Problem Solving

evansbd wrote:

I've followed the CRACKING THE GMAT guide to questions like this by plugging in an original number then solving the answer choices for the solution given by your particular number.

For example.

John and Mary were each paid X dollars in advance.

Let x = 80$

John worked 10 hrs. Mary worked 2 hrs less (8 hrs)

John 80$ for 10hrs ---> 8$/hr

Mary 80$ for 8hrs ---> 10$/hr

Mary gives John Y dollars of her payment so they recieve the same hourly wage.

Mary ---> lower her wage to 8$/hr ---> 8$*8hrs ---> 64 dollars.

She would need to give John 80 - 64 = 16 dollars.

Find x (80) in terms of y (16). 80 = 16*5 . So the answer is 5y.

I've highlighted the mistake in your work above. If Mary gives John money, Mary's hourly wage goes down, but John's hourly wage also goes up. If Mary gives John $16, then John made $96 in 10 hours, or $9.60 per hour. Using your numbers, for their hourly wage to be equal:

(80+y)/10 = (80 - y)/8
320 + 4y = 400 - 5y
9y = 80
y = 80/9

so we find that x = 9y.

I'm not convinced that choosing numbers here makes this any easier than just doing the question algebraically, however.

evansbd wrote:Ian that helps alot...could you share your insight on how question 3 was solved given the solution above?

Well, I can explain how I'd normally approach these sorts of problems. I tend to do these types of problems algebraically, because that's what I find easiest, but I also try to break the problems down into the simplest possible steps, as you'll see below. For some problems of these types, choosing numbers can be successful, but for Q2 below, it's not all that easy, as you saw above.


Q2. John and Mary were each paid x dollars in advance to do a certain job together. John worked on the job for 10 hrs and Mary worked for 2 hours less than him. If Mary gave John y dollars of her payment so that they receive the same hourly wage, what was the dollar amount, in terms of y, that John was paid in advance?

John received x dollars
Mary received x dollars

Mary gives John y dollars:

John has x+y dollars
Mary has x-y dollars

John worked for 10 hours, so he earned (x+y)/10 dollars per hour
Mary worked for 8 hours, so she earned (x-y)/8 dollars per hour.

They earned the same amount per hour:

(x+y)/10 = (x-y)/8

4x+4y = 5x - 5y
x = 9y


Q3. Seed mixture X is 40% ryegrass and 60% bluegrass; seed mixture Y is 25% ryegrass and 75% fescue. If a mixture of X and Y contains 30% ryegrass, what percent of the weight of the mixture is X?

a. 10% b.33 1/3% c.40% d.50% e.66 2/3%

This is a weighted average question, and if you understand weighted averages, you can see that to get a 30% rye mixture, you'll need twice as much of Y as you need of X, because 30 is twice as far from 40 as it is from 25- B must be the answer. That's by far the fastest way to answer the question. If you're not so confident with weighted averages, it can be done algebraically. Aleksander's solution above was good, except that the last line contained an error, which beeparoo corrected. I'll write that solution in greater detail:

Let x = the weight of X
Let y = the weight of Y
Then x+y is the weight of the mixture. The question asks us to find x/(x+y) (as a percentage).

X is 40% ryegrass: 0.4x is the amount of ryegrass in X
Y is 25% ryegrass: 0.25y is the amount of ryegrass in Y

The mixture contains 0.4x + 0.25y ryegrass.
The mixture weighs x +y in total.
The mixture is 30% ryegrass.
Thus:

(0.4x + 0.25y)/(x+y) = 30/100
0.4x + 0.25y = 0.3x + 0.3y
x = 0.5y
x/y = 1/2

So the ratio of x to y is 1 to 2, and the mixture is 1/3 X and 2/3 Y. That might look long, but obviously you don't need to write all of that down- I've done so for clarity.