Direct, Inverse, Joint, and Combined Variation

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A step-by-step guide to direct, inverse, joint, and combined variation

In a mathematical equation that establishes a relationship for some type of parameter, there are usually two types of quantities. One is a constant that does not change with the changes of other parameters in the equation and the other is a variable that changes for different situations.

Changing variable parameters is called variation. Variation can have different types according to the pattern of change or relationships of variables:

Direct variation:

Direct variation between two variables exists when one quantity is directly dependent on the other, i.e. If one quantity increases relative to the other and vice versa.

It is a relationship between two variables where one variable is a constant multiple of the other. Since these two variables are directly related to each other, it is also called directly proportional.

When two quantities are directly proportional to each other, they are represented using the symbol “(∝)”. Suppose there are two quantities (x) and (y) that are in direct variation, then they are expressed as follows:

(y∝x)

When the proportionality sign is removed, the direct variation formula is given as follows:

(color{blue}{y=kx})

Where (k) is the constant of proportionality.

Inverse variation:

If one quantity is increasing and the other quantity is decreasing at the same time, the two quantities are said to be in inverse variation. It is expressed as (y∝frac{1}{x}). The inverse variation formula is given by:

(color{blue}{y=frac{k}{x}})

Joint variation:

Joint variation describes a situation in which one variable is dependent on two (or more) other variables and, when the others are held constant, varies directly as each of them. We say that (z) varies jointly as (x) and (y) if:

(color{blue}{z=kxy})

For some constant (k).

Combined variation:

Combined variation describes a situation in which one variable is dependent on two (or more) other variables, varying directly with some of them and inversely with others (when all other variables are held constant). For example, if (z) varies directly as (x) and inversely as (y), we have the following combined variation equation:

(color{blue}{z=k(frac{x}{y})})

Direct, Inverse, Joint, and Combined Variation, Example 1:

Let x and y be in direct variation, (x = 6) and (y = 28). Find the equation of the direct variation.

Solution: As (x) and (y) are in a direct variation thus (y = kx) or (k =frac{y}{x}).

(k=frac{28}{6}=frac{14}{3})

The direct variation equation is (y=frac{14}{3}x)

How to Solve Any Variation Problem

Every variation problem follows the same four steps.

1. Read the wording and write the general equation with k.
2. Plug in the first set of values and solve for k.
3. Rewrite the equation using the value of k.
4. Plug in the new values and solve for the unknown.

Worked Examples: Inverse, Joint and Combined Variation

Inverse variation: y varies inversely with x, and y = 8 when x = 3. Find y when x = 6. Write y = k/x. Then 8 = k/3, so k = 24. Now y = 24/6 = 4.

Joint variation: z varies jointly with x and y, and z = 60 when x = 3 and y = 4. Find z when x = 2 and y = 7. Write z = kxy. Then 60 = k(3)(4), so k = 5. Now z = 5(2)(7) = 70.

Combined variation: z varies directly with x and inversely with y, and z = 12 when x = 6 and y = 2. Find z when x = 10 and y = 5. Write z = kx/y. Then 12 = 6k/2 = 3k, so k = 4. Now z = 4(10)/5 = 8.

Word problem: The time to finish a job varies inversely with the number of workers. Six workers take 10 hours. How long will 4 workers take? Write t = k/w. Then 10 = k/6, so k = 60. With 4 workers, t = 60/4 = 15 hours. Fewer workers means more time, which is what inverse variation predicts.

Frequently Asked Questions

What is the difference between joint and combined variation?

Joint variation means one quantity varies directly with two or more others multiplied together, as in z = kxy. Combined variation mixes direct and inverse variation in one equation, as in z = kx/y. If every variable is in the numerator, it is joint. If at least one variable is in the denominator, it is combined.

How do you find the constant of variation k?

Write the general equation, then plug in one known set of values and solve for k. For inverse variation y = k/x with y = 5 when x = 4, multiply both sides by 4 to get k = 20. Once you have k, you can find any other value in the same problem.

How can you tell direct from inverse variation in a table?

Divide y by x for each row. If y/x is the same every time, it is direct variation and that number is k. If instead the product xy is the same every time, it is inverse variation. For example, the pairs (2, 6), (3, 4) and (6, 2) all have xy = 12, so y = 12/x.

What is an example of joint variation in real life?

The area of a rectangle varies jointly with its length and width: A = lw, where k = 1. The volume of a cylinder, V = πr²h, varies jointly with the square of the radius and the height, with k = π. Simple interest, I = Prt, varies jointly with the principal, rate and time.

Can a variation involve squares or square roots?

Yes. A quantity can vary with a power or root of another. For example, ‘y varies directly with the square of x’ means y = kx². If y = 18 when x = 3, then 18 = 9k and k = 2, so y = 2x². The same four-step method still works.