Question 327·Hard·Math

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In a certain city, a house’s annual property tax is calculated in three steps:

The assessed value is set at $80\%$ of the house’s market value.
A special infrastructure levy then increases this assessed value by $5\%$ .
The property tax is finally computed as $1.4\%$ of the amount obtained in step 2.

If a homeowner paid $588 in property tax last year, what was the market value of the house?

Your answer

An assessed value based on market value, followed by a levy and a tax, forms a chain of percent calculations. If you're given the final tax and asked for market value, write the chain forward from the market value, then solve one equation with a Desmos regression. Don't apply every percent to the market value; each later percent uses the amount the preceding step produced.

Hints

Hint 1
The assessed value is $80\%$ of the market value, not the tax paid. If $x$ is the market value, what expression gives the assessed value?
Hint 2
An increase by $5\%$ keeps the whole current amount and adds $5\%$ of it. Which amount receives the levy, and what multiplier represents that increase?
Hint 3
The tax rate applies to the amount after the levy. Write $1.4\%$ of that amount, then set it equal to the tax the homeowner paid.

Step-by-step

Build the percent chain and solve
Step 1
Express the assessed value
Let $x$ be the market value in dollars. The assessed value is the value used to calculate the tax. The city sets it at $80\%$ of market value, so $\text{assessed value}=0.80x$ .
Step 2
Apply the levy to the assessment
The $5\%$ levy increases the assessed value. Keeping the whole assessment and adding $5\%$ gives $105\%$ of that assessment, so $\text{post-levy assessment}=1.05(0.80x)$ . Each percent acts on the amount from the step before it. Using $1.05x$ would apply the levy to market value instead.
Step 3
Connect the tax to the amount paid
The tax rate is $1.4\%$ of the post-levy assessment. That tax equals the homeowner's payment of $588, so $0.014\bigl(1.05(0.80x)\bigr)=588$ . Taking $1.4\%$ of $x$ would tax the market value, not the amount the city uses.
Step 4
Solve for the market value in Desmos
Type $0.014(1.05(0.80x_1))\sim588$ . Change $x$ to $x_1$ and $=$ to $\sim$ so Desmos solves for the unknown. Under PARAMETERS, it shows $x_1=50000$ . Since $x$ represents the market value, the house's market value was $50,000. Grid in 50000.

Question 327·Hard·Math

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