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Ratios are a way to compare two quantities by expressing how many times one value contains or is contained within the other. To calculate a ratio, you simply divide one number by another. This can be particularly useful to understand relationships between different values. For instance, in our earlier exercise, the ratios are found by dividing each consecutive number:

• \( \frac{10}{6} \) to get a ratio connecting 10 to 6.
• \( \frac{15}{10} \) to connect 15 to 10.
• \( \frac{25}{15} \) to join 25 to 15.

These ratios help us understand whether these numbers are in proportion to one another. It's all about measuring and understanding the size of the relationship itself, rather than the specific numbers involved.

Simplification of ratios is all about making these comparisons clearer and more straightforward. By breaking down the ratios to their simplest form, it's easier to evaluate their equivalency or inequality. Let's take a closer look at simplifying the fractions identified:

• \( \frac{10}{6} \) simplifies to \( \frac{5}{3} \) or approximately 1.67 when converted to a decimal.
• \( \frac{15}{10} \) simplifies to \( \frac{3}{2} \) or 1.5 as a decimal.
• \( \frac{25}{15} \) simplifies again to \( \frac{5}{3} \) or roughly 1.67 when expressed in decimal form.

To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor. This process gives a new fraction equivalent to the original. Simplified ratios allow us to compare these values without the clutter of larger numbers.

In the context of determining proportional relationships, comparing fractions is crucial. When checking for proportionality, we assess whether the simplified ratios are the same across the series of values. If the ratios are identical, the numbers are in proportion. Otherwise, they are not.
In our example, after simplification, the ratios were \( \frac{5}{3} \), \( \frac{3}{2} \), and \( \frac{5}{3} \). Clearly, not all the ratios are identical: 1.67 does not equal 1.5.
Therefore, the numbers \(6, 10, 15,\) and \(25\) are not proportional. By comparing each simplified ratio, we verify equivalency. Remember:

• For proportionality, all simplified ratios must match.
• Even a slight difference, as seen here, means the set isn't proportional.

This method of comparison is key to understanding the broader concept of proportionality.