Expressing a Ratio in Simplest Form

To express a ratio in its simplest form, we need to find the greatest common divisor (GCD) or greatest common factor (GCF) of the given numbers and divide both the numerator and denominator of the ratio by this factor. This process eliminates any common factors between the numerator and denominator, resulting in the simplest form of the ratio.

Step-by-Step Explanation:

1. Understand the Ratio: First, we need to understand the ratio we are given. A ratio is a comparison of two numbers or quantities and is often expressed using a colon (:) or a fraction. For example, if we have a ratio of 2:3, it means that the first quantity is twice the second quantity.

2. Find the GCD: Next, we need to find the greatest common divisor (GCD) or greatest common factor (GCF) of the two numbers in the ratio. The GCD is the largest number that divides both numbers without leaving a remainder. This can be found using various methods such as prime factorization, listing factors, or using the Euclidean algorithm.

3. Divide both numbers: Once we have determined the GCD, we divide both the numerator and denominator of the ratio by this factor. This ensures that the ratio is expressed in its simplest form.

4. Final Ratio: The resulting ratio, obtained after dividing both numbers by their GCD, is the simplest form of the given ratio. It is important to note that the value of the ratio remains the same, but it is expressed in a more concise and simplified manner.

Example:

Let's consider the ratio 6:9.

Step 1: Understand the Ratio
The given ratio is 6:9, which means that the first quantity is 6 and the second quantity is 9.

Step 2: Find the GCD
To find the GCD of 6 and 9, we can use the prime factorization method. The prime factors of 6 are 2 and 3, while the prime factors of 9 are 3 and 3. The common prime factors are 3 and 3. Therefore, the GCD is 3.

Step 3: Divide both numbers
We divide both the numerator (6) and denominator (9) by the GCD (3).
6 ÷ 3 = 2
9 ÷ 3 = 3

Step 4: Final Ratio
The simplest form of the ratio 6:9 is 2:3.

Summary:

Expressing a ratio in its simplest form involves finding the GCD of the given numbers and dividing both the numerator and denominator by this factor. This process simplifies the ratio by eliminating any common factors. Remember that the value of the ratio remains the same, but it is expressed in a more concise and simplified manner.

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