Video: AQA GCSE Mathematics Higher Tier Pack 4 β’ Paper 3 β’ Question 14

In the equation (π¦π₯Β²/π) = 1, π is a constant. Circle the correct statement. [A] π¦ is directly proportional to π₯Β² [B] π¦ is inversely proportional to π₯Β² [C] π₯ is directly proportional to π¦Β² [D] π¦ is inversely proportional to 1/π₯Β².

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Video Transcript

In the equation π¦π₯ squared over π equals one, π is a constant. Circle the correct statement. The options are π¦ is directly proportional to π₯ squared, π¦ is inversely proportional to π₯ squared, π₯ is directly proportional to π¦ squared, or π¦ is inversely proportional to one over π₯ squared.

Each of the four options describes a different type of proportional relationship between π¦ and π₯. So what we need to do in order to answer the question is determine how π¦ and π₯ are related to one another. In order to do this, we need to first rearrange the equation weβve been given to give π¦ in terms of π₯.

The first step is to multiply both sides of the equation by π to cancel out the π in the denominator on the left-hand side. This gives π¦π₯ squared is equal to π. The next step is to divide both sides of the equation by π₯ squared, giving π¦ equals π over π₯ squared. So we found π¦ in terms of π₯.

Now the fraction π over π₯ squared can be written as π multiplied by one over π₯ squared. So what we found is that π¦ is equal to a multiple of one over π₯ squared. Thatβs a multiple of the reciprocal of π₯ squared. We can express this using the proportionality symbol. π¦ is proportional to one over π₯ squared. Now as this is a reciprocal relationship, this means that as π₯ or π₯ squared increase, π¦ will decrease. So the two quantities π¦ and π₯ squared are inversely proportional to one another.

Be careful here because weβre given two possible answers involving inversely proportional relationships. The first is that π¦ is inversely proportional to π₯ squared, and the second is that π¦ is inversely proportional to one over π₯ squared. The correct answer is that π¦ is inversely proportional to π₯ squared because the inversely proportional part creates the reciprocal for us.

The statement π¦ is inversely proportional to one over π₯ squared would actually be written as π¦ is proportional to one over one over π₯ squared, which would actually mean that π¦ was proportional to π₯ squared because one over one over π₯ squared just simplifies to π₯ squared. So in this case, this would actually mean that π¦ was directly proportional to π₯ squared.

The correct answer found by rearranging the equation we were given to see that π¦ was equal to some multiple of one over π₯ squared is that π¦ is inversely proportional to π₯ squared.