Problem 38 - Prealgebra

Question

Simplify. $$\frac{48}{24}$$

Step-by-Step Solution

Answer

The simplified form of \( \frac{48}{24} \) is 2.

Step 1: Find the GCD

The first step is to find the greatest common divisor (GCD) of the numerator and the denominator. For 48 and 24, the GCD is 24 because it is the largest number that divides both 48 and 24 without a remainder.

Step 2: Divide Numerator and Denominator by GCD

Next, divide both the numerator (48) and the denominator (24) by their GCD (24). This yields: \( \frac{48 \div 24}{24 \div 24} = \frac{2}{1} \).

Step 3: Simplify the Fraction

Lastly, simplify the fraction \( \frac{2}{1} \) by recognizing that any number divided by 1 is itself. Therefore, \( \frac{2}{1} = 2 \).

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can divide two or more numbers without leaving a remainder. It's a crucial concept when simplifying fractions. By using the GCD, we ensure that fractions are reduced to their simplest forms.
For instance, when simplifying the fraction \( \frac{48}{24} \), we first need to find the GCD of 48 and 24. In this case, the GCD is 24. This is because 24 is the highest number that divides both 48 and 24 completely, without any remainder.
Understanding how to find the GCD:

• List all the factors of each number.
• Identify the largest factor that appears in both lists.

Finding the GCD helps in reducing the fraction to its simplest form, making it easier to understand and work with.

Numerator and Denominator

The terms numerator and denominator refer to the two parts of a fraction. Understanding these is vital to doing any operation with fractions.
The numerator is the top number in a fraction. It tells you how many parts of a whole you have. In \( \frac{48}{24} \), 48 is the numerator. It means you have 48 parts.
On the other hand, the denominator is the bottom number. It shows the total number of parts that make up a whole. So, in \( \frac{48}{24} \), 24 is the denominator. It means the whole is divided into 24 parts.
Using these concepts is crucial:

• By dividing both the numerator and the denominator by their GCD, we retain the value of the fraction while simplifying it.
• Understanding numerators and denominators allows you to compare and combine different fractions.

This knowledge forms the foundation for fraction operations and simplification.

Simplify Fractions

Simplifying fractions involves reducing them to their simplest form without changing their value. This process often involves finding the GCD of the numerator and denominator.
With our example \( \frac{48}{24} \), after determining the GCD is 24, you simplify by dividing both the numerator and the denominator by this number. This process transforms \( \frac{48}{24} \) into \( \frac{2}{1} \).
Why simplify fractions?

• It makes fractions easier to understand and compare.
• Simplified fractions are crucial in performing mathematical operations like addition and subtraction.
• A simpler form can often reveal properties of the fraction that are not obvious at first glance.

Simplifying fractions not only makes them easier to work with but also provides a clearer insight into their true value, as seen when \( \frac{2}{1} = 2 \).