Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Payroll functions Let the social security tax function SSTAX be defined as \(\operatorname{SSTAX}(x)=0.0765 x\), where \(x \geq 0\) is the weekly income. Let ROUND2 be the function that rounds a number to two decimal places. Find the value of \((\) ROUND2 ° SSTAX) \((525)\).

Step-by-Step Solution

Answer

The value is 40.16.

Step 1: Apply the Social Security Tax Function

First, we need to find the value of the Social Security Tax function for an income of $525. We use the formula \( \operatorname{SSTAX}(x) = 0.0765x \). Substitute \( x = 525 \):\[\operatorname{SSTAX}(525) = 0.0765 \times 525\]Calculate the result:

Step 2: Calculate Social Security Tax

Perform the multiplication to find the tax: \( 0.0765 \times 525 = 40.1625 \).

Step 3: Round to Two Decimal Places

Now apply the ROUND2 function, which rounds \( 40.1625 \) to two decimal places. This gives \( 40.16 \).

Step 4: Conclusion

The value of \((\text{ROUND2} \circ \text{SSTAX})(525)\) is \( 40.16 \).

Key Concepts

Social Security Tax

The Social Security Tax is a federal tax levied on both employers and employees to fund the Social Security program, providing benefits like retirement and disability income. In this context, the Social Security Tax function, denoted as \( \operatorname{SSTAX}(x) = 0.0765x \), is a mathematical expression that calculates the tax you owe based on your income. The coefficient 0.0765 represents the Social Security tax rate, which is currently set at 7.65%.

Understanding this function is crucial for estimating the tax amounts deducted from your earnings. For example, if your weekly income is \(525, the calculation for the tax you owe is straightforward.

• Multiply the income, \)525, by 0.0765.
• The result tells you the amount of Social Security tax you owe for that week.

Applying the formula helps you see exactly how your tax dues are determined and how they relate to your overall payroll. This understanding is not only foundational for payroll calculations but also serves in personal financial planning, ensuring you're aware of your net income after taxes.

Rounding Functions

Rounding functions are mathematical tools that adjust numbers to specified decimal places, making them easier to handle in everyday arithmetic. In this exercise, we're dealing with ROUND2, a function that rounds numbers to two decimal places. This is particularly useful when dealing with financial figures, where precision to cents is necessary.

Let's say you've calculated a Social Security tax amount of $40.1625. Financial standards require you to report this in a simple currency format, typically rounded to two decimal points. Here's how ROUND2 works:

• Identify the third decimal place.
• If it's 5 or greater, increase the second decimal place by 1.
• If it's less than 5, leave the second decimal place as is.

In our example, ROUND2 takes $40.1625 and adjusts it to $40.16, as the third decimal (2) is less than 5. This process ensures that all financial values are consistent and minimizes rounding errors in transactions and accounting records.

Weekly Income Functions

Weekly income functions allow you to determine your net and gross income over a specified period, taking into account any taxes or deductions applied. Calculating your weekly income is fundamental to managing finances, as it informs budgeting and savings.

In the given example, the weekly income is $525, before any taxes are taken out. Using functions like the Social Security Tax function, you can find out what's deducted from your income weekly. Here’s a simple process you might follow:

• Start with your gross weekly income.
• Apply the Social Security Tax function: compute \( \operatorname{SSTAX}(525) \) to find the amount taken out for taxes.
• Use rounding functions like ROUND2 where necessary to adjust final figures to an easy-to-understand value.
• Your final weekly net income is your gross income minus these deductions.

Understanding weekly income functions is crucial for accurate paycheck forecasting, financial planning, and meeting your financial goals effectively.