mmcooley33 wrote:
Alan’s regular hourly wage is 1.5 times Barney’s regular hourly wage, but Barney gets paid at twice his regular wage for any hours he works on Saturday. Both men work an integer number of hours on any given day. If Alan and Barney each worked for the same total non-zero number of hours last week, and earned the same total in wages, which of the following must be true?
I. Alan worked fewer hours Monday through Friday than did Barney.
II. Barney worked at least one hour on Saturday.
III. Barney made more money on Saturday than did Alan.
A.I only
B. II only
C. I and II only
D. I and III only
E. II and III only
Let Barney's hourly wage Monday through Friday = $2 per hour, implying that Barney's wage on Saturday = 2*2 = $4 per hour and that Alan's hourly wage = (1.5)(2) = $3 per hour.
For Barney to work the same number of hours as Alan and earn the same amount of money, Barney's AVERAGE hourly wage must be equal to Alan's hourly wage ($3 per hour).
Implication:
Since $3 is HALFWAY between Barney's two wages -- $2 and $4 -- Barney must work half of his total hours on WEEKDAYS (earning $2 per hour) and the other half on SATURDAY (earning $4 per hour), with the result that his AVERAGE hourly wage = $3 per hour.
Let the time for each worker = 2 hours, implying that Barney works half the time (1 hour) Monday through Friday and the other half (1 hour) on Saturday.
Alan can work his 2 hours any day of the week.
Total income for each worker = (average hourly wage)(number of hours) = 3*2 = $6.
I. Alan worked fewer hours Monday through Friday than did Barney. Whereas Alan can work all of his 2 hours Monday through Friday, Barney works only 1 weekday hour.
Thus, Statement I does not have to be true.
Eliminate A, C and D.
III: Barney made more money on Saturday than did Alan. Whereas Barney's 1 hour of work on Saturday earns him only $4, Alan's TOTAL income of $6 can be earned on Saturday.
Thus, Statement III does not have to be true.
Eliminate E.
Mathematical proof for Barney:Let w = Barney's weekday hours and s = Barney's Saturday hours.
Since Barney has a weekday rate of $2 per hour and a Saturday rate of $4 per hour, we get:
Total wages = 2w+4s
Since Barney must earn the same amount as Alan -- $6 -- we get:
2w+4s = 6
w+2s = 3Since Barney must work the same number of hours as Alan, we get:
w+s = 2Subtracting the blue equation from the red equation, we get:
(w+2s) - (w+s) = 3-2
s=1, implying that w=1.
Implication:
To earn the same amount as Alan, Barney must work ONE WEEKDAY HOUR AND ONE SATURDAY HOUR.