Problem 78 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{-1}{9}=\frac{k}{3} $$

Step-by-Step Solution

Answer

The value of \( k \) is \( \frac{-1}{3} \).

Step 1: Understand the Problem

We are given the equation \( \frac{-1}{9} = \frac{k}{3} \). Our goal is to find the value of \( k \).

Step 2: Cross-Multiplication

Cross-multiply the fractions to eliminate the denominators. This gives us:\[ -1 imes 3 = k imes 9 \]So, \( -3 = 9k \).

Step 3: Solve for k

To solve for \( k \), divide both sides of the equation by 9:\[ k = \frac{-3}{9} \]Simplify the fraction:\[ k = \frac{-1}{3} \]

Step 4: Conclusion

The solution to the equation \( \frac{-1}{9} = \frac{k}{3} \) is \( k = \frac{-1}{3} \).

Key Concepts

Cross-Multiplication

Cross-multiplication is a technique used to solve equations that involve fractions. It helps us eliminate denominators by turning them into simple multiplication problems. This process is particularly useful because it allows us to work with whole numbers instead of fractions, making calculations easier and more straightforward. Here is how cross-multiplication works:

• Given two fractions set equal to each other, such as \( \frac{a}{b} = \frac{c}{d} \), you multiply the numerator of each fraction by the denominator of the opposite fraction.
• This gives you the equation: \( a \times d = c \times b \).
• This new equation no longer has denominators, simplifying the problem into a simple equation of numbers or variables on both sides.

This method streamlines solving, as seen in the example \( \frac{-1}{9} = \frac{k}{3} \). By cross-multiplying, we obtained \( -1 \times 3 = k \times 9 \), simplifying our work greatly.

Fractions

Fractions are numbers that represent a part of a whole. They consist of a numerator, which is the top number, and a denominator, the bottom number. The numerator indicates how many parts we have, while the denominator shows into how many equal parts the whole is divided. Fractions are essential in mathematics, as they provide precision in representing parts of a whole object, time, or any divisible entity. When working with equations involving fractions, familiarizing oneself with their properties and operations is crucial. Some key points to remember about fractions are:

• Fractions with the same denominator can be easily added or subtracted by operating directly on the numerators.
• Multiplying fractions involves multiplying the numerators together and denominators together.
• Division of fractions is executed by multiplying by the reciprocal of the divisor fraction.

Understanding fractions and their properties is vital when we're cross-multiplying in the context of solving equations, as it helps simplify complex mathematical relationships into manageable steps.

Simplifying Fractions

Simplifying fractions means expressing them in their simplest form, where the numerator and denominator share no common factors other than 1. This process is essential to reduce fractions to their smallest possible representation, making them clearer and easier to interpret.Here's how to simplify fractions:

• Identify any common factors between the numerator and denominator.
• Divide both the numerator and the denominator by their greatest common factor (GCF).
• Check if the fraction can be further reduced, ensuring it's in its simplest form.

In the equation \( \frac{-3}{9} \), the fraction was simplified to \( \frac{-1}{3} \). This was achieved by recognizing that 3 is a factor of both 3 and 9, and dividing both by 3 yields the simplest form. Simplifying fractions is a crucial skill, especially when dealing with algebraic equations, as it often represents the final step in arriving at the most concise answer.