Problem 89 - Prealgebra

Question

Solve for \(n\). $$42 n=21$$

Step-by-Step Solution

Answer

The value of \( n \) is \( \frac{1}{2} \).

Step 1: Understand the Equation

The given equation is \( 42n = 21 \). We need to find the value of \( n \) that makes this equation true.

Step 2: Isolate the Variable

To solve for \( n \), we need to isolate \( n \) on one side of the equation. This can be done by dividing both sides of the equation by 42: \[ \frac{42n}{42} = \frac{21}{42} \]

Step 3: Simplify the Expression

After dividing, the equation simplifies to: \[ n = \frac{21}{42} \]. The fraction \( \frac{21}{42} \) can be simplified further. Divide the numerator and the denominator by their greatest common divisor, which is 21, to get: \[ \frac{21}{21} = \frac{1}{2} \].

Step 4: Final Answer

Thus, the value of \( n \) is \( \frac{1}{2} \).

Key Concepts

Exploring Prealgebra Concepts

Prealgebra encompasses some of the most fundamental skills needed to tackle algebra, and eventually more complex mathematics. It involves basic arithmetic, understanding numbers, and working with simple equations. When approaching a problem in prealgebra, it’s essential to identify the type of equation and recognize the steps needed to solve it. In our exercise with the equation \(42n = 21\), we see a direct relationship between multiplication and division, which are key operations in prealgebra. This type of equation, where a single variable is multiplied by a constant, is central to understanding how variables work.

• Recognizing different types of arithmetic operations and their inverses is crucial in prealgebra.
• Understanding the balance method can help in maintaining equality while solving equations.
• Becoming familiar with terms like “isolate the variable” or “simplify the equation” allows for smoother problem-solving.

Implementing these concepts means using inverse operations, such as division here, to untangle the equation and find \(n\). By mastering prealgebra, students build a solid foundation for future algebra concepts.

Fraction Simplification Fundamentals

Simplifying fractions is a critical skill in mathematics that helps to present answers in their simplest form. Simplifying a fraction means reducing it to its lowest terms, or making it as simple as possible. In our problem with \(\frac{21}{42}\), the idea is to find a number that divides evenly into both the numerator and the denominator.

• Identify the greatest common divisor (GCD) for both the numerator and the denominator.
• In this case, \(21\) is the GCD of \(21\) and \(42\).
• Divide both parts of the fraction by their GCD to simplify. Hence, \(\frac{21}{42}\) simplifies to \(\frac{1}{2}\).

Once simplified, fractions become easier to understand and work with, which can be especially helpful in complex mathematical problems. Simplifying fractions not only makes numbers easier to read but also aids in calculation accuracy.

Isolating the Variable in an Equation

The process of isolating the variable is a fundamental technique in solving algebraic equations. The goal is to get the unknown (usually \(n\) or \(x\)) by itself on one side of the equation, allowing for its value to be clearly identified.

• Begin by identifying what operation is currently bonding the variable to other numbers. In our exercise, \(n\) was multiplied by \(42\).
• Use the opposite operation to isolate the variable. Since multiplication is used here, division will counter it.
• Apply this operation consistently to both sides of the equation to maintain balance, resulting in \(n = \frac{21}{42}\).

By isolating the variable, we can determine that \(n\) equals a simplified fraction \(\frac{1}{2}\). Understanding this process is essential to solving not just simple equations, but more complex ones as well. This technique highlights a core algebraic principle: performing the same operation on both sides of an equation to maintain equality.