Economists work with supply and demand curves that show the price P of goods as a function of the quantity...
GMAT Data Sufficiency : (DS) Questions
Economists work with supply and demand curves that show the price \(\mathrm{P}\) of goods as a function of the quantity \(\mathrm{Q}\) of those goods supplied or demanded. For a certain product, the supply curve is \(\mathrm{P = aQ + b}\) and the demand curve is \(\mathrm{P = \frac{k}{Q}}\), where \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{k}\) are nonzero constants. The point at which these curves intersect in the \(\mathrm{(Q, P)}\) coordinate plane is referred to as the equilibrium point, and for this product the equilibrium point is \(\mathrm{(10,5)}\). For this product, what are the values of \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{k}\)?
- The point \(\mathrm{(8,1)}\) is on the supply curve.
- The point \(\mathrm{(25,2)}\) is on the demand curve.
Understanding the Question
We need to find the specific values of three constants: a, b, and k.
Given Information
- Supply curve: \(\mathrm{P = aQ + b}\) (linear function)
- Demand curve: \(\mathrm{P = k/Q}\) (inverse function)
- The equilibrium point (where both curves intersect) is \(\mathrm{(10, 5)}\)
- All constants are nonzero
What We Need to Determine
For sufficiency, we must be able to calculate the exact numerical values of a, b, and k.
Key Insight
Since both curves pass through the equilibrium point \(\mathrm{(10, 5)}\):
- For the demand curve: \(\mathrm{5 = k/10}\), which gives us \(\mathrm{k = 50}\) immediately
- For the supply curve: \(\mathrm{5 = 10a + b}\)
Notice: We already know k! But we have one equation with two unknowns (a and b). We need one more independent equation to solve for both values.
Analyzing Statement 1
Statement 1: The point \(\mathrm{(8, 1)}\) is on the supply curve.
What Statement 1 Tells Us
If \(\mathrm{(8, 1)}\) is on the supply curve \(\mathrm{P = aQ + b}\), then:
\(\mathrm{1 = 8a + b}\)
Mathematical Analysis
Now we have two equations:
- \(\mathrm{10a + b = 5}\) (from equilibrium point)
- \(\mathrm{8a + b = 1}\) (from Statement 1)
Subtracting the second equation from the first:
- \(\mathrm{(10a + b) - (8a + b) = 5 - 1}\)
- \(\mathrm{2a = 4}\)
- \(\mathrm{a = 2}\)
Substituting \(\mathrm{a = 2}\) back into either equation:
- \(\mathrm{10(2) + b = 5}\)
- \(\mathrm{20 + b = 5}\)
- \(\mathrm{b = -15}\)
Conclusion
We can determine all three values: \(\mathrm{a = 2, b = -15, k = 50}\).
[STOP - Statement 1 is Sufficient!]
This eliminates choices B, C, and E.
Analyzing Statement 2
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: The point \(\mathrm{(25, 2)}\) is on the demand curve.
What Statement 2 Provides
If \(\mathrm{(25, 2)}\) is on the demand curve \(\mathrm{P = k/Q}\), then:
\(\mathrm{2 = k/25}\)
This gives us \(\mathrm{k = 50}\).
The Critical Problem
But wait—we already knew \(\mathrm{k = 50}\) from the equilibrium point! Statement 2 merely confirms what we already had. It provides no new information.
We still have only one equation for a and b:
- \(\mathrm{10a + b = 5}\)
With one equation and two unknowns, we cannot determine unique values for a and b. For example:
- If \(\mathrm{a = 1}\), then \(\mathrm{b = -5}\)
- If \(\mathrm{a = 0.5}\), then \(\mathrm{b = 0}\)
- Many other combinations are possible
Conclusion
Statement 2 is NOT sufficient.
This eliminates choices B and D.
The Answer: A
Since Statement 1 alone is sufficient to determine all three values, but Statement 2 alone is not sufficient, the answer is A.
Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."