00:00

Try Full-Screen!

Question 24·Hard·Percentages

0 / 57

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

For chained percentage problems, translate every percentage into a decimal multiplier and multiply the factors to obtain one overall multiplier. Use the given final amount to form a one-variable equation, divide by the overall multiplier, and check that the original amount is reasonably larger than the final percentage amount.

Hints

Click me!

Represent each step with decimals

Let $x$ be the market value. Write each percentage step as a decimal multiplier: $80\%$ 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 multiply the original $x$ . Multiply the three decimal factors to find one overall multiplier from market value to final tax.

Use the given tax amount to form an equation

After finding a single multiplier $k$ , write an equation of the form $588=kx$ . Then solve the equation for $x$ .

Check the size of the result

Your market value should be much larger than $588, because the tax is only a small percentage of the house’s value.

Desmos Guide

Compute the required division

On a Desmos expression line, enter $\frac{588}{0.8\cdot1.05\cdot0.014}$ . The displayed result is the market value that produces the stated tax after all three percentage steps.

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 multiply by $1.05$ .
Step 3: The tax is $1.4\%$ of that amount, so multiply by $0.014$ .

Therefore, the property tax is

\text{tax}=x\cdot0.80\cdot1.05\cdot0.014.

Set up an equation using the tax payment

The homeowner paid $588 in tax, so

588=x\cdot0.80\cdot1.05\cdot0.014.

Combine the percentage factors

First, multiply the assessment and levy factors:

0.80\cdot1.05=0.84.

Then include the tax rate:

0.84\cdot0.014=0.01176.

So the equation becomes

588=0.01176x.

Solve for the market value

Divide both sides by $0.01176$ :

x=\frac{588}{0.01176}.

Because

0.01176=\frac{1176}{100000},

we have

x=588\cdot\frac{100000}{1176}.

Since

\frac{588}{1176}=\frac{1}{2},

x=\frac{1}{2}\cdot100000=50000.

The market value of the house was $50,000.

Question 24·Hard·Percentages

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation