[Math Revolution GMAT math practice question]
What is the standard deviation of a, b and c?
1) a^2+b^2+c^2 = 77
2) a+b+c =15
What is the standard deviation of a, b and c?
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- Max@Math Revolution
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Very nice problem, congrats Max!Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
What is the standard deviation of a, b and c?
1) a^2+b^2+c^2 = 77
2) a+b+c =15
\[? = \sigma \left( {a,b,c} \right)\]
\[\left( 1 \right)\,\,{a^2} + {b^2} + {c^2} = 77\,\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {a;b;c} \right) = \left( {\sqrt {\frac{{77}}{3}} \,;\sqrt {\frac{{77}}{3}} \,;\sqrt {\frac{{77}}{3}} \,} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{0}}\,\, \hfill \\
\,\,{\text{Take}}\,\,\left( {a;b;c} \right) = \left( {\sqrt {77} \,;0\,;0\,} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\text{?}}\,\, \ne \,\,{\text{0}}\,\,\,\, \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,a + b + c = 15\,\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {a;b;c} \right) = \left( {5\,;5\,;5\,} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{0}}\,\, \hfill \\
\,\,{\text{Take}}\,\,\left( {a;b;c} \right) = \left( {15\,;0\,;0\,} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\text{?}}\,\, \ne \,\,{\text{0}}\,\,\,\, \hfill \\
\end{gathered} \right.\]
\[\left( {1 + 2} \right)\,\,\,\,\mu = \frac{{a + b + c}}{3} = 5\]
\[? = \sqrt {\frac{{{{\left( {a - \mu } \right)}^2} + {{\left( {b - \mu } \right)}^2} + {{\left( {c - \mu } \right)}^2}}}{3}} \,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\boxed{\,\,\,? = {{\left( {a - 5} \right)}^2} + {{\left( {b - 5} \right)}^2} + {{\left( {c - 5} \right)}^2}\,\,\,}\]
\[?\,\,\, = \,\,\,{\left( {a - 5} \right)^2} + {\left( {b - 5} \right)^2} + {\left( {c - 5} \right)^2}\,\,\, = \,\,\,\,\underbrace {{a^2} + {b^2} + {c^2}}_{77} - 10\underbrace {\left( {a + b + c} \right)}_{15} + 3 \cdot 25\,\,\,\,\,{\text{unique}}\]
The correct answer is therefore [spoiler]__(C)_____[/spoiler] .
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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Question: SD of a, b and c?Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
What is the standard deviation of a, b and c?
1) a^2+b^2+c^2 = 77
2) a+b+c =15
Let's take each statement one by one.
1) a^2 + b^2 + c^2 = 77
Case 1: Say a = b = c = √(77/3); the answer is 0. Note that if the number in a set are equal, there is no deviation among them, thus, SD = 0.
Case 2: Say a = √(77/2); b = c = 0; the answer is NOT 0 but some positive value.
No unique value. Insufficient.
2) a + b + c = 15
Case 1: Say a = b = c = 5; the answer is 0.
Case 2: Say a = 15; b = c = 0; the answer is NOT 0 but some positive value.
No unique value. Insufficient.
(1) and (2) together
Though computation of SD is not within the scope of the GMAT, we can still discuss here.
SD = √[{(a - M)^2 + (b - M)^2 + (c - M)^2} / 3]; here M = Arithmetic mean of numbers a, b and c and 3 is the count of the total numbers of the elements in the set (there are three elements a, b and c)
If we get the unique value of (a - M)^2 + (b - M)^2 + (c - M)^2, we get the answer.
Question rephrased: What's the value of (a - M)^2 + (b - M)^2 + (c - M)^2?
Since a + b + c = 15, we have Arithmetic mean = M = (a + b + c)/3 = 5
(a - M)^2 + (b - M)^2 +(c - M)^2
= (a - 5)^2 + (b - 5)^2 +(c - 5)^2
= (a^2 - 10a + 25) + (b^2 - 10b + 25) + (c^2 - 10c + 25)
= (a^2 + b^2 + c^2) - 10(a + b + c) + 75
= 77 - 10*15 + 75
= A unique value
Sufficient.
The correct answer: C
Hope this helps!
-Jay
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- Max@Math Revolution
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=>
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
The mean of a, b and c is m = ( a + b + c ) / 3. So, a + b + c = 3m.
The variance of a, b and c, that is, the square of the standard deviation of a, b and c, is given by
VAR = SD^2 = { ( a - m )^2 + ( b - m )^2 + ( c - m )^2 } / 3
= { a^2 - 2am + m^2 + b^2 - 2bm + b^2 + c^2 - 2cm + m^2 } / 3
= { a^2 + b^2 + c^2 - 2m(a+b+c) + 3m^2 } / 3
= { a^2 + b^2 + c^2 - 2m*3m + 3m^2 } / 3
= { a^2 + b^2 + c^2 - 3m^2 } / 3
= ( a^2 + b^2 + c^2 ) / 3 - m^2
= ( a^2 + b^2 + c^2 ) / 3 - { ( a + b + c ) / 3 }^2
This value can be calculated using the information given in conditions 1) and 2). Thus, both conditions together are sufficient.
Therefore, C is the answer.
Answer: C
When we have a sum of data values and a sum of the squares of data values, we can always calculate the standard deviation.
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
The mean of a, b and c is m = ( a + b + c ) / 3. So, a + b + c = 3m.
The variance of a, b and c, that is, the square of the standard deviation of a, b and c, is given by
VAR = SD^2 = { ( a - m )^2 + ( b - m )^2 + ( c - m )^2 } / 3
= { a^2 - 2am + m^2 + b^2 - 2bm + b^2 + c^2 - 2cm + m^2 } / 3
= { a^2 + b^2 + c^2 - 2m(a+b+c) + 3m^2 } / 3
= { a^2 + b^2 + c^2 - 2m*3m + 3m^2 } / 3
= { a^2 + b^2 + c^2 - 3m^2 } / 3
= ( a^2 + b^2 + c^2 ) / 3 - m^2
= ( a^2 + b^2 + c^2 ) / 3 - { ( a + b + c ) / 3 }^2
This value can be calculated using the information given in conditions 1) and 2). Thus, both conditions together are sufficient.
Therefore, C is the answer.
Answer: C
When we have a sum of data values and a sum of the squares of data values, we can always calculate the standard deviation.
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