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Ratios are a way to compare two quantities by using division. To simplify a ratio means to make this comparison as straightforward as possible by reducing it to its smallest whole number terms. Let's look at the concept of simplifying ratios step by step.

When you have a ratio, it might not always be in its simplest form. For example, a ratio like 65:15 can be reduced. To simplify, you need to find the common factors of the two numbers in the ratio. Identifying a common factor helps reduce the numbers proportionately, keeping the relationship between them the same.

Let's say you find a common factor, like 5 in our example. Divide both numbers in the ratio by 5. Hence, 65:15 gets simplified to 13:3. Simplifying ratios makes it easier to understand and work with the quantities involved because smaller numbers are easier to handle and compare.

Decimals and fractions can sometimes seem tricky, but converting them is a manageable process. When asked to express decimals as a ratio, the first task is to get rid of the decimals by converting them into whole numbers, which makes them easier to compare.

In our scenario with the decimals 0.65 and 0.15, the goal is to express them as a simple ratio. Since both have two decimal places, the easiest method is to multiply each number by 100, getting rid of the decimals altogether. This results in 65 and 15, respectively.

Now, these numbers can be worked with as a simple ratio. Removing the decimals clears the pathway for further operations, like simplification, allowing you to see the numbers' relationship without the visual barrier of decimal points.

The greatest common divisor, or GCD, is an essential tool in simplifying ratios. It is the largest factor that two numbers share. The GCD allows you to reduce ratios effectively, ensuring they are in their simplest form.

For example, in the ratio 65:15, you need to find a common factor that both numbers can be divided by evenly. The process involves listing the factors of each number and identifying the greatest factor they share. For 65 and 15, this factor is 5.

By dividing both numbers in the ratio by this GCD (5), you end up with a simplified form: 13:3. This method ensures efficiency and accuracy when simplifying ratios. Using the GCD not only makes the ratio easier to work with but also retains the exact proportionate relationship between the numbers.