Question

The manager of a business is to receive a bonus amounting to \(20 \%\) of the profits which remain after deducting the business's income tax (but not the manager's bonus) from the gross profits. The income tax is \(40 \%\) of the profits which remain after deducting the manager's bonus (but, as required by law, not the tax) from the gross profits. The gross profits before deducting either bonus or tax are \(\$ 230,000\). Find the bonus and the tax.

Short answer

The bonus is \$30,000 and the tax is \$80,000.

Step-by-step solution

Define Variables

Let the manager's bonus be denoted by \( B \) and the income tax by \( T \).

Set Up Equations

Write the equations based on the given conditions. \( B = 0.2(GP - T)\) ewline and \( T = 0.4(GP - B) \), where GP is the gross profit.

Substitute Gross Profit Value

Substitute the given gross profits value into the equations: \( B = 0.2(230000 - T) \) ewline and \( T = 0.4(230000 - B) \).

Solve for Bonus

Substitute the second equation into the first equation: \( B = 0.2[230000 - 0.4(230000 - B)] \) ewline Simplify to find \( B \).

Simplify Equation

Simplify the equation: \[ B = 0.2[230000 - 92000 + 0.4B] \] \[ B = 0.2[138000 + 0.4B] \] \[ B = 27600 + 0.08B \] \[ 0.92B = 27600 \] \[ B = 30000 \]

Solve for Tax

Use \( B = 30000 \) to find \( T \): \[ T = 0.4(230000 - 30000) \] \[ T = 0.4(200000) \] \[ T = 80000 \]

Key concepts

income tax calculation

Income tax calculation involves determining how much tax needs to be paid based on certain percentages of profit. In this exercise, we learn that the income tax is 40% of the profits that remain after deducting the manager's bonus. To set this up, we defined the income tax as a variable, \( T \), and wrote an equation: \( T = 0.4(GP - B) \). Here, \( GP \) is the gross profit and \( B \) is the manager’s bonus. This translates to:

• First, identify the gross profit.
• Adjust the gross profit by subtracting the bonus.
• Apply the tax percentage to the adjusted profit.

In this context, ensuring the proper sequence of operations is crucial for accurate tax calculation.

bonus calculation

Calculating the manager's bonus requires understanding how the company's policies interact with their profits and taxes. The bonus is 20% of the profits after the income tax is deducted. This was represented by the equation: \( B = 0.2(GP - T) \). To solve for \( B \):

• Identify the gross profit \( GP \) and income tax \( T \).
• Subtract the tax from the gross profit to get the adjusted profit.
• Apply the bonus percentage to this adjusted profit.

By substituting \( GP = 230000 \) and solving the subsequent equations, we find that the bonus is $30,000.

system of equations

A system of equations consists of multiple equations that are solved together to find common solutions. In this problem, we dealt with two equations simultaneously:

• \( B = 0.2(230000 - T) \)
• \( T = 0.4(230000 - B) \)

To solve this system:

• Substitute one equation into the other.
• Simplify to isolate one variable first.
• Solve for the isolated variable.
• Use the calculated value to find the second variable.

This method of substitution allowed us to find both the bonus and tax amounts accurately.

gross and net profit calculation

Understanding the difference between gross and net profit is essential in financial calculations. Gross profit is the total profit before any deductions, while net profit is what remains after deducting expenses such as taxes and bonuses. In this exercise, the gross profit is \$230,000. From this, we calculate:

• Manager's bonus: \( B \)
• Income tax: \( T \)

To find the net profit:

• Subtract both the bonus and the tax from the gross profit.

By solving, we determined the bonus to be \$30,000 and the tax to be \$80,000. Thus, the net profit calculation is: \[ 230000 - (30000 + 80000) = 120000 \] This step-by-step breakdown simplifies the calculation of net profit from gross profit.