Problem 47 - Algebra for College Students

Question

Solve each percent problem. At the end of a day, Lawrence found that the total cash register receipts at the motel where he works were \(\$ 2725 .\) This included the \(9 \%\) sales tax charged. Find the amount of tax.

Step-by-Step Solution

Answer

The amount of tax is approximately \(\$ 225\).

Step 1: Identify the total amount and the sales tax percentage

The total cash register receipts, including the sales tax, is given as \(\$ 2725\). The sales tax charged is given as \(9\%\).

Step 2: Set up the equation

Let \(x\) be the amount before tax. The total amount including the sales tax can be represented by the equation: \[ x + 0.09x = 2725 \]

Step 3: Combine like terms

Combine the terms on the left side of the equation to get \[ 1.09x = 2725 \]

Step 4: Solve for \(x\)

Divide both sides of the equation by \(1.09\): \[ x = \frac{2725}{1.09} \] \[ x \approx 2500 \]

Step 5: Calculate the amount of tax

The amount of tax can be found by subtracting the amount before tax from the total amount: \[ 2725 - 2500 = 225 \]

Key Concepts

sales tax calculation

Calculating sales tax is a common task when managing finances or working in retail. In this problem, Lawrence noticed that the cash register receipts amounted to \(\text{\$2725}\) at the end of the day, which included a \(9\%\) sales tax. To find the amount of tax, we need to reverse-engineer the total to find out how much of it was tax. The total amount already includes the original price plus the tax, so we use the equation:
\( x + 0.09x = 2725 \), where \(x\) is the amount before tax.
Combining like terms, this becomes \( 1.09x = 2725 \).
Solving this by dividing both sides by 1.09 gives us the original amount.
Finally, subtracting the amount before tax from the total gives the tax amount.

solving linear equations

Solving linear equations is an essential skill in algebra, and it's used here to find the pre-tax amount. A linear equation is one in which the variable is raised to the power of one. In our problem, the equation is:
\(1.09x = 2725\).
The goal is to isolate the variable \(x\) by performing the same operation on both sides of the equation. Here, we divide both sides by 1.09 to get \( x = \frac{2725}{1.09} \).
This step-by-step simplification process helps us find that: \( x \approx 2500 \).
With this skill, you can solve more complex problems by breaking them down into simpler steps.

combining like terms

Combining like terms simplifies equations and expressions by grouping similar items. In our problem, the equation \( x + 0.09x = 2725 \) contains like terms on the left side: \(x\) and \(0.09x\).
These are like terms because both involve the variable \(x\).
Adding them together, we get: \( 1.09x = 2725 \).
Combining like terms is crucial as it transforms complex expressions into simpler ones, making it easier to solve for the variable.

proportion calculations

Proportion calculations help us find the relationship between different quantities. In sales tax problems, we often use proportions to find parts of wholes. For example, if you know that 9% of the total amount is tax, you can set up a proportion to find the tax amount.
Let’s say \(x\) is the amount before tax. We know that the total amount \(2725\) is the sum of the pre-tax amount and the tax amount.
The tax amount can be represented as \(0.09x\).
So our proportion equation is \(x + 0.09x = 2725\).
By solving for \(x\), we isolate the part without tax, and subtract it from the total to find the tax.
Proportions are handy because they keep relationships balanced, helpful in many real-life applications.