Question

If the quantities x, y and z are related by the equation$z=3\frac{x}{y}$, then we say that z is _____ _____to x_____ _____ to y.

Short answer

If the quantities x, y and z are related by the equation $z=3\frac{x}{y}$, then we say that z is directly proportional to x and inversely proportional to y.

Step-by-step solution

Step 1. Apply the concept of variation.

Two quantities are directly proportional if on increasing one quantity the other quantity also increases. And on decreasing one quantity the other quantity also decreases.

Mathematically it is expressed as $y=kx$ where y and x are quantities and k is the constant of proportionality along with the condition $k\ne 0$.

It is read as y varies directly with x or y is directly proportional to x.

Two quantities are inversely proportional if on increasing one quantity the other quantity also decreases. And on decreasing one quantity the other quantity increases.

Mathematically it is expressed as $y=\frac{k}{x}$ where y and x are quantities and k is the constant of proportionality along with the condition $k\ne 0$.

It is read as y varies inversely with x or y is inversely proportional to x.

Step 2. Example of Inverse variation and direct variation.

To denote the statement that the distance covered by a car is directly proportional to the fuel consumption, denote fuel consumption by f and distance covered by d. Therefore, it is expressed as $f=kd$ where k is the constant of proportionality along with the condition $k\ne 0$.

To denote the statement that a number is inversely proportional to the square of other number, denote first number by pand second number by q. Therefore, it is expressed as $p=\frac{k}{q^2}$ where k is the constant of proportionality along with the condition $k\ne 0$.

Step 3. Evaluate the expression.

Consider the provided expression $z=3\frac{x}{y}$.

Compare it with the expression $z=k\frac{x}{y}$.

This means z is inversely proportional to y.

And z is directly proportional to x.

When expressions are compared $k=3$.

Therefore, constant of proportionality is 3.

Thus, the complete statement is “If the quantities x, y and z are related by the equation $z=3\frac{x}{y}$, then we say that z is directly proportional to x and inversely proportional to y.”