Question

Rewrite each fraction or mixed number in lowest terms. \(\frac{9}{24}\)

Short answer

\( \frac{3}{8} \)

Step-by-step solution

Identify the Greatest Common Divisor (GCD)

Find the greatest common divisor of the numerator and the denominator. The GCD of 9 and 24 is 3.

Divide Both Numerator and Denominator by the GCD

Divide the numerator and the denominator by their GCD. \( \frac{9 \text{ (numerator)}}{3} \text{ ÷ } \frac{24 \text{ (denominator)}}{3} = \frac{3}{8} \)

Write the Final Simplified Fraction

The fraction \( \frac{9}{24} \) simplified to its lowest terms is \( \frac{3}{8} \).

Key concepts

greatest common divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. Think of it as the biggest number they both can share evenly. For example, in the fraction \(\frac{9}{24}\), both 9 and 24 can be divided by 3 without leaving leftovers. Step 1 in simplifying fractions is finding this GCD. To do this:

• List out the factors of each number
• Identify the largest factor they both share

In our example:
Factors of 9: 1, 3, 9
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
The greatest common factor is 3, so the GCD of 9 and 24 is 3.

numerator

The numerator is the top part of a fraction. It represents how many parts of the whole we are considering. For instance, in the fraction \(\frac{9}{24}\), the number 9 is the numerator. It tells us that we are looking at 9 parts out of a total of 24. When simplifying fractions, you'll often divide the numerator by the GCD to make the fraction simpler.

• In \(\frac{9}{24}\), we divide the numerator (9) by the GCD (3)
• This gives us \(\frac{9}{3} = 3\)

Thus, the numerator in the simplified fraction \(\frac{3}{8}\) is 3.

denominator

The denominator is the bottom part of a fraction. It represents the total number of equal parts the whole is divided into. In the case of the fraction \(\frac{9}{24}\), 24 is the denominator. It tells us there are 24 equal parts in total.

When simplifying fractions, you'll divide the denominator by the GCD to reduce the fraction to its simplest form. Let's consider our example again:

• The denominator in \(\frac{9}{24}\) is 24
• We divide 24 by the GCD, which is 3
• This gives us \(\frac{24}{3} = 8\)

So, the denominator in the simplified fraction \(\frac{3}{8}\) is 8.

lowest terms

When a fraction is in its lowest terms, it means it can't be simplified any more. Both the numerator and the denominator have no common factors other than 1. This makes the fraction easier to understand and work with.

To put a fraction in its lowest terms:

• Find the GCD of the numerator and denominator
• Divide both the numerator and the denominator by this GCD

In the example \(\frac{9}{24}\):

• We found the GCD to be 3
• Divided the numerator (9) and the denominator (24) by 3
• This gave us the simplified fraction \(\frac{3}{8}\)

Since 3 and 8 share no common factors other than 1, \(\frac{3}{8}\) is in its lowest terms.