If y varies directly as x, and y is 180 when x is n and y is n when x is 5, what is the value of n?
Answer (1)
Set up the direct variation equations: y = k x .
Use the given conditions to find two expressions for k : k = n 180 and k = 5 n .
Equate the two expressions and solve for n : n 180 = 5 n ⇒ n 2 = 900 .
Take the square root to find n : n = 900 = 30 .
The value of n is 30 .
Explanation
Understanding the Problem We are given that y varies directly as x. This means that there is a constant of proportionality, k, such that y = k x . We are given two conditions:
When x = n , y = 180 .
When x = 5 , y = n .
Our goal is to find the value of n .
Using the First Condition Using the first condition, we can write the equation 180 = kn . Solving for k , we get k = n 180 .
Using the Second Condition Using the second condition, we can write the equation n = k ( 5 ) . Solving for k , we get k = 5 n .
Equating the Expressions for k Since both expressions are equal to k , we can set them equal to each other: n 180 = 5 n
Solving for n To solve for n , we can cross-multiply: n 2 = 180 × 5 n 2 = 900
Finding the Value of n Taking the square root of both sides, we get: n = ± 900 n = ± 30 Since the problem does not specify that n must be positive, we should consider both positive and negative solutions. However, since y varies directly as x , if x is positive, y should also be positive, and vice versa. In the second condition, x = 5 is positive, so y = n should also be positive. Therefore, we take the positive root: n = 30
Final Answer Therefore, the value of n is 30.
Examples
Direct variation is a concept that shows up in many real-world scenarios. For example, the amount of money you earn might vary directly with the number of hours you work. If you earn $15 per hour, the more hours you work, the more money you'll make. Similarly, the distance a car travels at a constant speed varies directly with the time it travels. If a car travels at 60 miles per hour, the longer it drives, the farther it will go. Understanding direct variation helps us predict and calculate these kinds of relationships.
