Proportions & Variation
MATH 102
Contemporary Math
S. Rook
Overview
• Section 7.5 in the textbook:
– Ratios & proportions
– Direct & inverse variation
Ratios & Proportions
Ratios & Proportions
• 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
4
Proportions (Example)
Ex 1: Solve:
a)
b)
x

3
12
2
30
27
40

x
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
6
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
10
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
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
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)