Problem 97 - Algebra 1

Question

Solve the equation. \(\frac{n}{15}=\frac{3}{5}\)

Step-by-Step Solution

Answer

The solution to the equation \(\frac{n}{15}=\frac{3}{5}\) is \(n = 9\).

Step 1: Cross-Multiplication

Cross-multiplication is used when we have a proportion between two fractions, i.e., when one fraction equals another. It means that we multiply the numerator of the fraction on the left side by the denominator of the fraction on the right side and the numerator of the fraction on the right side by the denominator of the fraction on the left side. In our case, this gives: \(n*5 = 15*3\).

Step 2: Simplify and Solve for n

Now we simplify and solve the resulting equation for \(n\). So, \(5n = 45\) which simplifies to \(n = 45 / 5 = 9\).

Key Concepts

Cross-Multiplication

The concept of cross-multiplication is a quick and effective method for solving equations involving two equivalent fractions, which we also call proportions. It's based on the principle that if two fractions are equal, then their cross products (the product of the numerator of one fraction and the denominator of the other) will also be equal.

For example, in the equation \(\frac{n}{15} = \frac{3}{5}\), cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second and vice versa. This gives us the calculation \(n \times 5 = 15 \times 3\), leading to the equation \(5n = 45\). Now, the problem is reduced to a simple algebraic equation that needs to be solved.

Applying this method requires that you carefully align the terms and perform multiplication accurately to ensure a correct equation to solve.

Fraction Equivalence

Fraction equivalence is a fundamental concept in mathematics that allows us to understand when two fractions represent the same quantity or value, even if their numerators and denominators differ. It's this principle that underpins the process of cross-multiplication.

For two fractions to be equivalent, the ratios of their numerators to denominators must be the same. In the given problem, \(\frac{n}{15}\) and \(\frac{3}{5}\) are equivalent because they can be transformed into the same decimal or percentage, or because the cross-product is equal after cross-multiplication. Understanding fraction equivalence is crucial in a variety of mathematical contexts, from solving equations to simplifying expressions and comparing fractions.

Algebraic Equations

Algebraic equations are mathematical statements that use variables, like \(n\), to represent unknown values and include an equals sign to show that two expressions are equivalent. The goal when solving algebraic equations is to isolate the variable on one side to find its value. This often involves a series of steps, including simplifying expressions and performing operations such as addition, subtraction, multiplication, and division.

In the exercise example, we arrive at the algebraic equation \(5n = 45\) after cross-multiplication. To isolate \(n\), we divide both sides by 5, yielding \(n = \frac{45}{5}\), which simplifies to \(n = 9\). Solving algebraic equations provides the foundation for much of algebra and is a skill that's applied in many areas of mathematics and beyond.