consider this true proportion R/W=S/T.if the product of the means increases,what must happen to the product of the extremes?explain your reasoning.
To understand the relationship between the product of the means and the product of the extremes in a proportion, let's first clarify the terms:
In the proportion R/W = S/T, R and T are known as the "extremes," while W and S are known as the "means."
Let's consider the proportion in the form of fractions, with R/W on one side and S/T on the other:
R/W = S/T
When we cross-multiply, we multiply the numerator of the left fraction (R) by the denominator of the right fraction (T), and vice versa:
R * T = W * S
Now, let's examine the impact of the product of the means, W * S, on the product of the extremes, R * T.
If the product of the means, W * S, increases, let's say W * S = X.
Now, let's consider what happens when we apply the same increase to both W and S. Let's assume both W and S increase by a factor of Y, with new values of W' and S':
W' = W * Y
S' = S * Y
Now, let's substitute these new values into the proportion:
R * T = W' * S'
R * T = (W * Y) * (S * Y)
R * T = W * S * Y * Y
R * T = W * S * Y^2
We can see from this equation that the product of the extremes, R * T, is equal to the initial product of the means, W * S, multiplied by Y^2.
Based on this reasoning, the product of the extremes will also increase by the square of the increase in the product of the means.
In simpler terms, if the product of the means increases by a certain factor, the product of the extremes will increase by the square of that factor.