Problem Solving

Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machine Y. At these rates, if the two machines together produce (5/4)w widgets in 3 days, how many days would it take machine X alone to produce 2w widgets?
A. 4
B. 6
C. 8
D. 10
E. 12

I got the explanation of this math from previous posts. My question is, what level of difficulty this math may have? It seems very hard to me.

NaimaB wrote:Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machine Y. At these rates, if the two machines together produce (5/4)w widgets in 3 days, how many days would it take machine X alone to produce 2w widgets?
A. 4
B. 6
C. 8
D. 10
E. 12

I got the explanation of this math from previous posts. My question is, what level of difficulty this math may have? It seems very hard to me.

This is medium level difficulty problem as it doesn't involve too much creative thinking and is standard type of question.

The question should be put in 600-700 difficulty level.

NaimaB wrote:Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machine Y. At these rates, if the two machines together produce (5/4)w widgets in 3 days, how many days would it take machine X alone to produce 2w widgets?
A. 4
B. 6
C. 8
D. 10
E. 12

I got the explanation of this math from previous posts. My question is, what level of difficulty this math may have? It seems very hard to me.

Explanation in most basic and conventional manner:

Let, Machine Y takes D days to complete the work of w widgets
therefore, Machine X will take (D+2) days to complete the work of w widgets

1 Day work of Machine X = w/(D+2)
1 Day work of Machine Y = w/(D)

Total work of Machine X and Y for 1 day = {w/(D+2)}+{w/(D)} = 2w(D+1)/[D(D+2)]
Total work of Machine X and Y for 3 day = 6w(D+1)/[D(D+2)] = 5w/4
i.e. 24D+24 = 5[D(D+2)]
Solving them
D=4
i.e.
X takes 4+2=6 days to produce w widgets
i.e. Machine X will take 12 days to produce 2w widgets

Answer: Option E

NaimaB wrote:Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machine Y. At these rates, if the two machines together produce (5/4)w widgets in 3 days, how many days would it take machine X alone to produce 2w widgets?
A. 4
B. 6
C. 8
D. 10
E. 12

Let w = the LCM of all of the answer choices = 60.
In 3 days, the number of widgets that must be produced = (5/4)w = (5/4)60 = 75 widgets.
To produce 75 widgets in 3 days, the required rate = 75/3 = 25 widgets per day.

We can PLUG IN THE ANSWERS, which represent the time for X to produce 2w=120 widgets.
When the correct answer choice is plugged in, the combined rate for X and Y will be 25 widgets per day.

Answer choice D: 10 days for X to produce 120 widgets
Here, the time for X to produce 60 widgets = 5 days.
Since X takes 2 days longer than Y, the time for Y to produce 60 widgets = 3 days.
Rate for X = w/t = 60/5 = 12 widgets per day.
Rate for Y = w/t = 60/3 = 20 widgets per day.
Combined rate for X+Y = 12+20 = 32 widgets per day.

Since the required rate = 25 widgets per day, X and Y are working TOO FAST.
Since X and Y need to work more SLOWLY, X must take LONGER to produce 2w widgets.

The correct answer is E.

Answer choice E: 12 days for X to produce 120 widgets
Here, the time for X to produce 60 widgets = 6 days.
Since X takes 2 days longer than Y, the time for Y to produce 60 widgets = 4 days.
Rate for X = w/t = 60/6 = 10 widgets per day.
Rate for Y = w/t = 60/4 = 15 widgets per day.
Combined rate for X+Y = 10+15 = 25 widgets per day.
Success!

I would rate this as a 650-750 level problem.
I suspect that the average test-taker would find it quite challenging to solve via traditional methods.

Rate = w/time:
w/x = w/(y + 2) or w/y = w/(x - 2)
where x and y are measures of time

Together: w/x + w/(x - 2) = (5w/4)/3
So, 12(x -2) + 12x = 5x(x-2)
Rearrange to make a quadratic:
5x^2 - 34x + 24 = 0
(5x-4)(x-6) = 0
x = 4 or 6
As X takes 2 days longer than Y, then x = 6, y = 4 to make w widgets
Therefore, 2w widgets take X 12 days to make.

