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In mathematics, a ratio is a way to compare two quantities or values. It tells you how much of one thing there is in relation to another. For example, if you have a ratio of 1:2, it means for every 1 unit of a particular item, there are 2 units of another. This concept is fundamental in comparing, sharing, and analyzing relationships between numbers. In the given exercise, the problem presents the ratio of \( x \) to 8, which is equal to the ratio of 9 to 6. When these ratios are set equal to each other, they form a proportion. Proportions are equations that express the equality of two ratios, such as \( \frac{x}{8} = \frac{9}{6} \). Solving such proportions often involves finding the missing value (in this case, \( x \)) by using mathematical operations to manipulate the equation. This leads us to our next important technique: cross-multiplication.

The cross-multiplication technique is a powerful method to solve equations that involve proportions. When we write a proportion in the form \( \frac{a}{b} = \frac{c}{d} \), we can cross-multiply by multiplying across the equals sign, creating a new equation: \( a \times d = b \times c \). This method helps to eliminate the fractions in the equation, making it easier to solve for the unknown variable.In the provided exercise example, after simplifying the original ratio \( \frac{9}{6} \) to \( \frac{3}{2} \), we were left with \( \frac{x}{8} = \frac{3}{2} \). Cross-multiplying involves multiplying \( x \) by 2 and 8 by 3, giving us \( x \times 2 = 8 \times 3 \). This results in the linear equation \( 2x = 24 \), which can be solved in simple steps. Use this technique whenever you have a proportion to quickly and efficiently find the unknown value.

Simplifying fractions is a key skill in solving equations that contain ratios. It involves reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).In our exercise, the fraction \( \frac{9}{6} \) can be simplified by finding the GCD of 9 and 6, which is 3. Divide both 9 and 6 by 3 to get \( \frac{3}{2} \). Simplifying ratios before stepping into cross-multiplication makes calculations more manageable and helps in achieving accurate results. By mastering the simplification of fractions, you make solving proportions more straightforward and avoid unnecessary complications in your calculations.