Notes: Inverse Variation
Remember that two variables x and y show direct variation if y = ax and
a ≠ 0. The letter “a” represents the “Constant of Variation”. The equation for the
𝒚
Constant of Variation is 𝒂 =
𝒙
On the contrary, if two variables show inverse variation, then they have the equation
𝒂
y=
or a = yx .
𝒙
If two variables vary inversely, the products of every x and y combination are
the same.
For example,
x
y
-3
6
-2
9
2
-9
So, 6(-3) = -18, 9(-2) = -18, 2(-9) = -18
So this is another way to identify inverse variation. The Constant of Variation for this
−18
example would be a =6(-3) = -18 so the equation would be y =
.
𝑥
Does xy = 4 represents an inverse variation? Yes, because we can write it in the form
y=
𝑎
𝑥
or y =
The equation
4
𝑥
𝑦
2
.
= 𝑥 can be written as y = 2x so it represents a direct variation.
The equation y = 2x + 3 is neither, as this is not in the form of direct or inverse
variation.
Direct Variation
y = ax
𝑎=
𝑦
𝑥
Inverse Variation
𝑎
y=
𝑥
𝑎 = 𝑦𝑥
Real World Example:
a. A = $2,000 + n
b. $2,000 = A + n
c.
A = $2,000n
d. $2,000 = An
Ex.
For the Choir fundraiser, the number of tickets Allie can buy is inversely
proportional to the price of the tickets. She can afford 15 tickets that cost $5
each. How many tickets can Allie buy if each cost $3?
a = yx or a = 15(5) = 75
The equation is y =
So when x = 3
y=
𝟕𝟓
𝐱
𝟕𝟓
𝟑
,
y = 25
Then x = 25 or she can buy 25 tickets.
You Try:
1.
If x and y vary inversely and x = 1 when y = 11, find x when y = 5.5
2. In kick boxing, it is found that the force, f, needed to break a board, varies inversely
with the length, l, of the board. If it takes 5 lbs of pressure to break a board 2 feet long,
how many pounds of pressure will it take to break a board that is 6 feet long?
3. The cost per person to rent a mountain cabin is inversely proportional (varies
inversely) to the number of people who share the rent. If the cost is $36 per person
when 5 people share, what is the cost per person when 8 people share?
4. The amount of work time (h) in hours needed to prepare for the school play varies
inversely with the number of people working (p). Twenty five people working (p) can
complete the job in 300 hours (h). Write the inverse variation equation that relates h
and p. Then find how many hours it will take if there are 30 people working.
5. Heart rates and life spans of most mammals are inversely related. A cat lives for
about 15.2 years on average and has a heart rate of 126 beats per minute.
a. What is the constant of variation?
b. A hamster has a heart rate of about 634 beats per minute. About how long will a
hamster live?
c. An elephant lives for about 70 years. About how many times per minute does an
elephant’s heart beat?