Problem 64 - Elementary Algebra

Question

A solar technology distribution company expects a \(12 \%\) increase in first quarter sales as a result of a recently implemented rebate program. If the first quarter sales last year totaled \(\$ 350,000,\) then what are the sales projections for the first quarter of this year?

Step-by-Step Solution

Answer

The projected sales for the first quarter are $392,000.

Step 1: Understand the Problem

We need to find the projected sales for the first quarter of this year, given that there is a 12% increase compared to the last year's sales, which were $350,000.

Step 2: Calculate the Increase

First, calculate 12% of last year's sales. This is done by multiplying $350,000 by 0.12.\[\text{Increase} = 350,000 \times 0.12 = 42,000\]

Step 3: Calculate the Projected Sales

Add the increase to last year's sales to find the projected sales for this year.\[\text{Projected Sales} = 350,000 + 42,000 = 392,000\]

Key Concepts

Percentage Increase

Understanding percentage increase is essential in many aspects of life, including calculations related to finance or sales. A percentage increase shows how much a quantity has gone up in relation to its original size. To find a percentage increase, follow these steps:

• First, determine the percentage change you expect. In our case, this is a 12% increase.
• Second, convert the percentage to a decimal by dividing by 100; therefore, 12% becomes 0.12.
• Finally, multiply the original value by this decimal to find the increase amount.

For the given exercise, we calculated 12% of $350,000. The result was $42,000, representing the increase due to the rebate program.
The essence of percentage increase helps businesses and individuals to plan better by projecting future scenarios based on past data.

Sales Projections

Sales projections are crucial in business as they provide an estimate of future sales revenue. Accurately projecting sales allows companies to make informed decisions about inventory, staffing, and other operational requirements.
To project future sales, consider the following:

• Analyze past sales data to establish a baseline. Here, we used last year's first quarter sales of $350,000.
• Identify factors likely to affect sales in the future, such as economic conditions or new programs (like a rebate program).
• Quantify these factors as a percentage to understand their impact.
• Apply this percentage change to the baseline sales figure to project future sales revenue.

In this particular example, by applying the 12% increase to $350,000, we projected the first quarter sales for the current year to be $392,000. Such foretelling ensures businesses remain competitive and prepared for challenges and opportunities.

Basic Arithmetic Calculations

Basic arithmetic is foundational in solving mathematical problems, including those involving percentages and projections. It involves performing fundamental operations like addition, subtraction, multiplication, and division. Here’s a brief look at the calculations used in the exercise:

• Multiplication: This operation was used to find the increase. By multiplying $350,000 by 0.12, we determined the increase of $42,000.
• Addition: Once the increase was calculated, we added it to last year's sales to find the total projected sales. This was expressed as $350,000 plus $42,000, equaling $392,000.

These simple operations are the building blocks that support more complex mathematical procedures. Mastering them is crucial for understanding and accurately solving problems in algebra and beyond.