Problem 60 - Elementary Algebra

Question

In 2011, the total property tax millage rate for Fort Lauderdale, Florida, was \(20.1705\). (For every \(\$ 1000\) in taxable property, an owner owes a tax of \(\$ 20.1705\).) If a property owner pays the tax in four installments, a discount is applied to the first three installments. Find the total amount of tax paid by installments on taxable property of \(\$ 175,000\). Round to the nearest hundredth. $$ \begin{array}{|c|c|} \hline \text { Installment due date } & \text { Discount on the payment } \\ \hline \text { June 30 } & 6 \% \\ \text { September 30 } & 4.5 \% \\ \text { December 31 } & 3 \% \\ \text { March 31 } & \text { None } \\ \hline \end{array} $$ Sources: www.broward.org; www.bcpa.net.millage.asp

Step-by-Step Solution

Answer

The total amount of tax paid in installments is approximately \(\text{\$ 3410.89}\).

Step 1: Calculate the annual tax without discount

The millage rate is applied to each \(1000\) dollars of taxable property. Calculate the tax for \(175,000\) dollars with the given millage rate. The formula to use is: \[\text{Annual Tax} = \left( \frac{\text{Taxable Property}}{1000} \right) \times \text{Millage Rate}\] Substituting the values: \[\text{Annual Tax} = \left( \frac{175,000}{1000} \right) \times 20.1705 = 175 \times 20.1705 = 3530.8375\]

Step 2: Calculate the first installment with 6% discount

The first installment is paid with a 6% discount. Calculate 6% of the annual tax and subtract that from the tax to find the discounted amount for the first installment. \[\text{First Installment} = \frac{\text{Annual Tax}}{4} \times (1 - 0.06)\] Substituting the values: \[\text{First Installment} = \frac{3530.8375}{4} \times 0.94 = 3530.8375 \times 0.235 = 830.0715625\]

Step 3: Calculate the second installment with 4.5% discount

The second installment is paid with a 4.5% discount. Calculate 4.5% of the annual tax and subtract it to find the discounted amount for the second installment. \[\text{Second Installment} = \frac{\text{Annual Tax}}{4} \times (1 - 0.045)\] Substituting the values: \[\text{Second Installment} = \frac{3530.8375}{4} \times 0.955 = 3530.8375 \times 0.23875 = 842.067734375\]

Step 4: Calculate the third installment with 3% discount

The third installment is paid with a 3% discount. Calculate 3% of the annual tax and subtract that from the tax to find the discounted amount for the third installment. \[\text{Third Installment} = \frac{\text{Annual Tax}}{4} \times (1 - 0.03)\] Substituting the values: \[\text{Third Installment} = \frac{3530.8375}{4} \times 0.97 = 3530.8375 \times 0.2425 = 856.03901953125\]

Step 5: Calculate the final installment with no discount

The fourth installment is paid without any discount. It is one-fourth of the annual tax. \[\text{Fourth Installment} = \frac{\text{Annual Tax}}{4}\] Substituting the values: \[\text{Fourth Installment} = \frac{3530.8375}{4} = 882.709375\]

Step 6: Sum the discounted installments

Add up all the installment values to get the total tax paid: \[\text{Total Paid} = \text{First Installment} + \text{Second Installment} + \text{Third Installment} + \text{Fourth Installment}\] Substituting the values: \[\text{Total Paid} = 830.0715625 + 842.067734375 + 856.03901953125 + 882.709375 = 3410.88769140625\] Round to the nearest hundredth: \[\text{Total Paid} \approx 3410.89\]

Key Concepts

Millage Rate

The millage rate is a crucial concept in property tax calculation. It represents the amount of tax payable per \(1000\) dollars of the assessed property value. For instance, a millage rate of \(20.1705\) means that for every \(1000\) dollars of taxable property, \(20.1705\) dollars is owed in taxes.
The term 'millage' comes from the Latin word for thousand, 'mille', because it is based on each \(1000\) units of currency value. It's essential to understand because it determines how much the property owner needs to pay annually.
Examples:

• If a property's assessed value is \(175,000\) dollars and the millage rate is \(20.1705\), then the annual tax can be calculated by:
• \text{Annual Tax} = \frac{\text{Taxable Property}}{1000} \times \text{Millage Rate}
• Substituting the values, we have:
• \text{Annual Tax} = \frac{175,000}{1000} \times 20.1705 \text = 3530.8375

Knowing this calculation, you understand how the millage rate impacts your annual tax bill.

Installment Discount

Setting up property tax payments in installments can save you money due to installment discounts. These discounts incentivize early payments by reducing the amount owed.
For example, in Fort Lauderdale, the property owner receives discounts for paying in the first three installments:

• First installment (June 30): 6% discount
• Second installment (September 30): 4.5% discount
• Third installment (December 31): 3% discount
• Final installment (March 31): No discount

Let's calculate the installment payments for a property with an annual tax of \(3530.8375\) dollars:
Installment 1:

• \text{First Installment} = \frac{\text{Annual Tax}}{4} \times (1 - 0.06)
• \text{First Installment} = \frac{3530.8375}{4} \times 0.94 = 830.0715625

Installment 2:

• \text{Second Installment} = \frac{\text{Annual Tax}}{4} \times (1 - 0.045)
• \text{Second Installment} = \frac{3530.8375}{4} \times 0.955 = 842.067734375

Installment 3:

• \text{Third Installment} = \frac{\text{Annual Tax}}{4} \times (1 - 0.03)
• \text{Third Installment} = \frac{3530.8375}{4} \times 0.97 = 856.03901953125

Installment 4:

• \text{Fourth Installment} = \frac{\text{Annual Tax}}{4} = 882.709375

Annual Tax Calculation

Calculating the annual property tax is straightforward once you grasp the millage rate and the property's assessed value.
The key formula is:

• \text{Annual Tax} = \frac{\text{Taxable Property}}{1000} \times \text{Millage Rate}

Consider an example where the property is valued at \(175,000\) dollars with a millage rate of \(20.1705\):

• \text{Annual Tax} = \frac{175,000}{1000} \times 20.1705 = 3530.8375

Once you calculate the total annual tax without any discounts, you can use this value to determine the actual payments if you choose installment payments.
Remember, annual tax calculations form the baseline for how much you owe before factoring in any discounts or installment plans.

Discounted Payments

Discounted payments are a valuable method for reducing your overall tax liability. When paying in installments, you may receive discounts on the initial payments.
For example, assume an annual tax of \(3530.8375\) dollars. Let’s determine the total payment with installment discounts:

• First Installment: \text = 830.0715625
• Second Installment: \text = 842.067734375
• Third Installment: \text = 856.03901953125
• Fourth Installment: \text = 882.709375

Adding these up gives:

• \text{Total Paid} = 830.0715625 + 842.067734375 + 856.03901953125 + 882.709375 = 3410.88769140625

This amount is less than the total annual tax of \(3530.8375\) dollars you’d pay without discounts. Rounded to the nearest hundredth, this total is \(3410.89\) dollars.
Taking advantages of early payment discounts can result in substantial savings over the year.