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Ratios are mathematical expressions used to compare two or more quantities. In simplest terms, a ratio tells us how much of one thing there is compared to another. Imagine you have a recipe that uses 2 cups of flour for every 3 cups of sugar; the ratio is 2:3. In mathematics, ratios are often represented as fractions. For example, the ratio of 2 to 3 can be expressed as \(\frac{2}{3}\).

Ratios allow us to easily compare different things. They can be scaled up or down, meaning that if you double the amount of flour to 4 cups, you need to also double the sugar to 6 cups, keeping the ratio the same, \(\frac{4}{6}\), which simplifies back to \(\frac{2}{3}\). Understanding ratios is crucial for solving problems related to proportions, scaling, and comparisons in everyday life and various fields such as cooking, engineering, and more.

Cross multiplication is a handy method used to determine if two ratios are equal, forming a proportion. This process involves 'cross' multiplying the terms of the two ratios.

For example, if you have two ratios, \(\frac{a}{b}\) and \(\frac{c}{d}\), you can cross multiply by multiplying \(a\) by \(d\) and \(b\) by \(c\). The results of these multiplications are then compared. If the products are equal, the two ratios form a proportion. This can be mathematically expressed as \(a \cdot d = b \cdot c\).

This method is extremely useful because it simplifies the process of comparing whether two ratios are identical by avoiding the need to simplify each fraction. Let's say you want to check if \(\frac{9}{7}\) and \(\frac{81}{70}\) are equivalent. By cross multiplying, \(9 \cdot 70\) and \(7 \cdot 81\), you can directly compare the products to determine their equality.

Equality of ratios, or proportions, is a concept that occurs when two ratios are equivalent. This means that both ratios represent the same relationship between numbers, despite possibly looking different at first glance.

For instance, the ratios \(\frac{1}{2}\) and \(\frac{2}{4}\) might appear different but are equal because they simplify to the same value. To determine if two ratios are equal, you can simplify each ratio to its lowest terms and compare them, or use cross multiplication.

In the given problem, we determine the equality of \(\frac{9}{7}\) and \(\frac{81}{70}\). By cross multiplying, we compare \(9 \times 70\) with \(7 \times 81\). If these products are equal, then the ratios are equal, indicating that the original statement forms a proportion. Ensuring ratios are equal is essential in fields such as geometry, physics, and finance, where maintaining proportionality is crucial. It helps in tasks like scaling models or understanding the speed and distances in physics.