Video Transcript
In this video, we will learn how to
identify inverse proportion and write equations describing inverse variation to
solve problems. But before we discuss inverse
variation, let’s recap what we mean by direct variation. And we’ll also look at some of the
properties of variables that are directly proportional to one another.
We can say that two variables are
in direct variation, or direct proportion, if their ratio is constant. As an example, if we work a job
which pays an hourly wage, then the variables of hours worked and wages would be in
direct variation. The more hours that we work, then
the higher our wages will be. If we have two variables 𝑦 and 𝑥,
then we use the proportionality symbol to describe the relationship between the
variables. We can read this as a statement as
𝑦 varies directly with 𝑥 or 𝑦 is directly proportional to 𝑥.
Since we have the ratio between 𝑦
and 𝑥 is constant, then we can write that 𝑦 is equal to 𝑚𝑥, where 𝑚 is not
equal to zero. This value of 𝑚 is called the
constant of variation or the constant of proportionality. Sometimes we may see the letter 𝑘
being used instead. And this time 𝑘 would be the
constant of variation.
When it comes to drawing the graph
of a direct variation, we have a straight line which passes through the origin. But of course this is not the only
type of proportional relationship. Let’s take, for example, the
relationship between the velocity of a car and the time taken to reach a
destination. This can be given by the formula
that the time taken is equal to the distance divided by the velocity. But we know that as the velocity
would increase, then the time taken to reach the destination would decrease. This is an example of an inverse
variation.
We can define this by saying that
two variables 𝑦 and 𝑥 are in inverse variation if 𝑦 is in direct variation with
the reciprocal of 𝑥. And we write it using the same
proportionality symbol like this. When 𝑦 varies directly with one
over 𝑥, that’s equivalent to saying 𝑦 varies inversely with 𝑥.
So when we have an inverse
variation, for example, 𝑎 varies inversely with 𝑏, then we know that we will have
a proportionality symbol along with a reciprocal. And just like in direct variation,
we can write an inverse variation as an equation along with a constant of
proportionality. We can say that 𝑦 is equal to 𝑚
over 𝑥, where 𝑚 is not equal to zero and 𝑚 is the constant of
proportionality. Alternatively, the letter 𝑘 can be
used instead to represent the constant of proportionality.
Let’s now look at some example
questions. And in the first example, we’ll see
how we can determine the constant of proportionality.
𝑦 varies inversely with 𝑥. Given that 𝑦 equals eight when 𝑥
equals seven, what is the constant of proportionality?
Let’s recall that two variables 𝑦
and 𝑥 are said to vary inversely if 𝑦 is directly proportional to or in a direct
variation with the reciprocal of 𝑥. And we write that variation like
this: 𝑦 is proportional to one over 𝑥. This means that there is a nonzero
constant 𝑚 such that 𝑦 is equal to 𝑚 over 𝑥. 𝑚 is the constant of
proportionality. However, the answer to this
question is not simply 𝑚 or any other letter that we use to represent the constant
of proportionality. Instead, what we need to do here is
use the given information about these values of 𝑦 and 𝑥 to work out a numerical
value for the constant of proportionality.
We substitute 𝑦 is equal to eight
and 𝑥 is equal to seven into the equation. This gives us eight is equal to 𝑚
over seven. We can then multiply both sides of
this equation by seven. And since eight times seven is
equal to 56, we have that 𝑚 is equal to 56. And so we find that the constant of
proportionality is 56.
Earlier in this video, we recalled
that the graph of a direct variation is a linear graph which passes through the
origin. But what would the graph of an
inverse variation look like? Let’s take the two variables 𝑦 and
𝑥, where 𝑦 varies inversely with 𝑥. The graph of this will be an
equation of the form 𝑦 equals 𝑚 over 𝑥. This will be a reciprocal graph,
and it would look something like this. Notice that as the value of 𝑥
increases, the value of 𝑦 decreases. And similarly, as the value of 𝑥
decreases, the value of 𝑦 increases. We can use this information in the
next question to determine which one of several different graphs represents an
inverse variation.
Which of the following graphs
represents inverse variation?
We can remember that when we have
an inversely proportional relationship between two variables 𝑦 and 𝑥, we can write
this as 𝑦 is proportional to one over 𝑥. This means that we can also say
that there is some constant of proportionality 𝑚 such that 𝑦 is equal to 𝑚 over
𝑥. We can also recognize that as the
value of 𝑥 increases, the value of 𝑦 must decrease.
If we look at the graphs of the
functions that we’re given, the graphs of B, C, and D do not follow this
pattern. In fact, the graph represented by C
is actually the graph of a direct variation. But we can eliminate these three
options. This leaves us with the answer of
graph A. And we can indeed see that as 𝑥
increases, 𝑦 decreases. Similarly, as 𝑥 decreases, 𝑦
increases. We can also confirm the answer is
graph A because this is the graph of a reciprocal function.
In the next example, we’ll see a
very typical example of an inverse variation question where we first find the
constant of proportionality and then use this to solve a problem.
A group of scouts receives a
donation of 1000 dollars to fund places on an international jamboree. The amount each scout receives for
their trip varies inversely with the number of scouts from the group going to the
jamboree. Write an equation for 𝑚, the
amount each scout receives, in terms of 𝑛, the number of scouts from the group who
are going to the jamboree. If 25 scouts from the group are
going to the jamboree, how much will each scout receive from the donation?
