1. If P varies inversely as Q and P = 5 when Q = 3, find P when Q = 9.
Step 1: Write the inverse variation formula:
P=
k
Q
Step 2: Substitute known values of P and Q to solve for k:
k
3
k=15
5=
Step 3: Substitute the known Q-value, the value of k and solve for P:
15
9
5
P=
3
P=
2. If A varies jointly with b and h and A=6 when b=2 and h=9, find b when A=7 and h=3.
Step 1: Write the joint variation formula: A=kbh
Step 2: Substitute known values of A, b, and h to solve for k:
6  k  2  9 
6  18k
k
1
3
Step 3: Substitute the known values of A, h, and k and solve for b:
1
7     3 b
3
7  1 b
b7
3. Boyle’s law for gases states that the volume V of a gas varies inversely as the pressure P if the
mass and temperatures are constant. If the volume of a gas is 13 cubic feet when the pressure is
140 pounds per square foot, find the volume when the pressure is 185 pounds per square foot.
Also, use Geogebra to graph this variation plotting volume on the vertical axis and pressure on
the horizontal axis. Be sure to label the axes and title the graph.
Step 1: Write the inverse variation formula: V=
k
P
Step 2: Substitute known values of P and V to solve for k:
k
140
k  1820
13=
Step 3: Substitute the known P-value, the value of k and solve for V:
1820
185
31
V 9
cubic feet or  9.84 cubic feet (rounded to the nearest hundredth)
37
V=
Gas Volume vs Pressure
4. The time t required for a car to travel a fixed distance is inversely proportional to the rate of
the car is traveling r. If it takes a car traveling at 102 feet per second 52 seconds to travel a given
distance, what is the rate of a car if it travels the same distance in 84 seconds? Also, use
Geogebra to graph this variation plotting time on the vertical axis and rate on the horizontal axis.
Be sure to label the axes and title the graph
Step 1: Write the inverse variation formula: t=
k
r
Step 2: Substitute known values of t and r to solve for k:
k
102
k  5304
52=
Step 3: Substitute the known t-value, the value of k and solve for r:
5304
r
84r=5304
84=
r  63
1
feet per second or  63.14 feet per second (rounded to the nearest hundredth)
7
Stopping Time vs Rate of a Car
4. The force f on an object is jointly proportional to its mass m and its acceleration a. If an
object experiences 15 N of force when its mass is 5 kg and its acceleration is 3 m/s2, what is its
mass when it experiences 20 N of force and its acceleration is 6 m/s2?
Step 1: Write the inverse joint variation formula: F  kma
Step 2: Substitute known values of f, m and a to solve for k:
15  k  5  3
15  15k
k 1
Step 3: Substitute the known f-value, a-value and the value of k and solve for m:
20  1 6  a
20  6a
10
a
kg or  3.33 kg (rounded to the nearest hundredth)
3
5. The simple interest I earned by a savings deposit in a given time varies jointly as the principal
P and the interest rate r. If a deposit of $450 earns $56 interest at 6% in a given time, what is
interest rate if a deposit of $890 earns $62 in interest?
Step 1: Write the inverse joint variation formula: I=kpr
Step 2: Substitute known values of I, p and r to solve for k:
I  kpr
56  k  450 .06  Be sure to change the interest rate to its decimal form!
56=27k
k = 2.07 (Rounded to the nearest hundredth)
Step 3: Substitute the known I-value, p-value and the value of k and solve for r be sure to
change r to its percentage form in the final answer:
62  2.07 890  r
62=1842.3r
r  0.034 (Rounded to the nearest thousandth) or 3.4%