A batch of widgets costs

[saifullahkhan7]

I am a bit confused about the explanation of Workout Set 1, Question Number 2 on Page 231 of the Advanced GMAT Quant, Strategy Supplement. The Question Reads:

A batch of widgets costs p+15 dollars for a company to produce and each batch sells for p(9-p) dollars. For which of the following values of p does the company make a profit?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

After putting values in the formula for profits, we get and solve a quadratic equation with values of p=5 or p=3.

Now the interesting part in the explanation is "For p>5, both (p-5) and (p-3) are positive. In that case, the profits are negative, i.e. the company losses money. The profit is only positive if (p-5) and (p-3) have opposite signs, which occurs when 3<p<5". (from Page 270 of the strategy supplement.

My specific question is "Why is difference of sign of p a requirement for profitability. While simple multiplication tells us that profits will be positive when the signs of p are similar".


Regards

[georgepa]

• Profit = (How much the batch sells for) - (How much it costs to make the batch)


• => Profit = Revenue - Cost


• => Profit = p(9-p) - (p + 15)

We need to find values where Profit is > 0.

Strategy: Plug in #s

• p=3 =>Profit = 3(9-3) - (3+15) = (3*6) - 18 = 18 - 18 = 0


• p=4 =>Profit = 4(9-4) - (4+15) = (4*5) - 19 = 20 - 19 > 0


• p=5 =>Profit = 5(9-5) - (5+15) = (5*4) - 20 = 20 - 20 = 0


• p=6 =>Profit = 6(9-6) - (6+15) = (6*3) - 21 = 18 - 21 < 0


• p=7 =>Profit = 7(9-7) - (7+15) = (7*2) - 22 = 14 - 22 < 0

[georgepa]

Algebra:

• Profit = (How much the batch sells for) - (How much it costs to make the batch)


• => Profit = Revenue - Cost


• => Profit = p(9-p) - (p + 15)


• => Profit = -p^2 +8p -15


• => Profit = (-1)(p-3)(p-5)


• => Profit = (3-p)(p-5)

Need to find numbers where Profit is +ve


Solution:

• When p=3 or p=5, Profit = 0 (since they are roots)


• When p > 5, Profit = (3-p)(p-5) = (-ve)x(+ve) = -ve


• When p < 3, Profit = (3-p)(p-5) = (+ve)x(-ve) = -ve


• The only other option is when p=4. (3-p)(p-5)=(-1)(-1) = 1 = +ve

[RonPurewal]

I am a bit confused about the explanation of Workout Set 1, Question Number 2 on Page 231 of the Advanced GMAT Quant, Strategy Supplement. The Question Reads:

A batch of widgets costs p+15 dollars for a company to produce and each batch sells for p(9-p) dollars. For which of the following values of p does the company make a profit?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

After putting values in the formula for profits, we get and solve a quadratic equation with values of p=5 or p=3.

Now the interesting part in the explanation is "For p>5, both (p-5) and (p-3) are positive. In that case, the profits are negative, i.e. the company losses money. The profit is only positive if (p-5) and (p-3) have opposite signs, which occurs when 3<p<5". (from Page 270 of the strategy supplement.

My specific question is "Why is difference of sign of p a requirement for profitability. While simple multiplication tells us that profits will be positive when the signs of p are similar".


Regards

Profit means that revenues are greater than costs. Here, that means p(9 - p) is greater than p + 15.
If you set up that inequality, and then factor it, you'll get (p - 5)(p - 3) < 0.
Negative product, so that's why the answer key is talking about opposite signs.

[danr969]

for this problem: revenue = p(9-p). Cost = p + 15.
We can solve the inequality p(9-p) > p + 15 in a simple way.
The left hand side is a "frown" with roots at 0 and 9. Thus, its max value occurs at 4.5. 4.5 is not an answer choice, but 4 and 5 are both choices, and they are each 0.5 units away from 4.5. Since cost
increases with p (cost = p + 15), we should choose 4 instead of 5:
They both have the same revenue, but p = 4 has lower cost.

We don't have to find the value of p that maximizes profit, only the single value from the answer choice that gives +ve profit.

[jnelson0612]

Let me just add that if I had this on the actual GMAT I would test the answer choices. It would take less than a minute to determine which one allows a profit.

[RonPurewal]

danr969, do you have a question?

Since you're responding to a thread that...
"- is six months old,
- has already been answered by our staff,
... I'd guess you have some sort of question. But I don't see one.

Please clarify, thanks.