Let’s note that this question is
all about inverse variation. We are told that the amount that
each scout receives varies inversely with the number of scouts going. And that makes sense that it is an
inverse variation. Let’s imagine that just one scout
is going. They would have all of this 1000
dollars to themselves. But if more and more scouts decide
that they are going to go to the jamboree, then the less money that each individual
scout would have towards the travel cost.
We are given here the letters to
assign for each variable. 𝑚 is the amount each scout
receives, and 𝑛 is the number of scouts who are going. If we have two variables 𝑚 and 𝑛
which vary inversely, then we can write that 𝑚 is directly proportional to the
reciprocal of 𝑛. We can then immediately say that
there is some constant of proportionality 𝑘, which is not equal to zero, such that
𝑚 is equal to 𝑘 over 𝑛. 𝑘 is called the constant of
proportionality. And so, in order to find an
equation for 𝑚 in terms of 𝑛, we need to work out this value of 𝑘.
Sometimes in questions like this,
we are given a known pair of values for 𝑛 and 𝑚. We are not given this here, but we
can work some out. Remember, we’ve already noted that
if there’s just one student, then they will get all of the 1000 dollars. One student means that 𝑛 is equal
to one. The amount of money 𝑚 would be
1000. We can then substitute these values
into the proportionality equation. This gives us 1000 is equal to 𝑘
over one. So 𝑘 is equal to 1000. We can then substitute this value
of 𝑘 into this proportionality equation. And so we get the answer for the
first part of this question: 𝑚 is equal to 1000 over 𝑛.
In order to answer the second part
of this question, we use the equation that we’ve worked out in part one to help
us. We know that the relationship is
that the amount of money that any scout receives, which is 𝑚, is equal to 1000 over
the number of scouts. If 25 scouts are going, then that
means that 𝑛 is equal to 25. And we need to work out the value
of 𝑚, the amount of money that each scout would receive. Substituting 𝑛 equals 25 into this
equation, we have 𝑚 is equal to 1000 over 25, so 𝑚 is equal to 40. So the answer is that if 25 scouts
are going to the jamboree, then each scout would receive 40 dollars.
Let’s take a look at one final
example.
The number of hours 𝑛 needed for
carrying out a certain task varies inversely with the number of workers who carry
out the task. If the task is carried out by 23
workers in 35 hours, what is the time needed for 115 workers to carry out the
task?
In this problem, we are
investigating the relationship between the number of hours it takes workers to do a
task and the number of workers doing that task. We are told that these two
variables are related with an inverse variation. It can be tempting to think that as
the number of workers increases, then the time taken also increases. But this would be incorrect, but
it’s actually the reverse. As the number of workers increases,
then the time taken will decrease. This is why this is an inverse
variation. We can describe the time taken or
the number of hours using the variable 𝑛. And let’s denote the number of
workers with the variable 𝑤.
Recall that if we have two
variables 𝑛 and 𝑤 which are in an inverse variation, then we can write this as 𝑛
is directly proportional to one over 𝑤. We can then say that there is some
constant of proportionality 𝑘 such that 𝑛 is equal to 𝑘 over 𝑤. And in order to find what the value
of 𝑘 will be, we’re given a piece of information that it takes 23 workers 35 hours
to do the task. So we can substitute 𝑛 is equal to
35 and 𝑤 is equal to 23 into this equation. This gives us that 35 is equal to
𝑘 over 23. When we multiply both sides of this
equation by 23, we have 35 times 23 is equal to 𝑘. We know then that 𝑘 is equal to
805. We then substitute this value of 𝑘
into the equation of proportionality. We have that 𝑛 is equal to 805
over 𝑤.
We now have an equation that
relates the number of hours 𝑛 with the number of workers 𝑤. We could use this equation to work
out the time needed for any number of workers. And we can use it here to work out
the time needed for 115 workers. The value of 𝑤 is the number of
workers, and that’s equal to 115. And we need to work out 𝑛, the
number of hours. Substituting this into the
equation, we have 𝑛 is equal to 805 over 115, which simplifies to seven. And so the answer is that the time
taken for 115 workers to carry out the task is seven hours.
But there is an alternative way in
which we could’ve worked out this inverse variation problem. And this is by using the property
that if two variables vary inversely with each other, then their product remains
constant. In this context, we’d be saying
that the number of hours 𝑛 multiplied by the number of workers 𝑤 is equal to some
constant 𝑘. In the first part of this problem,
we were told that it takes 23 workers 35 hours to complete the task. And that product will be the same
as 115 workers multiplied by 𝑛, the number of hours. To solve for 𝑛, we would divide
through by 115. And when we do that, we get a value
𝑛 is equal to seven. And so we have confirmed the
earlier answer that it takes seven hours for 115 workers.
We can now summarize the key points
of this video. Two variables 𝑦 and 𝑥 are said to
be in an inverse variation, or inverse proportion, if 𝑦 is proportional to one over
𝑥. This also means that their product
remains constant. If 𝑦 and 𝑥 are in an inverse
variation, this is equivalent to 𝑦 equals 𝑚 over 𝑥 for some constant 𝑚, which is
not equal to zero. We call 𝑚 the constant of
proportionality. Finally, we also saw that the graph
of variables in an inverse variation is a reciprocal graph.
