# Video: AQA GCSE Mathematics Higher Tier Pack 4 • Paper 3 • Question 14

In the equation (𝑦𝑥²/𝑘) = 1, 𝑘 is a constant. Circle the correct statement. [A] 𝑦 is directly proportional to 𝑥² [B] 𝑦 is inversely proportional to 𝑥² [C] 𝑥 is directly proportional to 𝑦² [D] 𝑦 is inversely proportional to 1/𝑥².

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### Video Transcript

In the equation 𝑦𝑥 squared over 𝑘 equals one, 𝑘 is a constant. Circle the correct statement. The options are 𝑦 is directly proportional to 𝑥 squared, 𝑦 is inversely proportional to 𝑥 squared, 𝑥 is directly proportional to 𝑦 squared, or 𝑦 is inversely proportional to one over 𝑥 squared.

Each of the four options describes a different type of proportional relationship between 𝑦 and 𝑥. So what we need to do in order to answer the question is determine how 𝑦 and 𝑥 are related to one another. In order to do this, we need to first rearrange the equation we’ve been given to give 𝑦 in terms of 𝑥.

The first step is to multiply both sides of the equation by 𝑘 to cancel out the 𝑘 in the denominator on the left-hand side. This gives 𝑦𝑥 squared is equal to 𝑘. The next step is to divide both sides of the equation by 𝑥 squared, giving 𝑦 equals 𝑘 over 𝑥 squared. So we found 𝑦 in terms of 𝑥.

Now the fraction 𝑘 over 𝑥 squared can be written as 𝑘 multiplied by one over 𝑥 squared. So what we found is that 𝑦 is equal to a multiple of one over 𝑥 squared. That’s a multiple of the reciprocal of 𝑥 squared. We can express this using the proportionality symbol. 𝑦 is proportional to one over 𝑥 squared. Now as this is a reciprocal relationship, this means that as 𝑥 or 𝑥 squared increase, 𝑦 will decrease. So the two quantities 𝑦 and 𝑥 squared are inversely proportional to one another.

Be careful here because we’re given two possible answers involving inversely proportional relationships. The first is that 𝑦 is inversely proportional to 𝑥 squared, and the second is that 𝑦 is inversely proportional to one over 𝑥 squared. The correct answer is that 𝑦 is inversely proportional to 𝑥 squared because the inversely proportional part creates the reciprocal for us.

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

The correct answer found by rearranging the equation we were given to see that 𝑦 was equal to some multiple of one over 𝑥 squared is that 𝑦 is inversely proportional to 𝑥 squared.