}}\,\, = {{11} \over 5}\,\,\, \hfill \cr \\
C + 1\,\,\mathop > \limits^{\left( * \right)} \,\,1\,\,\,\, \Rightarrow \,\,\,\,\,0 < {1 \over {C + 1}} < 1 \hfill \cr \\
{{11} \over 5} = {{10 + 1} \over 5} = 2 + {1 \over 5} \hfill \cr} \right\}\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,B = 2\)
\({1 \over 5} = {1 \over {C + 1}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,C = 4\)
\(? = 2\left( 6 \right) + 3\left( 2 \right) - 4\left( 4 \right) = 2\)
The correct answer is (C).
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
General Discussion
07 Aug 2025, 23:34
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The Solution
1. Isolate A: Convert `71/11` into a mixed number.
• `71 / 11` is `6` with a remainder of `5`. So, `71/11 = 6 + 5/11`.
• By comparing this to `A + 1/(...)`, we can see that A = 6.
2. Isolate B: Now, we work with the rest of the equation.
• `5/11 = 1 / (B + (C+1)^-1)`.
• Flip both sides: `11/5 = B + 1/(C+1)`.
• Convert `11/5` to a mixed number: `11/5 = 2 + 1/5`.
• By comparing, we see that B = 2.
3. Isolate C: Finally, compare the remaining fractions.
• `1/5 = 1/(C+1)`.
• This means `5 = C+1`, so C = 4.
4. Calculate the Final Value:
• Plug the values into `2A + 3B - 4C`.
• `2(6) + 3(2) - 4(4)`
• `12 + 6 - 16`
• `18 - 16 = 2`.
The correct answer is (C) 2.
1. Isolate A: Convert `71/11` into a mixed number.
• `71 / 11` is `6` with a remainder of `5`. So, `71/11 = 6 + 5/11`.
• By comparing this to `A + 1/(...)`, we can see that A = 6.
2. Isolate B: Now, we work with the rest of the equation.
• `5/11 = 1 / (B + (C+1)^-1)`.
• Flip both sides: `11/5 = B + 1/(C+1)`.
• Convert `11/5` to a mixed number: `11/5 = 2 + 1/5`.
• By comparing, we see that B = 2.
3. Isolate C: Finally, compare the remaining fractions.
• `1/5 = 1/(C+1)`.
• This means `5 = C+1`, so C = 4.
4. Calculate the Final Value:
• Plug the values into `2A + 3B - 4C`.
• `2(6) + 3(2) - 4(4)`
• `12 + 6 - 16`
• `18 - 16 = 2`.
The correct answer is (C) 2.
