Question

Write each ratio in simplest form. \(\frac{4}{2}\)

Short answer

The simplified form of \( \frac{4}{2} \) is \( \frac{2}{1} \).

Step-by-step solution

Identify the Common Factor

To simplify the ratio \( \frac{4}{2} \), we need to find the greatest common factor (GCF) of the numerator (4) and the denominator (2). Both numbers can be divided by their shared factors.

Divide the Numerator and Denominator by the GCF

The GCF of 4 and 2 is 2. Divide both the numerator and the denominator by this factor: \( \frac{4}{2} = \frac{4 \div 2}{2 \div 2} = \frac{2}{1} \).

Write the Simplified Ratio

The simplified form of the ratio \( \frac{4}{2} \) is \( \frac{2}{1} \). This means the original ratio, when simplified, represents two units of the numerator for every one unit of the denominator.

Key concepts

greatest common factor

When simplifying a ratio, the greatest common factor (GCF) is your best friend. The GCF is the largest number that divides evenly into both the numerator and the denominator. In other words, it's the biggest number they both have in common. Finding the GCF is crucial because it helps you simplify the ratio to its smallest form.

To find the GCF:

• List the factors of each number. Factors are numbers that can be multiplied together to get the original number.
• Identify the common factors between the two numbers.
• The greatest number among these is your GCF.

For example, in the ratio \( \frac{4}{2} \), the factors of 4 are 1, 2, and 4. The factors of 2 are 1 and 2. The common factors are 1 and 2, and the greatest one is 2. That makes 2 the GCF in this case. Once found, the GCF helps break down the ratio to its simplest terms by dividing both parts of the ratio by this number.

numerator

The numerator is the top number in a ratio or a fraction. It signifies how many parts or units you have. In our specific example, \( \frac{4}{2} \), the numerator is 4. Think of the numerator as the part you are focusing on in any given scenario. It represents the number of parts being considered, counted, or evaluated in the whole set described by the ratio.

To simplify using the numerator:

• Divide the numerator by the GCF found previously.
• This step reduces the numerator to its simplest form by eliminating the common factor it shares with the denominator.

So, when simplifying \( \frac{4}{2} \), we divide the 4 by the GCF of 2 to get 2. This process helps define a clearer and more straightforward ratio, which is easier to interpret and use.

denominator

The denominator is the bottom number of the ratio or fraction and it tells you how many parts make up a whole. In the ratio \( \frac{4}{2} \), the denominator is 2. It sets the stage by specifying how many equal parts your numerator is being divided into. In real-world scenarios, it often represents a total or a group that is being partitioned.

Here's how to handle the denominator when simplifying a ratio:

• Divide the denominator by the same GCF used for the numerator.
• This action ensures that both parts of the ratio are reduced proportionately.

The purpose of simplifying the denominator is to make the ratio barer and reveal the simple relationship it has to the numerator. In our example with \( \frac{4}{2} \), dividing the denominator 2 by the GCF 2 results in 1, simplifying the whole expression to \( \frac{2}{1} \), thus clarifying the underlying proportionality.