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Chapter 7: Proportional Reasoning
7.2 Proportional Variation and Solving Proportions
7.2.1. Recognizing proportional variation
7.2.1.1. ratio remains constant – Fundamental Theorem of Fractions idea
7.2.1.2. Definition of quantities varying proportionally: Two quantities vary
proportionally if and only if, as their corresponding values increase or decrease,
the ratios of the two quantities are always equivalent
7.2.1.3. Tables are an effective means of showing proportional variation
7.2.2. Characteristics of quantities that vary proportionally
7.2.2.1. A multiplicative relationship
ac
7.2.2.1.1. another use of the Fundamental Theorem of Fractions - ba  bc
7.2.2.1.2. Tables helpful in seeing these relationships
7.2.2.1.3. Multiplicative property of quantities that vary proportionally: When
quantities a and b vary proportionally, a nonzero number k exists such that
a
 k , or a  b  k for all corresponding values a and b. k is called the
b
constant of proportionality
7.2.2.1.4. This type of variation is called direct proportionality
7.2.2.2. A constant change relationship
7.2.2.2.1. numerator increases by causes constant increase in denominator
7.2.2.2.2. Constant change property of quantities that vary proportionally: When
quantities a and b vary proportionally and the ratio of a to b is 1 to n, a unit
change in a value of a always evokes a constant change of n in the
corresponding value of b that is – unit changes in a result in constant changes
in b
7.2.2.3. The idea of a proportion
7.2.2.3.1. setting two ratios equal to each other produces an equation known as a
proportion
7.2.2.3.2. Definition of a proportion: A proportion is an equation stating that two
ratios are equivalent
7.2.3. Proportional variation in geometry
7.2.3.1. Similar figures and proportional variation
7.2.3.1.1. Geometric figures with the same shape are called similar figures
7.2.3.1.2. Lengths of corresponding sides of similar figures are proportional to each
other
7.2.3.1.3. These relationships are used to solve many different types of problems
7.2.3.1.4. All squares are similar
7.2.3.1.5. All equilateral triangles are similar
7.2.3.1.6. Any two regular polygons are similar
7.2.3.1.7. Only some rectangles are similar
7.2.3.1.8. See figure 7.1 and 7.2 p. 384-5
7.2.3.1.9. See example 7.8 p. 386
7.2.4. Properties of proportions
7.2.4.1. First property of proportions: Cross-Product property of proportions: For
integers a, b, c, and d (b  d  0), ba  cd if and only if ad = bc
7.2.4.2. Second property of proportions: Reciprocal property of proportions: For
nonzero integers a, b, c, and d, ba  cd if and only if ba  cd
7.2.5. Solving proportional problems
7.2.5.1. can be solved in a variety of ways
7.2.5.2. methods are related but require different procedures
7.2.5.3. Which method works best depends on what is needed and of course your
comfort level in using that method
7.2.5.4. Using properties of equations to solve proportions
7.2.5.4.1. multiply both sides of the proportion by the same number
7.2.5.4.2. See example 7.9 p. 388
7.2.5.5. Using cross products to solve proportions
7.2.5.5.1. cross multiply to simplify the proportion into a one or two step equation
7.2.5.5.2. See example 7.10 p. 389
7.2.5.6. Equating numerators or denominators to solve proportions
7.2.5.6.1. Similar to the common denominator method for division
7.2.5.6.2. See example 7.11 p. 390
7.2.5.7. Converting one ratio to a decimal to solve proportions
7.2.5.7.1. Aha! A calculator method! 
7.2.5.7.2. convert one ratio to a decimal (usually the one consisting of just numbers)
7.2.5.7.3. See example in the middle of p. 391 also example 7.12 p. 392
7.2.6. Problems and Exercises p. 394
7.2.6.1. Home work: 1, 2-5, 6-9, 11, 13, 15, 19, 21, 24, 29, 30, 31, 34, 35, 39, 41, 42