1. Proportions & Variation MATH 102 Contemporary Math S. Rook

2. Overview Section 7.5 in the textbook: – Ratios & proportions – Direct & inverse variation

3. Ratios & Proportions

4. 4 Ratio – a quotient of two quantities e.g. 1 / 5 or x / (x – 1) – An expression Proportion – a mathematical equation of two equal ratios Ex: 2 / 3 = 6 / 9 or x / (x – 1) = 7 / 8 – Only one fraction on each side of the = We cross-multiply to solve a proportion e.g. x / 4 = 1 / 2

5. Proportions (Example) Ex 1: Solve: a) b)

6. 6 Using Proportions to Solve Word Problems Key is to extract information from the word problem to set up proportions Proportions compare two units – e.g. cups of sugar to batches, number of cards to people, etc Align the units – e.g. Put cups of sugar is in the numerator and batches in the denominator for both sides of the proportion – DO NOT mix up the units on each side

7. Proportions & Word Problems (Example) Ex 2: The dosage of a particular drug is proportional to the patient’s body weight. If the dosage for a 150-pound woman is 6 milligrams, what would the dosage be for her daughter who weighs 65 pounds?

8. Proportions & Word Problems (Example) Ex 3: If 36 cookies require 1.5 cups of sugar, how many cookies can be made with 4 cups of sugar?

9. Direct & Inverse Variation

10. Variation in General Given two quantities that are related, variation refers to how either increasing or decreasing the first quantity affects the second quantity Because the two quantities are related, they differ by only a constant value – This constant is called the constant of proportionality and is often denoted by k We can model variation by equations 10

11. Direct Variation Direct variation: situations that can be modeled with the formula y = kx where x and y represent the two quantities k is the constant of proportionality The following statements are all equivalent and indicative of direct variation (also in the book): y varies directly as x y is directly proportional to x y = kx for some nonzero constant k 11

12. Inverse Variation Inverse variation: situations that can be modeled with the formula y = k ⁄ x where x and y represent the two quantities k is the constant of proportionality The following statements are all equivalent and indicative of inverse variation (also in the book): y varies inversely as x y is inversely proportional to x y = k ⁄ x for some nonzero constant k 12

13. Direct & Inverse Variation (Example) Ex 4: Solve: a) Assume that y varies directly as x. If y = 37.5 when x = 7.5, what is the value for y when x = 13? b) Assume that r varies inversely as s. If r = 12 when s = 2 ⁄ 3, what is the value for r when s = 8?

14. Direct & Inverse Variation (Example) Ex 5: The volume of a cylinder varies directly with the square of its radius AND its height. If the volume of a cylinder with a radius of 5 cm and height of 2 cm is 157.08 cm 2, find the volume of a cylinder with a radius of 7 cm and a height of 10 cm.

15. Summary After studying these slides, you should know how to do the following: – Solve proportions and word problems containing proportions – Solve problems involving direct and inverse variation Additional Practice: – See problems in Section 7.5 Next Lesson: – Percent Change & Taxes (Section 9.1)