Proportions & Variation

MATH 102Contemporary Math

S. Rook

Overview

• Section 7.5 in the textbook:– Ratios & proportions– Direct & inverse variation

Ratios & Proportions

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Ratios & Proportions

• Ratio – a quotient of two quantitiese.g. 1/5 or x/(x – 1) – An expression

• Proportion – a mathematical equation of two equal ratiosEx: 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

Proportions (Example)

Ex 1: Solve:

a)

b)

2

3

12

x

x

27

40

30

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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

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?

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?

Direct & Inverse Variation

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

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Direct Variation

• Direct variation: situations that can be modeled with the formula y = kx where

x and y represent the two quantitiesk is the constant of proportionality

• The following statements are all equivalent and indicative of direct variation (also in the book):

y varies directly as xy is directly proportional to xy = kx for some nonzero constant k

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Inverse Variation

• Inverse variation: situations that can be modeled with the formula y = k⁄x where

x and y represent the two quantitiesk is the constant of proportionality

• The following statements are all equivalent and indicative of inverse variation (also in the book):

y varies inversely as xy is inversely proportional to x y = k⁄x for some nonzero constant k

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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?

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 cm2, find the volume of a cylinder with a radius of 7 cm and a height of 10 cm.

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)