Direct, and Inverse Proportion; Invariants.

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In summary, the conversation discusses the concept of direct proportionality and its relation to invariants. It is explained that if two quantities are directly proportional, their quotient is a constant. This also holds true for another pair of directly proportional quantities, and when their quotients are multiplied, the result is a third invariant. However, this method cannot be applied to the given example since the constants in the equations may depend on other variables. Therefore, the correct solution involves finding the individual constants for each pair of directly proportional quantities and then using them to calculate the final value of x.
  • #1
Faizan Sheikh
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If two quantities are directly proportional then their quotient is an invariant(it does not change, it is constant). Further, if we have another pair of two quantities that are directly proportional then their quotient is also an invariant. Moreover, if you multiple these two quotients, you end up with a third invariant.
But, consider the following example.
Given that x is directly proportional to y and to z and is inversely propotional to w, and that x=4 when (w,y,z)=(6,8,5), what is x when (w,y,z)=(4,10,9)?

Correct Solution:
xw is a contant, x/z is a constant, and x/y is a constant.
Thus, xw/yz is constant. (They just combined the constant terms)

so, xw/yz=(4)(6)/(8)(5)=3/5.

Thus, when (w,y,z)=(4,10,9), we find

x=3yz/5w=27/2.


Wrong Solution a.k.a My Solution:
Since xw is a contant, x/z is a constant, and x/y is a constant. Therefore if we multiply all these constant terms we will get a constant.

Thus, (x^3)w/yz is a constant. The rest is immaterial since I do not end up with 27/2.

I guess there is something wrong with (x^3)w/yz being a constant. Can anybody please explain to me why (x^3)w/yz is not a constant? Hence explain why I do not get 27/2 if my method is followed?
 
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  • #2
I would like to see how the "combined the constants" in the first solution!

The problem with your solution is that "x is directly proportional to z" means
x/z= k1 (constant) for fixed y and w. "x is directly proportional to y" means x/y= k2 (constant) for fixed z and w. k1 may depend on y and w, k2 may depend on z and w. That's why you cannont just multiply the two equations and say
x2/yz= constant.
 
  • #3



Your understanding of direct and inverse proportion is correct, but your approach to solving the problem is not entirely accurate. Let's break it down step by step to understand where the mistake lies.

Firstly, the given information states that x is directly proportional to y and z, and inversely proportional to w. This means that the ratio of x to y is always constant, as is the ratio of x to z. However, the ratio of x to w is not constant - it changes depending on the value of w. This is because in inverse proportion, as one quantity increases, the other decreases. So, the constant term in this case is not xw, but rather x/y and x/z.

Now, let's look at the given values. When (w, y, z) = (6, 8, 5), we know that x = 4. This means that x/y = 4/8 = 1/2 and x/z = 4/5. We also know that xw is a constant, which we can represent as k. So, we have the following equations:
x/y = 1/2
x/z = 4/5
xw = k

To find the value of x when (w, y, z) = (4, 10, 9), we can use the constant terms we already know. We have x/y = 1/2 and x/z = 4/5, but we need to find the value of xw. To do this, we can use the given value of x when (w, y, z) = (6, 8, 5) to solve for k.
xw = k
4w = k (since xw = 4w when (w, y, z) = (6, 8, 5))

Now, we can substitute this value of k into our original equation xw = k to find the value of x when (w, y, z) = (4, 10, 9).
xw = k = 4w
x/y = 1/2
x/z = 4/5

xw/yz = (4w)/(10*9) = 2/9

Thus, x = (2/9)(10)(9) = 20/3 = 6 2/3.

In conclusion, your
 

1. What is direct proportion?

Direct proportion is a relationship between two variables where an increase in one variable leads to a corresponding increase in the other variable, and a decrease in one variable leads to a corresponding decrease in the other variable.

2. What is inverse proportion?

Inverse proportion is a relationship between two variables where an increase in one variable leads to a corresponding decrease in the other variable, and a decrease in one variable leads to a corresponding increase in the other variable.

3. How do you determine if two variables are in direct proportion?

To determine if two variables are in direct proportion, you can create a table or a graph and check if the values of one variable increase or decrease in proportion to the values of the other variable.

4. How do you determine if two variables are in inverse proportion?

To determine if two variables are in inverse proportion, you can create a table or a graph and check if the values of one variable increase while the values of the other variable decrease, or vice versa.

5. What are invariants in direct and inverse proportion?

Invariants are values that remain constant in a direct or inverse proportion relationship. In direct proportion, the product of the two variables remains constant, while in inverse proportion, the ratio of the two variables remains constant.

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