Problem 43 - Prealgebra

Question

Find each quotient. Write in simplest form. $$\frac{c}{8} \div \frac{c d}{5}$$

Step-by-Step Solution

Answer

The simplest form of the quotient is \( \frac{5}{8d} \).

Step 1: Understand Division of Fractions

When dividing by a fraction, you multiply by its reciprocal. The original expression is \( \frac{c}{8} \div \frac{cd}{5} \). This can be rewritten as \( \frac{c}{8} \times \frac{5}{cd} \).

Step 2: Multiply the Fractions

To multiply the fractions, multiply the numerators together and the denominators together. This gives us \( \frac{c \times 5}{8 \times cd} = \frac{5c}{8cd} \).

Step 3: Simplify the Expression

Simplify the fraction by canceling out common factors in the numerator and the denominator. The common factor here is \( c \), so \( \frac{5c}{8cd} = \frac{5}{8d} \).

Step 4: Write the Simplified Form

The simplest form of the quotient \( \frac{c}{8} \div \frac{cd}{5} \) is \( \frac{5}{8d} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is a fundamental concept in fraction arithmetic.
The idea is to reduce a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. The process involves:

• Identifying common factors in the numerator and the denominator.
• Dividing both the numerator and denominator by their greatest common factor (GCF).

For instance, in the step-by-step solution of the exercise, the fraction \( \frac{5c}{8cd} \) is simplified by canceling the common factor \(c\), resulting in \( \frac{5}{8d} \). This makes the fraction easier to work with and interpret.
Remember that when simplifying fractions, if one or both terms become 1, you generally do not need to explicitly write them unless clarity calls for it.

Reciprocal in Division

When dividing fractions, the key is to use the reciprocal. The reciprocal of a fraction is switching its numerator and denominator. For example, the reciprocal of \( \frac{cd}{5} \) is \( \frac{5}{cd} \).
In the division process, you convert the division of two fractions into a multiplication problem by multiplying the first fraction by the reciprocal of the second.

• This changes the expression from \( \frac{c}{8} \div \frac{cd}{5} \) to \( \frac{c}{8} \times \frac{5}{cd} \).
• Now you have a standard fraction multiplication problem, which is typically easier to handle.

The use of reciprocals this way provides a straightforward method to solve division problems in fraction arithmetic. Always remember, when dividing fractions, "flip and multiply."

Fraction Multiplication

The multiplication of fractions involves an easy, two-step process:

• Multiply the numerators of the fractions to get the new numerator.
• Multiply the denominators of the fractions to get the new denominator.

For example, using the division problem rewritten as a multiplication problem \( \frac{c}{8} \times \frac{5}{cd} \), you multiply the numerators: \( c \times 5 = 5c \), and the denominators: \( 8 \times cd = 8cd \).
Thus, forming the product \( \frac{5c}{8cd} \). This process keeps fractions straightforward and is a core technique in fraction arithmetic.
Ensuring you correctly multiply fractions is crucial for accurate results, setting the stage for any further simplification.

Common Factors in Fractions

Identifying and canceling common factors is key to simplifying fractions. Common factors are numbers or algebraic terms that divide both the numerator and the denominator evenly.
To simplify a fraction, you look for these common terms.

• For example, with the fraction \( \frac{5c}{8cd} \), \( c \) is a common factor.
• You divide both the numerator and the denominator by \( c \) to simplify the fraction, resulting in \( \frac{5}{8d} \).

This approach makes fractions cleaner and more accessible for additional operations.
Always check for any factors that can be eliminated to make arithmetic more manageable. Common factors make a significant appearance throughout math, simplifying expressions alike.