It is a data sufficiency question. We have to tell whether the question can be solved using one of the two options given below, both of them, either of them, or if it can't be solved even if we use both the options together:
Q. Last year the average (arithmetic mean) salary of the 10 employees of Company X was $42,800. What is the average salary of the same 10 employees this year?
1) For 8 of the 10 employees, this year's salary is 15 percent greater than last year's salary.
2) For 2 of the 10 employees, this year's salary is the same as last year's salary.
I know the answer is that it cannot be solved even if we combine both of the options. My doubt is regarding the explanation given in my book for option 1, which goes thus: Since all 10 employees didn't receive the same 15% increase, it can't be assumed that the mean this year is 15% higher than last year (I understood this part). It remains unknown whether these 8 salaries were the top 8 salaries, the bottom 8, or somewhere in-between. Without this type of information from last year, the mean for this year can't be determined.
I have a doubt regarding the part in bold. Even if I knew whether the 8 salaries were top 8 or bottom 8, how would it help me to determine the mean for this year? We don't know anything about the rest of the 2 salaries, whether they increased or decreased, and even if they remained the same, how would that help me?
For example, if the mean of 10 nos. (1,2,3,4,5,6,7,8,9,10) is 5.5, and if I increase the bottom 8 nos. by 50%, the new mean becomes 6.3. The new mean is increased by 14.5% from the old mean. Had I not known the individual nos., how would I get to know what the new mean would be?
You don't actually have to know the individual numbers, but you do have to know what percentage of the total payroll increased by $15\%.$ If we know that the salaries of those who get an increase represent $90\%$ of the total payroll, then the new payroll will be $$.9\cdot1.15+.1\cdot1.0=113.5\%$$ of the old payroll, so that the average salary will increase by $13.5\%$, because the number of employees doesn't change.