Problem 24 - Prealgebra

Question

Divide the numerator and the denominator of each of the following fractions by 2 . $$\frac{10}{12}$$

Step-by-Step Solution

Answer

The fraction \( \frac{10}{12} \) simplifies to \( \frac{5}{6} \).

Step 1: Identify Numerator and Denominator

In the fraction \( \frac{10}{12} \), the numerator is 10 and the denominator is 12.

Step 2: Divide the Numerator by 2

Divide the numerator, which is 10, by 2. \( 10 \div 2 = 5 \). This becomes the new numerator.

Step 3: Divide the Denominator by 2

Divide the denominator, which is 12, by 2. \( 12 \div 2 = 6 \). This becomes the new denominator.

Step 4: Form the New Fraction

Using the results from the previous steps, form the new fraction. The fraction is now \( \frac{5}{6} \).

Key Concepts

Numerator

In a fraction, the numerator is the top number. It shows how many parts of a whole we have. For instance, in the fraction \( \frac{10}{12} \), the numerator is 10. This means we have 10 parts of something that is divided into 12 equal parts.
You can think of the numerator as the "counting" number in a fraction. It's similar to counting the number of pieces of pizza you have, while the denominator tells you what kind of slice each piece is.
This value is crucial when simplifying or adjusting fractions, as any change in the numerator will directly affect the overall value of the fraction.

Denominator

The denominator is the bottom number in a fraction, and it tells us into how many equal parts the whole is divided. In \( \frac{10}{12} \), the denominator is 12. This implies that the whole (let’s say a pie) has been divided into 12 equal pieces.
The denominator sets the "type" or "size" of each piece. Just like the number of total pages in a book tells us about the complexity of the book, the denominator conveys the scale of each fraction part.
Understanding the denominator helps in understanding why fractions can mean different things even if the numerators are the same. For example, \( \frac{1}{2} \) is not the same as \( \frac{1}{3} \), since the halves are larger than thirds.

Division

Division is a mathematical process used to break down numbers into smaller, equal parts. When we divide in mathematics, we are essentially asking how many times a number can be subtracted from another.
For fractions, division plays a key role when we simplify them by dividing both the numerator and the denominator by the same number. This maintains the equality because we are scaling down both components of the fraction evenly.
In our exercise, we divided the numerator 10 by 2 to get 5, and the denominator 12 by 2 to get 6. This step of division has helped us find a simpler form of our original fraction, without changing its value.

Fraction Reduction

Fraction reduction, also known as simplifying fractions, involves making a fraction as simple as possible. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD).
The fraction \( \frac{10}{12} \) is simplified by dividing both numbers by 2, which gives \( \frac{5}{6} \).

• We began by identifying the GCD of the numerator and denominator. In this case, it was 2.
• Then, we divided both the top and bottom by this number.

The result is a fraction in its simplest form, meaning it can't be simplified further using whole numbers. Simplifying fractions is an important skill because it makes numbers easier to understand and compare.