Mathsbuddy wrote: So, 12(x -2) + 12x = 5x(x-2)
Rearrange to make a quadratic:
5x^2 - 34x + 24 = 0
(5x-4)(x-6) = 0
x = 4 or 6
As X takes 2 days longer than Y, then x = 6, y = 4 to make w widgets
Therefore, 2w widgets take X 12 days to make.

The highlighted step should be
x = 4/5 or 6

GMATinsight wrote:

Mathsbuddy wrote: So, 12(x -2) + 12x = 5x(x-2)
Rearrange to make a quadratic:
5x^2 - 34x + 24 = 0
(5x-4)(x-6) = 0
x = 4 or 6
As X takes 2 days longer than Y, then x = 6, y = 4 to make w widgets
Therefore, 2w widgets take X 12 days to make.

The highlighted step should be
x = 4/5 or 6

Well spotted... oops!
Thanks for that.

Brent@GMATPrepNow wrote:

Mathsbuddy wrote:

GMATinsight wrote:

Mathsbuddy wrote: So, 12(x -2) + 12x = 5x(x-2)
Rearrange to make a quadratic:
5x^2 - 34x + 24 = 0
(5x-4)(x-6) = 0
x = 4 or 6
As X takes 2 days longer than Y, then x = 6, y = 4 to make w widgets
Therefore, 2w widgets take X 12 days to make.

The highlighted step should be
x = 4/5 or 6

Well spotted... oops!
Thanks for that.

Since there are TWO solutions (x = 4/5 or x = 6) to the equation, it might be worthwhile explaining WHY we have concluded that x = 6 (and then double that to get answer choice E) and not x = 4/5.

Notice that both are solutions to the equation 5x² - 34x + 24 = 0,
HOWEVER, you already say that x = the number of days it takes Machine X to produce w widgets.
And we're told that it takes Machine Y 2 days LESS to produce w widgets.
So, if x = 4/5, it means it takes Machine X 4/5 days to produce w widgets.
That means it takes Machine Y -6/5 days to produce w widgets (since it takes Machine Y 2 days LESS to produce w widgets)
Since this makes no sense, we can RULE OUT x = 4/5 as a possible answer, which means x MUST EQUAL 6....

Cheers,
Brent

Thanks Brent. As usual you are quite correct!

Here's another (similar) approach:

Rate = w/time:
w/x = w/(y + 2) or w/y = w/(x - 2)
where x and y are measures of time

Together: w/y + w/(y + 2) = (5w/4)/3
So, 12(y+2) + 12y = 5y(y+2)
Rearrange to make a quadratic:
5y^2 - 14y - 24 = 0
(5y+6)(y-4) = 0
y = -6/5 or 4
As y > 0, then y = 4,
So x = 4 + 2 = 6 to make w widgets
Therefore, 2w widgets take X 12 days to make.

Here's a another way of working backwards from the answers.
Answers changed to reflect manufacturing just w widgets:
A. x = 2, y = 0: 1/2 + 1/0 = UNDEFINED
B. x = 3, y = 1: 1/3 + 1/1 = 4/3 in 1 day, so 4 in 3 days
C. x = 4, y = 2: 1/4 + 1/2 = 3/4 in 1 day, so 9/4 in 3 days
D. x = 5, y = 3: 1/5 + 1/3 = 8/15 in 1 day, so 8/5 in 3 days
E. x = 6 ,y = 4: 1/6 + 1/4 = 10/24 in 1 day, so 5/4 in 3 days
Only ANSWER E complies.

Mathsbuddy wrote:Here's a another way of working backwards from the answers.
Answers changed to reflect manufacturing just w widgets:
A. x = 2, y = 0: 1/2 + 1/0 = UNDEFINED
B. x = 3, y = 1: 1/3 + 1/1 = 4/3 in 1 day, so 4 in 3 days
C. x = 4, y = 2: 1/4 + 1/2 = 3/4 in 1 day, so 9/4 in 3 days
D. x = 5, y = 3: 1/5 + 1/3 = 8/15 in 1 day, so 8/5 in 3 days
E. x = 6 ,y = 4: 1/6 + 1/4 = 10/24 in 1 day, so 5/4 in 3 days
Only ANSWER E complies.

Once the above is understood, we can take a shortcut:
Only C and E produce a denominator which is a multiple of 4.
Now work out 9/4 and 5/4 respectively. Answer E.

This could be a fast way of solving all such rates problems.