If x varies as yz, then y varies inversely as
Variation Explained: Understanding 'x Varies as yz'
The core of this problem lies in understanding the concept of variation, specifically joint variation. We are given the relationship where one quantity, '\(x\)', changes proportionally with the product of two other quantities, '\(y\)' and '\(z\)' (\(yz\)). Our task is to figure out how '\(y\)' varies inversely with other variables based on this initial condition.
Mathematical Equation for Variation: x = kyz
The statement "\(x\) varies as \(yz\)" means that \(x\) is directly proportional to the product \(yz\). This relationship can be written as an equation using a constant of proportionality, \(k\):
$\( x = k \cdot yz $\)
In this equation, \(k\) is a constant value that does not change.
Determining the Inverse Variation for y
The question asks us to find what '\(y\)' varies inversely as. This means we need to express \(y\) in a form like \(y = \frac{C}{A}\), where \(C\) is a constant and \(A\) is the expression we're looking for. Let's rearrange our equation to solve for \(y\).
Step-by-Step Derivation
Starting with the equation:
$\( x = k yz $\)
We need to isolate \(y\).
- Divide both sides by \(yz\) to get \(k\): $\( \frac{x}{yz} = k $\)
- Rearrange this to express \(y\). We can write it as: $\( y = \frac{x}{kz} $\)
- This equation tells us \(y\) depends on \(x\) and \(z\). Specifically, \(y\) increases if \(x\) increases (direct variation) and \(y\) decreases if \(z\) increases (inverse variation). We can combine this as \(y \propto \frac{x}{z}\).
- To match the format "y varies inversely as A", we need \(y = \frac{C}{A}\). Let's rewrite \(y = \frac{x}{kz}\).
- Let's define a new constant \(C = \frac{1}{k}\). Since \(k\) is constant, \(C\) is also constant. The equation becomes: $\( y = \frac{C \cdot x}{z} $\)
- Now, we need to put this into the form \(y = \frac{C}{A}\). We can rewrite \(\frac{Cx}{z}\) as \(\frac{C}{(z/x)}\).
- This gives us: $\( y = \frac{C}{\left(\frac{z}{x}\right)} $\)
- By comparing this equation to the general form \(y = \frac{C}{A}\), we identify \(A\) as \(\frac{z}{x}\).
Conclusion: y Varies Inversely as z/x
The derivation shows that \(y\) is inversely proportional to the expression \(\frac{z}{x}\).
| Given Relationship | Derived Relationship |
|---|---|
| \(x = k yz\) | \(y = \frac{C}{(z/x)}\) where \(C = 1/k\) |
| Interpretation | \(y\) varies inversely as \(\frac{z}{x}\) |
Therefore, '\(y\) varies inversely as \(\frac{z}{x}\)'.
Similar Questions
Consider the following statements:
1. If x is directly proportional to z and y is directly proportional to z, then (x 2- y 2) is directly proportional to z 2.
2. If x is inversely proportional to z and y is inversely proportional to z, then (xy) is inversely proportional to z 2.
Which of the above statements is/are correct?
If (a3+ b3) is proportional to (a2 – b2), then (a2 - ab + b2) is proportional to
Given y is inversely proportional to √x, and x = 36 when y = 36. What is the value of x when y = 54?
A variable \(y\) is directly proportional to a variable quantity \(x^n\). Given that when \(x=2\), \(y=20.8\) and when \(x=3\), \(y=105.3\). Which one of the following is the constant of proportionality if it is greater than \(1\)?
Important Questions from Direct or Indirect Proportion
Meena sells 70 red marbles and 105 blue marbles in boxes without mixing such that each box has x number of marbles. What is the value of x?
If 3A = 4B = 5C, then A : B : C is equal to:
Divide Rs. 156 in the ratio 1 : 2 : 4 : 5. The rupees in the respective ratios are given by:
A. 13, 26, 53 & 64
B. 13, 26, 51 & 66
C. 13, 26, 52 & 65
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Mrs. Deepa Devi saves 30% of her salary. If she receives Rs. 42,000 per month as her salary, what is her monthly expenditure?
