$y$ is directly proportional to $x$. When $y=2, x=1$. Find the value of $x$ when $y=8$.
Answer (1)
Establish the direct proportionality relationship: y = k x .
Determine the constant of proportionality: k = x y = 1 2 = 2 .
Calculate x when y = 8 : x = k y = 2 8 = 4 .
State the final answer: 4 .
Explanation
Understanding the Problem We are given that y is directly proportional to x . This means that there exists a constant k such that y = k x . We are also given that when y = 2 , x = 1 . We need to find the value of x when y = 8 .
Finding the Constant of Proportionality First, we need to find the constant of proportionality k . We can use the given values y = 2 and x = 1 to find k . Substituting these values into the equation y = k x , we get 2 = k ( 1 ) . Solving for k , we find that k = 2 .
Finding the Value of x Now that we have the value of k , we can use the equation y = k x to find the value of x when y = 8 . Substituting y = 8 and k = 2 into the equation y = k x , we get 8 = 2 x . Solving for x , we divide both sides of the equation by 2: x = 2 8 = 4 .
Final Answer Therefore, when y = 8 , the value of x is 4.
Examples
Direct proportionality is a fundamental concept in many real-world scenarios. For instance, the distance you travel at a constant speed is directly proportional to the time you spend traveling. If you travel 100 miles in 2 hours, your speed is 50 miles per hour. Using this, you can determine that in 4 hours, you would travel 200 miles, assuming the same constant speed. This concept is also applicable in calculating currency exchange rates, scaling recipes, and understanding simple machines.
