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.

I only

II only

I and II only

I and III only

II and III only

I would just like to see the equation written out, so I can visualize the problem. I am having trouble conceptualizing the problem. I understand the a=1.5b but I would like to see how the second part is written when they have equal hours and equal pay. Thanks,

Although I don't think that algebraic way is the best for this problem, here you go:

Let Barney's regular hourly wage be

x, then his Saturday wage will be

2x and Alan's hourly wage will be

1.5x;

Let the # of hours Barney worked Monday through Friday be

m and on Saturday be

n and the # of hours Alan worked Monday through Friday be

p and on Saturday be

q;

Given:

xm+2xn=1.5x(p+q) and

m+n=p+q.

xm+2xn=1.5x(p+q) -->

m+2n=1.5(m+n) -->

m=n --> Barney worked the equal # of hours Monday-Friday and on Saturday.

The above directly tells us that II must be true (as Barney worked total non-zero # of hours and he worked an integer # of hours on any given day then he must have been worked at least one hour on Saturday.)

As for I: Alan may have worked ALL his hours Monday through Friday so in this case this statement is not true (p=total>m). Alan also may have worked all his hours on Saturday. Or algebraically: there are any distribution possible between p and q, p=0 and q=total or p=total and q=0 or any other;

The above means that III is also not always true: if Alan worked all his hours on Saturday then he made all his money on Saturday thus he made more money on Saturday than Barney did.

Answer: B (II only).

But the above can also be done with much less algebra:As Alan and Barney worked the same # of hours and earned the same amount of money, then their hourly average wages must have been the same:

(average wage)=(total amount earned)/(# of hours worked). Now, Alan has constant hourly wage which is

1.5*x and Barney's average (

\frac{xm+2xn}{m+n}) to be equal to this he must have been worked the equal # of hours Monday-Friday and on Saturday, so

m=n.

Hope it's clear.