Question

in a car dealership, all of the vehicles are either a sedan or a suv. if 36 sedans are sold and 36 suvs are added, there will be an equal number of sedans and suvs. if 8 suvs are sold and 8 sedans are added, there will be twice as many sedans as suvs. how many sedans were at the dealership before any vehicle was sold?
a) 132
b) 144
c) 156
d) 168

Explanation:

Step1: Define Variables

Let \( s \) be the number of sedans and \( u \) be the number of SUVs initially.

Step2: Set Up First Equation

After selling 36 sedans (\( s - 36 \)) and adding 36 SUVs (\( u + 36 \)), they are equal:
\( s - 36 = u + 36 \)
Simplify: \( s - u = 72 \) (Equation 1)

Step3: Set Up Second Equation

After selling 8 SUVs (\( u - 8 \)) and adding 8 sedans (\( s + 8 \)), sedans are twice SUVs:
\( s + 8 = 2(u - 8) \)
Expand: \( s + 8 = 2u - 16 \)
Simplify: \( s - 2u = -24 \) (Equation 2)

Step4: Solve the System

Subtract Equation 2 from Equation 1:
\( (s - u) - (s - 2u) = 72 - (-24) \)
\( s - u - s + 2u = 96 \)
\( u = 96 \)

Substitute \( u = 96 \) into Equation 1:
\( s - 96 = 72 \)
\( s = 72 + 96 = 168 \)? Wait, no—wait, let's check again. Wait, Equation 1: \( s - u = 72 \), so \( s = u + 72 \). Substitute into Equation 2:
\( (u + 72) - 2u = -24 \)
\( -u + 72 = -24 \)
\( -u = -96 \)
\( u = 96 \). Then \( s = 96 + 72 = 168 \)? Wait, but let's verify with the second condition.

Check second condition:
Sedans after adding 8: \( 168 + 8 = 176 \)
SUVs after selling 8: \( 96 - 8 = 88 \)
\( 176 = 2 \times 88 \), which works.

Wait, but let's check the first condition:
Sedans after selling 36: \( 168 - 36 = 132 \)
SUVs after adding 36: \( 96 + 36 = 132 \), which also works. Wait, but the options include 168 (D). Wait, but let me recheck the equations.

Wait, Equation 1: \( s - 36 = u + 36 \) → \( s = u + 72 \). Correct.
Equation 2: \( s + 8 = 2(u - 8) \) → \( s + 8 = 2u - 16 \) → \( s = 2u - 24 \). Wait, earlier I had \( s - 2u = -24 \), which is correct. Then substituting \( s = u + 72 \) into \( s = 2u - 24 \):
\( u + 72 = 2u - 24 \)
\( 72 + 24 = 2u - u \)
\( u = 96 \), so \( s = 96 + 72 = 168 \). Yes, that's correct. Wait, but let's check the answer options. Option D is 168. Wait, but when I first solved, I thought maybe a miscalculation, but verification works.

Wait, but let's check again. Wait, the problem says "how many sedans were at the dealership before any vehicle was sold?" So \( s = 168 \).

Answer:

D) 168