What is the average of numbers a and b?

Question

(1) The median of a+2, a-3 and a+5 is 14.
(2) The only mode of a, b and 14 is 14.

Bunuel · Answer

What is the average of numbers a and b?

(1) The median of a+2, a-3 and a+5 is 14. The median of a set with odd # of terms is just a middle term (when ordered in ascending/descending order). So, the median of a-3, a+2, and a+5 is a+2, thus a+2=14 --> a=12. We don't know the value of b. Not sufficient.

(2) The only mode of a, b and 14 is 14. The mode is the number that occurs the most frequently in a data set. Since a, b, and 14 has only one mode then a or b or both must also be 14. Not sufficient.

(1)+(2) From (1) a=12, then from (2) b=14 --> the average = 13. Sufficient.

Answer: C.

OptimusPrepJanielle · Answer

Statement 1: The median of a+2, a-3 and a+5 is 14
This does not tell us anything about the value of b.
Insufficient

Statement 2: The only mode of a, b and 14 is 14
 Mode is the number that occurs the most number of times in a series.

Case 1: The series is 14, 14, 14. Here a = b = 14
Average = 14
Case 2: The series is 10, 14, 14. Here a = 10, b = 14
Average = 12
Hence Insufficient

Combining both statements:
 Median = middle number when series is arranged in ascending/desscending order for odd number of terms
Median = average of middle two terms when series is arranged in ascending/descending order for even number of terms

From statement 1, the median = the middle term of (a-3, a + 2 and a+5 )
Median = a + 2 = 14
Hence a = 12

From statement 2, mode of 12, b and 14 = 14
Since mode is the number that is repeated most number of times in a series, b has to be 14

Average of a and b = (12 + 14)/2 = 13
Sufficient

Correct Option: C

MathRevolution · Answer

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

What is the average of numbers a and b?

(1) The median of a+2, a-3 and a+5 is 14.
(2) The only mode of a, b and 14 is 14.
 
In the original condition, there are 2 variables, which should match with the number of equations. So you need 2 equations. For 1) 1 equation, for 2) 1 equation, which is likely to make C the answer.
When 1) & 2), from a+2=14 -> a=12 and b=14. Since a+b=12+14=26, (a+b)/2=13 is derived, which is unique and the answer is C.

 For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.

What is the average of numbers a and b?

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GMAT Data Sufficiency (DS) Questions

What is the average of numbers a and b?

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