Video: Solving Proportion Equations Involving Direct Variation

If 𝑦 ∝ π‘₯ and π‘₯ = 75 when 𝑦 = 25, find the value of 𝑦 when π‘₯ = 30.

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Video Transcript

If 𝑦 varies directly or is proportional to π‘₯ and π‘₯ equals 75 when 𝑦 equals 25, find the value of 𝑦 when π‘₯ equals 30.

As 𝑦 and π‘₯ are proportional to each other, or as 𝑦 varies directly with π‘₯, 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 π‘˜. We’ll replace the proportion sign with equals π‘˜, giving us the equation 𝑦 equals π‘˜ multiplied by π‘₯.

Our next step is to substitute in the values that we’re given in the question. We are told that when π‘₯ is 75, 𝑦 is 25. Therefore, 25 equals π‘˜ multiplied by 75 or 75π‘˜. Balancing this equation by dividing both sides by 75 leaves us with 25 over 75 equals π‘˜. Simplifying this fraction gives us a final answer of π‘˜ 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 π‘₯, or another way of saying this would be that π‘₯ is always three times 𝑦. To complete the question find the value of 𝑦 when π‘₯ equals 30, we now need to substitute the value of 30 into the equation 𝑦 equals one-third π‘₯. The calculation 𝑦 equals a third of 30 or a third multiplied by 30, therefore, when π‘₯ equal 30, 𝑦 equals 10.

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