Problem 44 - Prealgebra

Question

Find each quotient. Write in simplest form. $$\frac{5 s}{t} \div \frac{6 r s}{t}$$

Step-by-Step Solution

Answer

The simplest form of the expression is \(\frac{5}{6r}\).

Step 1: Rewrite the Division as Multiplication

To divide by a fraction, we multiply by its reciprocal. Here, rewrite the division of \(\frac{5s}{t}\) by \(\frac{6rs}{t}\) as: \[\frac{5s}{t} \times \frac{t}{6rs}\]

Step 2: Simplify the Expression

Multiply the numerators and the denominators separately: \[\frac{5s \cdot t}{t \cdot 6rs}\] Next, cancel common terms in the numerator and the denominator. Notice both numerator and denominator have a 't'. Cancel 't' from both:\[\frac{5s}{6rs}\]

Step 3: Cancel Common Factors

Now cancel the common factor 's' from the numerator and the denominator:\[\frac{5}{6r}\] There are no additional common factors, so the fraction is now in its simplest form.

Key Concepts

Quotients in Algebra

In algebra, a quotient is the result of dividing one expression by another. When you encounter a division problem involving algebraic expressions, like \( \frac{5s}{t} \div \frac{6rs}{t} \), it is important to understand that the word "quotient" signifies the outcome of this division.

To simplify the problem, we need to tackle the task of division, which ultimately might involve steps like flipping the fraction we are dividing by and multiplying instead. This process will help you determine how the numerator and the denominator relate and how they reduce. Understanding how to manage these operations with fractions and variables helps build a strong foundation for more complex algebraic manipulations.

Fraction Simplification

When simplifying fractions, the goal is to reduce the expression to its simplest form while preserving its value. This means getting rid of any factors that the numerator and the denominator have in common.

For example, in step 2 of the given solution, we transform the division into multiplication using the reciprocal. But, to simplify further, focus on:

• Multiplying the numerators and denominators separately.
• Identifying and canceling common terms.

This reduces the fraction to a simpler, more manageable form. By simplifying fractions, you make calculations easier and results more understandable.

Reciprocal in Algebra

Understanding reciprocals is crucial to mastering algebra. A reciprocal of a fraction is achieved by swapping its numerator and denominator. This means the reciprocal of \(\frac{6rs}{t}\) is \(\frac{t}{6rs}\).

When dividing by fractions, instead of directly performing the division, you multiply by the reciprocal. This transformation simplifies the operation because multiplication is typically easier to handle than division in algebra, especially when dealing with multiple variables.

This concept is regularly used in algebraic simplifications and is essential for reducing complexity in expressions.

Common Factors

Common factors in algebraic expressions are elements found in both the numerator and the denominator that can be divided out to simplify the expression. In the division problem, this step is key. Look at all terms and identify any variables or numbers they share.

For instance, in the example \(\frac{5s}{6rs}\), both the numerator and the denominator share the variable 's'. By canceling 's', you are left with \(\frac{5}{6r}\), simplifying the algebraic expression.

Recognizing and canceling common factors aid in achieving the simplest form of a fraction, making complex problems become straightforward and clear.