Video Transcript
๐ฆ is inversely proportional to ๐ฅ. When ๐ฅ equals three, then ๐ฆ equals six. Find the value of ๐ฆ when ๐ฅ equals eight.
Weโre told that ๐ฆ is inversely proportional to ๐ฅ, which means that as ๐ฅ increases, ๐ฆ decreases at the same rate. When two variables are inversely proportional to one another, this means that they are directly proportional to each otherโs reciprocal. So we can write ๐ฆ is directly proportional to one over ๐ฅ and then express this as an equation ๐ฆ equals ๐ over ๐ฅ, where ๐ represents the constant of proportionality.
Weโre then told that when ๐ฅ is equal to three, ๐ฆ is equal to six. So we have a pair of values that we can use to determine this constant ๐. Substituting three for ๐ฅ and six for ๐ฆ gives the equation six equals ๐ over three. And then we can multiply both sides of this equation by three to find that ๐ is equal to 18. Weโve found then the relationship between ๐ฅ and ๐ฆ is ๐ฆ is equal to 18 over ๐ฅ.
Finally, weโre asked to find the value of ๐ฆ when ๐ฅ is equal to eight. Substituting ๐ฅ equals eight into the given equation gives ๐ฆ 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.
Weโve found then that for this inversely proportional relationship between ๐ฅ and ๐ฆ, when ๐ฅ is equal to eight, ๐ฆ is equal to two and one-quarter.
