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In mathematics, a ratio is a way to compare two quantities by showing the relative size of one quantity to another. A ratio can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are the two quantities being compared. For example, in the exercise above, the ratio of the numbers 1 and 2 is written as \( \frac{1}{2} \). This means that 1 is half the size of 2.
Understanding ratios is essential for solving problems involving comparisons between different quantities. Ratios can be simplified by dividing both terms by their greatest common divisor. For instance, the ratio \( \frac{10}{20} \) can be simplified to \( \frac{1}{2} \), since both 10 and 20 can be divided by 10.

Proportional pairs are pairs of numbers that maintain the same ratio. To determine if two pairs of numbers are proportional, you compare their ratios. In the exercise, the pairs 1, 2 and 10, 20 are considered proportional because their ratios are equal:
\(\frac{1}{2} = \frac{10}{20} \). This equivalence shows that both pairs form the same fraction.
Proportionality is used in various aspects of mathematics, including scaling, finding missing values in similar figures, and solving real-world problems. When two pairs are proportional, it means that the relationship between the numbers in one pair is the same as the relationship in the other pair.

Comparing fractions involves determining whether two fractions represent the same value. In the exercise example, we compared the fractions \( \frac{1}{2} \) and \( \frac{10}{20} \). By simplifying \( \frac{10}{20} \) to \( \frac{1}{2} \), we see that they are equal. This process shows that the two fractions represent the same part-to-whole relationship.
To compare fractions effectively, it's often helpful to simplify them to their lowest terms. This method can reveal whether two fractions are indeed equivalent. If the simplified forms of two fractions are identical, they are the same. On the other hand, if they are different, the fractions are not equivalent. Simplification helps in making these comparisons clear and understandable.