Question

Translate to an equation and solve. A company spends \(\$ 245,000\) on advertising during the first quarter of the year. If this represents \(49 \%\) of the company's advertising budget for the year, what is the company's advertising budget for the year?

Short answer

Answer: The company's yearly advertising budget is \$500,000.

Step-by-step solution

Translate the problem into an equation

Let \(x\) be the advertising budget for the year. According to the problem, \(49 \%\) of \(x\) is equal to \(\$ 245,000\). So, the equation will be: $$0.49x = 245,000$$

Solve the equation for x

To find the yearly advertising budget, we need to solve the equation for \(x\). In order to do that, we'll divide both sides of the equation by \(0.49\): $$x=\frac{245,000}{0.49}$$

Calculate the value of x

Now we'll calculate the value of \(x\): $$x=\frac{245,000}{0.49}=500,000$$ So, the company's advertising budget for the year is \(\$ 500,000\).

Key concepts

Equation Solving

The process of equation solving is about finding the values that make an equation true. When solving equations, we aim to make the variable (often denoted as \( x \)) stand alone on one side of the equation. In this exercise, we start with the equation \( 0.49x = 245,000 \). Here, \( 0.49 \) is the percentage in its decimal form, \( x \) is our unknown variable which we need to solve, and \( 245,000 \) is the amount spent in the first quarter.

To isolate \( x \), we use inverse operations to undo the multiplication by \( 0.49 \). This means we will divide both sides of the equation by \( 0.49 \). So,\( x = \frac{245,000}{0.49} \). After performing this division, we find that \( x = 500,000 \). Therefore, the company's total advertising budget for the year is \$500,000.

Understanding how to arrange and rearrange equations is essential in prealgebra. It gives you the ability to solve for unknown numbers in a logical and systematic way. Mastering this skill involves practicing simple to complex equations by applying addition, subtraction, multiplication, and division.

Percentage Problems

Percentage problems often involve finding what percentage one quantity is of another or finding the total quantity given a part and its percentage. In our exercise, we are told that \( 245,000 \) is \( 49\% \) of the company's advertising budget. Thus, we translate this into an equation using the concept of percentage: \( 0.49x = 245,000 \).

To comprehend this better, remember that percentage itself is simply a number out of 100. So \( 49\% \) is the same as 49 out of 100, or \( 0.49 \) as a decimal. Knowing how to convert percentages into decimals is helpful in setting up your equations.

To solve the problem, knowing which value represents the percentage (in this case \( 0.49 \)) and which represents the total amount (\( 245,000 \)) allows you to set up an equation properly. Solving this equation tells you the entire budget needed using the information provided.

Budget Calculation

Budget calculation involves determining the financial plan for a specific period. In the exercise you've encountered, the purpose is to calculate the total advertising budget when given that a specific amount corresponds to a part of it.

Starting with the partial expense, the \$245,000 spent in the first quarter, you relate it to the percentage of the whole budget it represents. The equation \( 0.49x = 245,000 \) helps depict the relationship between the partial spending and the full budget.

By solving for \( x \), which represents the total budget, you'll determine how much the company needs for the entire year based on their initial spending. Budget calculations are crucial in prealgebra because they help you understand how various costs add up and how they can be managed efficiently. Tools from solving equations and understanding percentages are put together here to form a comprehensive understanding of financial planning.