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Ratios are a way to compare two quantities by showing the relative size of one quantity to the other. They express a relationship, showing how much of one thing there is compared to another. Ratios can be written in several different forms: for example, as "3 to 4," "3:4," or "\( \frac{3}{4} \)."

Understanding ratios is key in interpreting data, making predictions, and solving problems in mathematics. For instance, in our exercise, the ratio \( \frac{16}{n} \) is set equal to \( \frac{25}{40} \), indicating that their values form a proportion and can be solved similarly. When ratios are equal, as they are in this problem, we can use cross multiplication to solve for the unknown variable."

Fractions describe how a part of something relates to the whole. They consist of two numbers: the numerator (top number) and the denominator (bottom number). In our exercise, fractions are central to understanding the relationships between the numbers provided: \( \frac{16}{n} \) and \( \frac{25}{40} \).

Fractions are useful in comparing sizes, performing arithmetic, and tackling algebraic expressions. Simplifying fractions makes them more manageable—a process seen when reducing \( \frac{25}{40} \) by dividing both the numerator and denominator by their greatest common divisor. Recognizing when and how to perform operations with fractions, like cross-multiplying, helps in efficiently solving equations.

Cross multiplication is a handy technique for solving proportions, where two ratios are set to be equal. This method involves multiplying the numerator of one fraction by the denominator of the other, across the "equals" sign. In our example, we have \( \frac{16}{n} = \frac{25}{40} \).

• First, cross multiply: \( 16 \times 40 = 25 \times n \).
• This results in the equation \( 640 = 25n \).
• Next, isolate the variable by dividing both sides by 25: \( n = \frac{640}{25} \).
• Calculate the division to find \( n \approx 25.60 \).

Understanding cross multiplication enables you to solve for unknowns within fraction-based equations quickly, using straightforward arithmetic and algebra principles. Remember to always round your final answer if required by the problem, as seen in rounding to the nearest hundredth.