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Question Video: Creating Inverse Variation Equations Mathematics • 9th Grade

Given that π‘Ž varies inversely with 𝑏, and π‘Ž = 5 when 𝑏 = 3/4, write an equation for π‘Ž in terms of 𝑏.

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Video Transcript

Given that π‘Ž varies inversely with 𝑏 and π‘Ž equals five when 𝑏 equals three-quarters, write an equation for π‘Ž in terms of 𝑏.

In this question, we are told that a variable π‘Ž varies inversely with variable 𝑏 and that when π‘Ž is equal to five, we have that 𝑏 is equal to three-quarters. We want to use this to find an equation for π‘Ž 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 π‘Ž is proportional to one over 𝑏. We can then recall that this is the same as saying that π‘Ž is equal to π‘š divided by 𝑏 for some constant π‘š, where 𝑏 is nonzero. We call π‘š the constant of proportionality.

To find an equation involving π‘Ž and 𝑏, we need to find the value of π‘š. We can do this by substituting the known values of π‘Ž and 𝑏 into this equation. Substituting π‘Ž equals five and 𝑏 equals three-quarters into the equation gives us five equals π‘š divided by three-quarters. We can solve for π‘š by multiplying both sides of the equation by three-quarters. We get π‘š equals five times three-quarters, which we can evaluate is equal to 15 over four. We can now substitute π‘š equals 15 over four into our proportionality equation to obtain that π‘Ž equals 15 over four divided by 𝑏. This is an equation for π‘Ž 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 π‘Ž is equal to 15 over four multiplied by one over 𝑏, which we can simplify to give us the equation π‘Ž equals 15 over four 𝑏.

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