Problem 12 - Elementary Algebra

Question

A company must increase production by \(12 \%\) over last year's production. The new output will be 56 items. What was last year's output?

Step-by-Step Solution

Answer

Answer: The last year's output was approximately 50 items.

Step 1: Define the variables

Let x be the last year's output. The new output is an increase of 12% over last year's output, which can be represented as $$1.12x$$. Since the new output is 56 items, we can set up the equation: $$1.12x=56$$.

Step 2: Solve the equation

To solve the equation for x, we need to isolate the x variable by dividing both sides of the equation by 1.12: $$x = \frac{56}{1.12}$$ Now we can calculate the value of x: $$x \approx 50$$

Step 3: Interpret the result

Since x represents the last year's output, we can conclude that the output for last year was approximately 50 items.

Key Concepts

Equation Solving

Understanding how to solve an equation is a fundamental skill in algebra. In this exercise, we are presented with an equation that involves a percentage increase to determine a past value. The equation given is \(1.12x = 56\), which represents a 12% increase in last year's production, resulting in a new output of 56 items.

To solve for \(x\), you need to isolate it on one side of the equation. You can do this by performing the same operation on both sides—in this case, dividing both sides by 1.12. This step undoes the multiplication by 1.12, effectively 'removing' the increase influence from the equation. Here is the key step:

• Divide both sides by 1.12

Hence, the equation becomes:

• \(x = \frac{56}{1.12}\)

Once you calculate the division, you determine that \(x \approx 50\). This means that the production was approximately 50 items in the previous year. By understanding the principle of solving equations, you are equipped to tackle more complex problems in algebra.

Algebraic Expressions

Algebraic expressions are key components in understanding math problems and equations. They consist of variables, numbers, and operations. In this specific exercise, the algebraic expression is used to model the real-world scenario of production change.

The expression \(1.12x\) refers to last year's output multiplied by 1.12, indicating an increase of 12%.

• The number 1.12 in the expression represents the total production compared to the original, including the 12% increase.
• The variable \(x\) stands for last year's output.

This expression helps us to easily form an equation that can be solved to find unknown values, providing a connection between abstract math and practical situations. By manipulating algebraic expressions, we translate real-life conditions into solvable equations, leading us to solutions.

Variable Definition

Defining variables is crucial for setting up equations correctly and understanding how they operate in algebra. In the given problem, we define \(x\) as last year’s output. Understanding what a variable represents allows us to interpret its role in the context of the problem.

In algebra:

• Variables act as placeholders for unknown values.
• They enable the creation of general solutions that can be applied to different scenarios by adjusting the variable's value.

In this exercise, defining \(x\) as last year's output clarifies the problem's focus, allowing us to construct the equation \(1.12x = 56\), where solving for \(x\) reveals the quantity produced last year. Mastering variable definition is integral to seamlessly transitioning between problem description and equation formulation, forming the foundation for solving mathematical problems.