In a certain city, a house’s annual property tax is calculated in three steps: 1. The assessed value is set at of the house’s market value. 2. A special infrastructure levy then increases this assessed value by . 3. The property tax is finally computed as of the amount obtained in step 2. If a homeowner paid \$5,292 in property tax last year, what was the market value of the house?

Solution available with step-by-step explanation

SAT Percentages Question 24 | Free SAT Prep | Aniko

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Question 24·Hard·Percentages

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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 $5,292 in property tax last year, what was the market value of the house?

Your answer

(Express the answer as an integer)

For chained percentage problems, always translate each step into a decimal multiplier and multiply them together to get one overall factor from the original amount to the final amount. Then set up a single equation where (overall factor) × (original amount) = (given final amount), solve by dividing, and finally check whether your answer’s size makes sense given that the final percent is relatively small.

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Represent each step with decimals

Let $x$ be the market value. Write what happens to $x$ at each step using decimal multipliers instead of percentages: $80\%$ becomes $0.80$ , an increase of $5\%$ becomes multiplication by $1.05$ , and $1.4\%$ becomes $0.014$ .

Combine the percent steps

All three steps are multiplications applied to the original $x$ . Multiply the three decimal factors together to get one overall multiplier that takes you directly from market value to the final tax.

Use the given tax amount to form an equation

Once you have a single multiplier $k$ , you should have an equation of the form $5292 = kx$ . How do you solve this kind of equation for $x$ ?

Check whether your answer size makes sense

Whatever value you get for the market value should be much larger than $5{,}292$ , because the tax is only a small percentage (around $1\%$ ) of the house's value.

Desmos Guide

Compute the required division

In Desmos, type the expression 5292 / (0.8 * 1.05 * 0.014) and look at the output value. That value is the market price that, after all three percentage steps, produces a tax of $5{,}292$ .

Step-by-step Explanation

Translate each step into a multiplication factor

Let $x$ be the market value of the house.

Step 1: The assessed value is $80\%$ of $x$ , which is $0.80x$ .
Step 2: This amount is increased by $5\%$ , so you multiply by $1.05$ : the new amount is $0.80x \cdot 1.05$ .
Step 3: The tax is $1.4\%$ of that amount, so you multiply by $0.014$ .

So the property tax in terms of $x$ is:

\text{tax} = x \cdot 0.80 \cdot 1.05 \cdot 0.014.

Set up the equation using the known tax amount

You are told the tax paid was $5{,}292$ , so set the expression equal to $5{,}292$ :

5292 = x \cdot 0.80 \cdot 1.05 \cdot 0.014.

Now you need to simplify the product of the decimal factors.

Combine the percentage factors into a single multiplier

First multiply $0.80$ and $1.05$ :

0.80 \times 1.05 = 0.84.

Then multiply that result by $0.014$ :

0.84 \times 0.014 = 0.01176.

So the equation becomes

5292 = 0.01176x.

Solve for the market value

To find $x$ , divide both sides of the equation by $0.01176$ :

x = \frac{5292}{0.01176}.

Write $0.01176$ as a fraction to simplify:

0.01176 = \frac{1176}{100000}, \quad\text{so}\quad x = 5292 \cdot \frac{100000}{1176}.

Now simplify $\dfrac{5292}{1176}$ :

\frac{5292}{1176} = \frac{441}{98} = \frac{9}{2} = 4.5.

x = 4.5 \times 100000 = 450000.

The market value of the house was $\boxed{450000}$ .

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Step-by-step Explanation

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Step-by-step Explanation