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A fraction represents a part of a whole. It consists of a numerator and a denominator, where the numerator indicates how many parts are taken from the whole, and the denominator shows into how many equal parts the whole is divided. For example, in the fraction \( \frac{9}{10} \), the numerator is 9 and the denominator is 10, meaning we have 9 parts of something that is divided into 10 equal parts.

When working with fractions, it's often necessary to create fractions that have the same value but look different on paper. These are called equivalent fractions. They may represent the same portion of a whole, but they have different numerators and denominators. Knowing how to find equivalent fractions is essential in many areas of mathematics, including solving problems that involve fractions of different denominators.

The 'scale factor' is a term used in mathematics to describe the multiplier necessary to increase or decrease the size of an object or a number without changing its shape or essence. When dealing with fractions, the scale factor is the number by which both the numerator and the denominator are multiplied to find an equivalent fraction.

For instance, in the exercise \( \frac{9}{10} = \frac{ }{80} \), to find the equivalent fraction with 80 as the denominator, we need to determine the scale factor that transforms the denominator 10 into 80. This factor is 8 because \( 10 \times 8 = 80 \). If we multiply both the numerator and the denominator by 8, we ensure that the value of the fraction remains unchanged, which is the key to finding equivalent fractions.

The numerator and denominator of a fraction are like partners in a dance—they must move together to maintain the fraction's value. The numerator, the top part of the fraction, tells you how many parts you have. The denominator, the bottom part, tells you the size of those parts (the number of parts the whole is divided into).

When you multiply the numerator and denominator by the same number (the scale factor), you're changing the size of the 'dance floor' without changing the 'dance' itself. This means the relationship between the numerator and denominator stays the same, and thus, the value of the fraction remains constant. In our previous example, we found that \( \frac{9}{10} = \frac{72}{80} \) by multiplying both the numerator (9) and the denominator (10) by the scale factor (8). The ratio of the numerator to the denominator, and thereby the value of the fraction, does not change.