Problem 25 - Prealgebra

Question

Divide the numerator and the denominator of each of the following fractions by 2 . $$\frac{86}{94}$$

Step-by-Step Solution

Answer

The simplified fraction is \( \frac{43}{47} \).

Step 1: Identify the Numerator and the Denominator

The given fraction is \( \frac{86}{94} \). Here, 86 is the numerator, and 94 is the denominator.

Step 2: Divide the Numerator by 2

Take the numerator, which is 86, and divide it by 2. \[ 86 \div 2 = 43 \] The result is 43.

Step 3: Divide the Denominator by 2

Take the denominator, which is 94, and divide it by 2. \[ 94 \div 2 = 47 \] The result is 47.

Step 4: Write the Reduced Fraction

After dividing both the numerator and the denominator by 2, the new fraction is \( \frac{43}{47} \). This fraction is simplified.

Key Concepts

Understanding Numerator and Denominator

In any fraction, there are two key components: the numerator and the denominator. A fraction, like \( \frac{86}{94} \), represents a division between two numbers:

• The numerator is the top part of the fraction. It indicates how many parts we have.
• The denominator is the bottom part of the fraction. It indicates the total number of equal parts that make up a whole set.

In our fraction \( \frac{86}{94} \), 86 is the numerator, and 94 is the denominator. Understanding these roles helps us work with the fraction, whether we're simplifying, adding, subtracting, multiplying, or dividing fractions. When the numerator is larger than the denominator, it’s an improper fraction, which can sometimes be a separate point of confusion.

Dividing Fractions Effectively

Dividing fractions can initially seem tricky, but it gets much simpler with practice. To simplify a fraction, like \( \frac{86}{94} \), we identify a common number that can divide both the numerator and the denominator evenly. In this case, that number is 2.

• Step 1: Divide the numerator (86) by 2, which gives you 43.
• Step 2: Divide the denominator (94) by 2, resulting in 47.

Thus, \( \frac{86}{94} \) simplifies to \( \frac{43}{47} \). By doing this, we reduce the fraction to its simplest form, which often makes it easier to work with in mathematical operations. Simplification doesn’t change the value of the fraction, it simply provides a cleaner, more understandable version.

Exploring Proper Fractions

A proper fraction is a type of fraction where the numerator is less than the denominator. This means the fraction represents a value less than one. In our simplified example, \( \frac{43}{47} \), we have a proper fraction since 43 is indeed smaller than 47.

• Proper fractions are always less than 1, representing parts of a whole.
• They contrast with improper fractions, which have numerators greater than or equal to denominators, often representing values equal to or greater than one.

Understanding proper fractions is crucial when performing various mathematical operations and comparing sizes of different fractional values. They help in visualizing parts of a whole more effectively.