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3.1.1 Ratios and Ratio Notation

Understanding ratios as comparisons between quantities and expressing them in different forms (a:b, a/b, a to b).

定义

A ratio is a comparison between two quantities, showing the relative size of one quantity to another. Ratios can be expressed in three equivalent forms: (1) using a colon notation \(a:b\), (2) as a fraction \(\frac{a}{b}\), or (3) using the word "to" as \(a \text{ to } b\). All three forms represent the same relationship. For example, if there are 3 red balls and 5 blue balls, the ratio of red to blue is \(3:5\), which can also be written as \(\frac{3}{5}\) or "3 to 5". Ratios are typically expressed in simplest form by dividing both quantities by their greatest common divisor (GCD). A ratio \(a:b\) can be scaled by multiplying both terms by the same non-zero constant \(k\), giving the equivalent ratio \(ka:kb\). Ratios are dimensionless when comparing quantities of the same type, but can have units when comparing different types of quantities (such as miles per hour).

核心公式

  • \(\text{Ratio} = a:b = \frac{a}{b} = a \text{ to } b\)
  • \(\text{Equivalent Ratio: } a:b = ka:kb \text{ for any non-zero constant } k\)
  • \(\text{Simplest Form: } \frac{a}{b} = \frac{a \div \gcd(a,b)}{b \div \gcd(a,b)}\)
  • \(\text{If } a:b = c:d, \text{ then } ad = bc \text{ (Cross-multiplication property)}\)
  • \(\text{Scaling a ratio: If the ratio is } a:b \text{ and we scale by factor } k, \text{ the new quantities are } ka \text{ and } kb\)

易错点

  • ⚠️ Confusing the order of quantities in a ratio: Students often reverse the order (writing 5:3 instead of 3:5) when the problem specifies 'the ratio of red to blue.' Always pay careful attention to the order specified in the problem statement.
  • ⚠️ Failing to simplify ratios to lowest terms: Many students leave ratios like 6:9 unsimplified instead of reducing to 2:3. AP exams expect ratios in simplest form unless otherwise specified.
  • ⚠️ Mixing up ratios with fractions of the whole: A ratio 3:5 does NOT mean 3/8 of the total. The ratio compares two parts to each other, not a part to the whole. Students often incorrectly interpret 3:5 as '3 out of 8 total.'
  • ⚠️ Incorrectly applying ratio scaling: When scaling a ratio by a factor, students sometimes multiply only one term instead of both. Remember that equivalent ratios require multiplying BOTH terms by the same constant.