If demand for house cleaning services is given by D = 45000 minus 21P and supply is given by S = 15000 plus 9P, what is the equilibrium quantity of services?

Introduction / Context:
This question tests your ability to use simple algebra to find market equilibrium when linear demand and supply functions are given. In microeconomics, the equilibrium price and quantity are found at the point where quantity demanded equals quantity supplied. Being comfortable with this calculation is important for understanding how markets work and for solving many numerical problems on demand and supply.

Given Data / Assumptions:

• Demand function: D = 45000 minus 21P, where D is quantity demanded and P is price.
• Supply function: S = 15000 plus 9P, where S is quantity supplied.
• Equilibrium occurs where D equals S.
• We are asked for the equilibrium quantity, not directly for the equilibrium price.

Concept / Approach:
To find equilibrium in a competitive market, we set quantity demanded equal to quantity supplied and solve the resulting equation for price. Once the equilibrium price is obtained, we substitute it back into either the demand or supply equation to find the corresponding equilibrium quantity. This is a direct application of solving simultaneous equations in one variable.

Step-by-Step Solution:
1. Write the demand equation: D = 45000 minus 21P. 2. Write the supply equation: S = 15000 plus 9P. 3. At equilibrium, set D equal to S. 4. So 45000 minus 21P = 15000 plus 9P. 5. Rearrange terms: 45000 minus 15000 = 21P plus 9P. 6. This gives 30000 = 30P. 7. Solve for price: P = 30000 divided by 30 = 1000. 8. Substitute P = 1000 into the demand equation: D = 45000 minus 21 multiplied by 1000. 9. Calculate: 21 multiplied by 1000 = 21000, so D = 45000 minus 21000 = 24000. 10. Check using the supply equation: S = 15000 plus 9 multiplied by 1000 = 15000 plus 9000 = 24000, which matches.

Verification / Alternative check:
The fact that both equations give the same quantity when P is 1000 confirms that we have found the correct equilibrium. Also note that if you had solved directly for quantity by eliminating P incorrectly, you would get inconsistent results, so always use the method of setting D equal to S and solving for price first in one variable problems like this.

Why Other Options Are Wrong:
Option A: 12000 units would result from an incorrect calculation or mis substitution of price into the demand or supply equation.
Option B: 6000 units is far below the correct equilibrium and would not satisfy both equations simultaneously.
Option D: 30000 units does not satisfy both demand and supply at any single price using the given equations.
Option E: 18000 units once again cannot be obtained from both demand and supply equations with the same price, so it is not an equilibrium quantity.

Common Pitfalls:
Students sometimes forget to equate demand and supply and instead try to average them or guess the quantity. Another common mistake is algebraic, such as mis moving terms or dividing incorrectly, which leads to wrong values for price and quantity. Always carefully perform each step and, most importantly, substitute your answer back into both equations to check that they match.

Final Answer:
24000 units of services