Combined Variation:
- when a quantity varies directly (or jointly) with one or more variables
and inversely with one or more variables
𝑘𝑥 𝑚 𝑧 𝑛
- described by formulas of the form 𝑦 = 𝑤 𝑝
o 𝑦 varies directly with 𝑥 and 𝑧 and inversely with 𝑤
 depending on how the variables in the numerator and/or
denominator change (increasing or decreasing), the
dependent variable could increase, decrease, or remain
unchanged
- Example:
o Newton’s law of universal gravitation
𝐺𝑚 𝑚
 𝐹 = 𝑑12 2 ; the constant of variation 𝑘 = 𝐺, the
gravitational constant
Example 1: Express the following statement as a formula that involves
the given variables and a constant of proportionality 𝑘, and then determine
the value of 𝑘 from the given conditions.
𝑟 varies directly as 𝑠 and indirectly as 𝑡. If 𝑠 = 2 and 𝑡 = 4, then 𝑟 = 7.
Example 2: Express the following statement as a formula that involves
the given variables and a constant of proportionality 𝑘, and then determine
the value of 𝑘 from the given conditions.
𝑦 is directly proportional to the square of 𝑥 and inversely proportional to
𝑧. If 𝑥 = 5 and 𝑧 = 3, then 𝑦 = 25.
Example 3: Express the following statement as a formula that involves
the given variables and a constant of proportionality 𝑘, and then determine
the value of 𝑘 from the given conditions.
𝑦 is directly proportional to the square root of 𝑥 and inversely
proportional to the cube of 𝑧. If 𝑥 = 9 and 𝑧 = 2, then 𝑦 = 5.
Example 4: The centrifugal force 𝐹 of a body moving in a circle varies
jointly with the radius 𝑟 of the circular path and the body’s mass 𝑚, and
inversely with the square of the time 𝑡 it takes to move about one full
circle.
a. Express the statement above as a formula.
b. A 6-gram body moving in a circle with radius 100 centimeters at a
rate of 1 revolution every 2 seconds has a centrifugal force of 6,000
dynes. Use this information to determine the value of 𝑘.
c. Find the centrifugal force of an 18-gram body moving in a circle with
radius 100 centimeters at a rate of 1 revolution every 3 seconds?
Example 5: In baseball, a pitcher’s earned-run average 𝐴 varies directly as
the number of earned runs 𝑅 allowed and inversely as the number of
innings pitched 𝐼.
a. Express the previous statement as a formula.
b. If a pitcher has an earned-run average of 3.6 after pitching 95 innings
and allowing 38 earned-runs, what is the value of 𝑘?
c. What is the earned-run average of a pitcher who gave up 69 earned
runs in 308 innings? Round to the hundredths place.
Answers to Examples:
1. 𝑟 =
𝑘𝑠
𝑡
4a. 𝐶 =
5a. 𝐸 =
; 𝑘 = 14 ; 2. 𝑦 =
𝑘∙𝑟∙𝑚
𝑡2
𝑘∙𝑅
𝐼
𝑘𝑥 2
𝑧
; 4b. 𝑘 = 40 ; 4c.
; 𝑘 = 3 ; 3. 𝑦 =
𝐶 = 8,000 ;
; 5b. 𝑘 = 9 ; 5c. 𝐸 = 2.02 ;
𝑘 √𝑥
𝑧3
;𝑘=
40
3
;