Question Video: Calculating the Value of One Variable in an Inversely Proportional Relationship Mathematics • Third Year of Preparatory School

Video Transcript

𝑦 is inversely proportional to 𝑥. When 𝑥 equals three, then 𝑦 equals six. Find the value of 𝑦 when 𝑥 equals eight.

We’re told that 𝑦 is inversely proportional to 𝑥, which means that as 𝑥 increases, 𝑦 decreases at the same rate. When two variables are inversely proportional to one another, this means that they are directly proportional to each other’s reciprocal. So we can write 𝑦 is directly proportional to one over 𝑥 and then express this as an equation 𝑦 equals 𝑘 over 𝑥, where 𝑘 represents the constant of proportionality.

We’re then told that when 𝑥 is equal to three, 𝑦 is equal to six. So we have a pair of values that we can use to determine this constant 𝑘. Substituting three for 𝑥 and six for 𝑦 gives the equation six equals 𝑘 over three. And then we can multiply both sides of this equation by three to find that 𝑘 is equal to 18. We’ve found then the relationship between 𝑥 and 𝑦 is 𝑦 is equal to 18 over 𝑥.

Finally, we’re asked to find the value of 𝑦 when 𝑥 is equal to eight. Substituting 𝑥 equals eight into the given equation gives 𝑦 equals 18 over eight. Simplifying by dividing both the numerator and denominator by two gives the simplified fraction nine over four. And then we can convert this to the mixed number two and one-quarter if we wish.

We’ve found then that for this inversely proportional relationship between 𝑥 and 𝑦, when 𝑥 is equal to eight, 𝑦 is equal to two and one-quarter.