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Ratios are a way of expressing a relationship between two numbers, indicating how many times one value contains or is contained within the other. In the exercise, the sentence "5 is to 6 as 15 is to \(x\)" highlights two important ratios. The ratio of 5 to 6 can be written as \(\frac{5}{6}\), and similarly, the ratio of 15 to \(x\) is \(\frac{15}{x}\).

Ratios serve as the foundation for proportions. When two ratios are equal, they form a proportion. This is the case in the exercise as it assumes the ratio \(\frac{5}{6}\) is equal to the ratio \(\frac{15}{x}\).

Understanding ratios and proportions can be helpful in many different real-life situations, such as cooking recipes, mixing paints, and comparing distances. By comparing different amounts in a way that makes sense, we can better understand and manage relationships between numbers.

Equivalent fractions are different fractions that represent the same portion of a whole. In a proportion, like \( \frac{5}{6} = \frac{15}{x} \), the idea is that the two fractions are equivalent, meaning they have the same value when simplified or calculated.

To check if two fractions are equivalent, you can cross-multiply. In the context of our exercise, this means multiplying 5 by \(x\) and 6 by 15. If these two products are equal, the fractions are equivalent, confirming the proportion:

• \(5 \times x = 6 \times 15\)

Dealing with equivalent fractions helps us in understanding and solving mathematical problems involving proportions by allowing us to find unknown values and compare different relationships.

A mathematical equation is a statement that asserts the equality of two expressions. In our exercise, we set up the equation \( \frac{5}{6} = \frac{15}{x} \) to represent the given proportion.

To solve for \(x\), we utilize the cross-multiplication method because it helps us eliminate the fractions, making it easier to find unknown values. Cross-multiplying means:

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Multiply the denominator of the first fraction by the numerator of the second fraction.

For this exercise, it becomes:

• \(5 \times x = 6 \times 15\)

This equation simplifies to:

• \(5x = 90\)

Afterward, to find the value of \(x\), divide both sides by 5 resulting in \(x = 18\). By setting up mathematical equations like this one, we can solve various real-world problems efficiently and accurately.