Video Transcript
Given that 𝑎 varies inversely with
𝑏 and 𝑎 equals five when 𝑏 equals three-quarters, write an equation for 𝑎 in
terms of 𝑏.
In this question, we are told that
a variable 𝑎 varies inversely with variable 𝑏 and that when 𝑎 is equal to five,
we have that 𝑏 is equal to three-quarters. We want to use this to find an
equation for 𝑎 in terms of 𝑏. To do this, we first recall that we
say a variable varies inversely with another variable if it varies proportionally to
its reciprocal. So, we must have that 𝑎 is
proportional to one over 𝑏. We can then recall that this is the
same as saying that 𝑎 is equal to 𝑚 divided by 𝑏 for some constant 𝑚, where 𝑏
is nonzero. We call 𝑚 the constant of
proportionality.
To find an equation involving 𝑎
and 𝑏, we need to find the value of 𝑚. We can do this by substituting the
known values of 𝑎 and 𝑏 into this equation. Substituting 𝑎 equals five and 𝑏
equals three-quarters into the equation gives us five equals 𝑚 divided by
three-quarters. We can solve for 𝑚 by multiplying
both sides of the equation by three-quarters. We get 𝑚 equals five times
three-quarters, which we can evaluate is equal to 15 over four. We can now substitute 𝑚 equals 15
over four into our proportionality equation to obtain that 𝑎 equals 15 over four
divided by 𝑏. This is an equation for 𝑎 in terms
of 𝑏.
We could leave our answer like
this. However, we can simplify the
equation by noting that instead of dividing by 𝑏, we can multiply by the reciprocal
of 𝑏. This gives us that 𝑎 is equal to
15 over four multiplied by one over 𝑏, which we can simplify to give us the
equation 𝑎 equals 15 over four 𝑏.
