A manufacturer wants to produce x balls and y boxes. Resource constrai

Question

7x + 6y \leq 38,000

4x + 5y \leq 28,000

A manufacturer wants to produce x balls and y boxes. Resource constraints require that x and y satisfy the inequalities shown. What is the maximum number of balls and boxes combined that can be produced given the resource constraints?

A.  5,000
B.  6,000
C.  7,000
D.  8,000
E. 10,000

PS30421.02

chetan2u · Accepted Answer

I believe the equations must be talking of 28,000 and 38000 and not 3800 and 2800 because we have a comma after sets of 3 digit in GMAT, and also answers are in 1000s.

7x + 6y \leq 38,000

4x + 5y \leq 28,000

Add both the inequalities..
 7x+6y+4x=5y\leq{38000+28000}

11x+11y\leq{66000}....x+y\leq{6000}

B

siglon · Answer

HOW TO ADD AND SUBTRACT INEQUALITIES 
First and foremost, these are NOT equations, but inequalities. You cannot simply add and subtract inequalities in the same way as you can with equations. But let us see how we can...

This particular question asks for the maximum value of x+y.

1) We can ADD inequalities if the the signs face the same way:
  7x+6y\leq{38000}   and   4x+5y\leq{28000}  
...can be written as   (7x+6y)+(4x+5y)\leq{38000+28000}  
...which simplifies to   11x+11y\leq{76000}  
  x+y\leq{\approx{6900}}  
Notice that the sign STAYS the same way.

2) We can SUBTRACT inequalities if the signs face the opposite way:
  7x+6y\leq{38000}   and   28000\geq{4x+5y}   (we switch the direction of the last inequality)
...can be written as   (7x+6y)-28000\leq{38000-(4x+5y)}  
...which simplifies to   7x+6y-28000\leq{38000-4x-5y}   and   11x+11y\leq{76000}  
  x+y\leq{\approx{6900}}  
Notice that the we keep the first inequality's direction of the sign (the one we subtract FROM).

We get the same result either way. But of course, ADDING the inequalities makes the most sense for this particular question.

Nerdy01 · Answer

I was thinking about how to do it fast.

We simply have to look at the options.

According to the second expression, the sum of x and y should be multiplied on at least 4.

C, D and E give us a result that is more than 28k. As a result, the only two options are A and B. We choose the B because we are looking for Max.

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Pksri · Answer

Since we have to produce both balls and boxes to a maximum capacity, we can't neglect any of the terms.

Constraint 1:

we need to have a value such that 7x+6y<=38.
2 Possible situations are 7*1 + 6*5 <=38 or 7*4+6*1 <= 38
So, according to this equation, we can produce 6000 (1000+5000) or 5000 (4000+1000)

Constraint 2
Similarly, we have 2 possibilities to get close to 28. 
4*2+5*4 <= 28 or 4*5+5*1 <= 28
According to this equation, we can produce 6000 (2000+4000) or 6000 (5000+1000)

Combining both the constraints, the maximum we can produce is 6000. Answer is B

Abhishek009 · Answer

7x + 6y \leq 38,000 ---------------->(I)
 4x + 5y \leq 28,000 ---------------->(II)

(I) + (II) =>  11x + 11y \leq 66000  Or,  x + y \leq 6000 , Answer must be (B)

bluefalkin · Answer

I solved this question by solving for two simultaneous equations - apart form taking too long, any other potential issues with this approach?

EMPOWERgmatRichC, would appreciate your input on how to solve a question like this as fast as possible

RaghavKhanna · Answer

Adding the two INEQUALITIES together: 
 11x + 11y <= 66000
x + y <= 6000

Answer: B

Ruchihirawat · Answer

If we subtract the two equations, we get 3x + y <= 10,000
After further solving we get x = 1000 and Y = 7000

The total combined Value of balls and boxes would be 8000.
Could someone shed a light on where did I go wrong?

ksramanv · Answer

Hello,
If x+y is lesser than or equal to 6000, why couldn't the answer be A)5000? How did you eliminate it?

chetan2u · Answer

We are looking at the MAXIMUM number, and 6000 is the maximum value of x+y.

mwauthor · Answer

We can add inequalities if the inequality sign faces the same way. Therefore add equations 1 & 2 11x+11y

poojaarora1818 · Answer

We can add the inequalities as their signs are in the same direction. So, adding both we get:

7x + 6y \leq 38,000 
 4x + 5y \leq 28,000

11x + 11y \leq 66000

Then, divide the equation by 11

x + y \leq 6,000

So, correct answer is 6000

heyferno · Answer

I don't understand where this question implies that we have to produce both boxes and balls. I should be able to produce 7,000 balls if we produce 0 boxes. Nowhere in the question stem does it make it clear that we couldn't produce 0 boxes, y could just be zero.

In my opinion, the sentence "What is the maximum number of balls and boxes combined... " is not sufficient to imply that y could not be zero, especially when we are analyzing two inequalities which could absolutely be satisfied by setting y to zero.

I guess I'll have to learn to play by the GMAT's rules.

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