Richland Community College The Inverse and Direct Variation Problems

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Inverse Variation
The flowchart below shows how to decide whether a relationship between two
variables is a direct variation, inverse variation, or neither.
How does the value of x change?
increases
decreases
How does the value of y change?
How does the value of y change?
increases
decreases
decreases
increases
Test x and y values in direct
variation model k =
Test x and y values in inverse
variation model: xy = k.
Is each ratio of y to x equal
to the same value, k?
no
yes
Is each product of x and y equal
to the same value, k?
no
yes
neither
direct variation
inverse variation
neither
Problem
Do the data in the table represent a direct variation, inverse variation, or neither?
x 1 2 4 5
y 20 10 5 4
As the value of x increases, the value of y decreases, so test the table values in
the inverse variation model: xy=k: 1.20 = 20, 2 – 10 = 20, 4.5=20,
5.4 = 20. Each product equals the same value, 20, so the data in the table
model an inverse variation.
Do the data in the table represent a direct variation, inverse variation, or neither?
1.
2.
x 5 10 15 20
x 1 3 4 6
y 10 20 30 40
y 12 4 3 2
Pearson Texas Algebra II
Inverse Variation
To solve problems involving inverse variation, you need to solve for the constant
of variation k before you can find an answer.
Problem
The time that is necessary to complete a task varies inversely as the number of
people p working. If it takes 4 h for 12 people to paint the exterior of a house, how
long does it take for 3 people to do the same job?
k
р
Write an inverse variation. Because time is dependent on people, t is the
dependent variable and p is the independent variable.
4
k
12
Substitute 4 fort and 12 for p.
48=k
Multiply both sides by 12 to solve for k, the constant of variation.
4-
48
р
Substitute 48 for k. This is the equation of the inverse variation.
48
= 16
3
Substitute 3 for p. Simplify to solve the equation.
It takes 3 people 16 h to paint the exterior of the house.
3. The time t needed to complete a task varies inversely as the number of people p.
It takes 5 h for seven men to install a new roof. How long does it take ten men
to complete the job?
4. The time t needed to drive a certain distance varies inversely as the speed r. It
takes 7.5 h at 40 mi/h to drive a certain distance. How long does it take to drive
the same distance at 60 mi/h?
5. The cost of each item bought is inversely proportional to the number of items
when spending a fixed amount. When 42 items are bought, each costs $1.46.
Find the number of items when each costs $2.16.
6. The length 1 of a rectangle of a certain area varies inversely as the width w. The
length of a rectangle is 9 cm when the width is 6 cm. Determine the length if
the width is 8 cm.