The ratio $8 w^2: 6 t w$ can be written in the form $1: n$. Write an expression for $n$, in terms of $t$ and $w$, in its simplest form.
Answer (1)
We start with the ratio 8 w 2 : 6 tw , write it as a fraction 6 tw 8 w 2 , simplify the fraction to 3 t 4 w , and then set it equal to n 1 . Finally, we solve for n to get n = 4 w 3 t . Therefore, the expression for n in terms of t and w is 4 w 3 t .
Explanation
Understanding the Problem We are given the ratio 8 w 2 : 6 tw and we want to express it in the form 1 : n . This means we need to find an expression for n in terms of t and w .
Writing the Ratio as a Fraction First, let's write the given ratio as a fraction: 6 tw 8 w 2 .
Simplifying the Fraction Now, we simplify the fraction by canceling out common factors. Both the numerator and the denominator have a factor of 2, a factor of w . Thus, we have: 6 tw 8 w 2 = 3 t 4 w .
Relating to the Desired Form We are given that this ratio can be written in the form 1 : n , which means 3 t 4 w = n 1 .
Solving for n To find n , we take the reciprocal of both sides of the equation: n = 4 w 3 t .
Examples
In real-world applications, ratios like the one in this problem are often used to compare different quantities. For example, if w represents the width of a rectangle and t represents its thickness, the ratio 8 w 2 : 6 tw could describe a relationship between the area of the rectangle's face and its volume. Simplifying this ratio to find n allows us to easily compare how the dimensions affect these quantities. If we know that the ratio of the face area to some reference area is 1 : n , then we can quickly determine the actual face area by knowing n and the reference area. This kind of proportional reasoning is useful in fields like engineering, architecture, and manufacturing, where scaling and comparing dimensions are common tasks.
