Problem 21 - Algebra A Combined Function

Question

Simplify each fraction. See Examples 8 through \(10 .\) $$ \frac{3}{7} $$

Step-by-Step Solution

Answer

The fraction \( \frac{3}{7} \) is already in its simplest form.

Step 1: Identify the Numerator and Denominator

For the fraction \( \frac{3}{7} \), the numerator is \(3\), and the denominator is \(7\).

Step 2: Determine the Greatest Common Divisor (GCD)

Identify the greatest common divisor of \(3\) and \(7\). Since both numbers are prime and have a GCD of \(1\), no further reduction is possible.

Step 3: Simplify the Fraction (if needed)

Divide both the numerator and the denominator by their GCD. Since the GCD is \(1\), the fraction \( \frac{3}{7} \) remains the same.

Key Concepts

Numerator

When we look at a fraction, the number above the line is called the numerator. For the fraction \( \frac{3}{7} \), the numerator is the number \(3\). This number represents the part of a whole that you have. Imagine slicing a pizza into an equal number of pieces. If you eat three of those slices, the number \(3\) is the numerator, telling you how many parts are being considered or taken.

• The numerator indicates the portion of the whole.
• It is always placed above the fraction line.
• In our example \(3\), it tells us about the parts we have out of a total.

Remember, changing the numerator changes how much of the whole you are considering. If you ate another slice, the numerator would increase, changing the fraction to represent the new amount.

Denominator

The denominator is the number below the fraction line. It shows into how many equal parts the whole is divided. For \( \frac{3}{7} \), the denominator is \(7\). This number tells you the total number of equal slices that could exist in our pizza example.

• The denominator gives the total possible size of the pie.
• It sets the context for what the numerator is counting.
• In \( \frac{3}{7} \), \(7\) means the whole item is divided into seven equal parts.

The denominator stays constant unless the whole is divided differently. For instance, if the pizza had been cut into 8 slices instead of 7, the denominator would be \(8\). This changes the context of the fraction, making it essential for accurately communicating the portion you are considering.

Greatest Common Divisor

The greatest common divisor (GCD) is a key tool in simplifying fractions. It is the largest number that can perfectly divide both the numerator and the denominator without leaving any remainder. In mathematical terms, the GCD of two numbers is their largest shared factor. For instance, when simplifying \( \frac{3}{7} \), the GCD of \(3\) and \(7\) is \(1\) because both numbers are prime and don't share any other factors beyond \(1\).

• The GCD helps determine if a fraction can be reduced further.
• It is especially useful in cases where numbers are large or complex.
• For prime numbers like \(3\) and \(7\), the GCD will always be \(1\).

Finding the GCD means looking for common factors between the numerator and denominator. If the GCD is more than \(1\), you can divide both by this number to simplify the fraction. However, if it's \(1\), as in our example, the fraction is already in its simplest form. Simplifying fractions makes them easier to work with and can reveal simple, elegant mathematical relationships.