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Explore the concept of direct proportionality and its inverse relationship in mathematical terms with x, y, 1/x, and 1/y.
If x and y are in direct proportion then 1/x and 1/y are
A. in direct proportionB. in inverse proportionC. neither in direct nor in inverse proportionD. sometimes in direct and sometimes in inverse proportion
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Answer : in inverse proportion
If two quantities, x and y, are directly proportional, it means that they increase or decrease together at the same rate. This relationship can be expressed mathematically as:
x = ky
where k is a constant of proportionality.
Now, let’s consider the reciprocals of x and y, which are 1/x and 1/y, respectively. We want to determine how these reciprocals relate to each other.
To do this, we can take the reciprocal of both sides of the equation that represents the direct proportion between x and y:
1/x = 1 / (ky)
Using the distributive property of division, we can rearrange the right side of the equation:
1/x = (1/k) * (1/y)
This equation shows that the product of 1/x and 1/y is equal to a constant value, which is (1/k). Since the product of two quantities is constant, they are in inverse proportion.
Therefore, the correct answer is:
B. in inverse proportion
This means that if 1/x increases, 1/y will decrease proportionally, and vice versa. They are not directly proportional, nor do they have a relationship that varies depending on the specific values.
Invertendo and Alternendo Property
The terms “invertendo property” and “alternendo property” typically refer to properties of proportions or equations.
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Invertendo Property: This property states that if in a proportion, the antecedent (the first term) and the consequent (the second term) are interchanged, the resulting proportion is still true. Mathematically, if you have a proportion a/b = c/d, then the invertendo property states that b/a = d/c. This property is based on the fact that when you invert both sides of a true proportion, the result remains true.
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Alternendo Property: This property states that if in a proportion, the terms are alternated (i.e., the first with the third and the second with the fourth), the resulting proportion is still true. Mathematically, if you have a proportion a/b = c/d, then the alternendo property states that a/c = b/d. This property essentially shows that the product of the means is equal to the product of the extremes.
These properties are commonly used in solving equations involving proportions or in proving certain mathematical statements involving proportions.
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