If 𝑦 varies directly or is proportional to 𝑥 and 𝑥 equals 75 when 𝑦 equals 25, find the value of 𝑦 when 𝑥 equals 30.
As 𝑦 and 𝑥 are proportional to each other, or as 𝑦 varies directly with 𝑥, as one of them increases, so does the other. In order to solve this question, we firstly need to find the constant of proportionality, which we will call the letter 𝑘. We’ll replace the proportion sign with equals 𝑘, giving us the equation 𝑦 equals 𝑘 multiplied by 𝑥.
Our next step is to substitute in the values that we’re given in the question. We are told that when 𝑥 is 75, 𝑦 is 25. Therefore, 25 equals 𝑘 multiplied by 75 or 75𝑘. Balancing this equation by dividing both sides by 75 leaves us with 25 over 75 equals 𝑘. Simplifying this fraction gives us a final answer of 𝑘 equals a third. The constant of proportionality is equal to a third.
In practical terms, this means that our value for 𝑦 is always a third of the value of 𝑥, or another way of saying this would be that 𝑥 is always three times 𝑦. To complete the question find the value of 𝑦 when 𝑥 equals 30, we now need to substitute the value of 30 into the equation 𝑦 equals one-third 𝑥. The calculation 𝑦 equals a third of 30 or a third multiplied by 30, therefore, when 𝑥 equal 30, 𝑦 equals 10.