Variation

Two quantities may be such that as one quantity changes in value, the other quantity also changes in value. This variation in value may be:

• Directly Proportional
• Directly Related
• Indirectly Proportional
• Indirectly Related
• A mixture of the above (Joint Variation)

Direct Variation

It may be of two types: Directly Proportional and Directly Related

Directly Proportional

One quantity B is directly proportional to another quantity A if their relation is such that if A is increased in a certain ratio, B also increases in the same ratio and if A is decreased in a certain ratio, B also decreases in the same ratio.

E.g. if A becomes three times, B also becomes thrice and if A becomes 2.6 times, B also becomes 2.6 times

Relationship of direct proportion is denoted as B α A (B varies directly as A).

Directly Proportional means Ratio is constant

If B α A, then B = kA, where k is a constant (constant of proportionality).
So, B/A = k or the ratio of the two quantities is a constant.
Conversely, when the ratio of two quantities is a constant, we can conclude that they vary directly with each other.

Two pairs of directly proportional variables

If B varies directly with A and we have two sets of values of the variables B and A, A1 corresponding to B1 and A2 corresponding to B2, then, since B α A, we can write down:
B1/A1 = B2/A2
Or B1/B2 = A1/A2

Real-life examples of of direct proportion

A few instances of direct proportion are:

a. Expenditure α price per unit (if number of units bought is constant)

b. Amount of work done α Number of men working (rate of work and time for which work done remaining constant)

c. Distance covered α Speed (time remaining constant)

Q. The expense on a ride in an adventure park is directly proportional to the number of people riding on it. When 15 people take a ride then expense is Rs. 300. What would be the expense if 45 people take a ride?

Explanation:

Special Cases of direct proportion

A quantity B can also be directly proportional to some power of A.

That is, B α $A^n$ or B = k $A^n$

So, the ratio $\frac{B}{A^n}$ = k (i.e. constant)

Q. The height of a boy is in direct proportion to the square root of his age. If at age of 4 years he was 2 feet tall, then what will be his height 12 years later?

Explanation:

Directly Related

If B is directly related to A (not directly proportional), then as A increases, B also increases but not proportionally.

B = $k_1$ A + $k_2$ (where $k_1$ and $k_2$ are constants.)

In direct proportion the line passes through origin (i.e. when the value of A is zero, B is also zero).
But in the case of direct relation, even though A = 0, B is not zero.

Real-life examples of of direct proportion

Total cost of production = fixed cost + variable cost = fixed cost + (cost per unit × number of units produced)

Total cab fare = fixed cost + (fare per km × number of kilometers travelled)

Variable cost - Cost per unit (will be incurred only if any unit is produced, e.g. raw material) or Fare per km (will have to be given only if cab moves)

Fixed cost - Cost even if no unit is produced (e.g. rent of place, cost of machinery, salaries) or no distance travelled by the cab.

Q. There is a ride in an adventure park. When 10 people take a ride then expense is Rs. 300, while when 15 people take a ride then expense is Rs. 400. What would be the expense if 25 people take a ride?

Explanation:

Inverse Variation

It may be of two types: Inversely Proportional and Inversely Related

Inversely Proportional

One quantity B is inversely proportional to another quantity A if their relation is such that if A is increased in a certain ratio, B gets decreased in the same ratio and if A is decreased in a certain ratio, B also gets increased in the same ratio.

Inversely Proportional means Product is constant

B being inversely proportional to A is the same as saying that B varies in direct proportion with 1/A.
So, B α 1/A
Or B = k/A, where k is the constant of proportionality.
So, AB = k, a constant

Conversely, if the product of two quantities is a constant, we can conclude that they vary inversely with each other.

Two pairs of inversely proportional variables

If B varies inversely with A and we have two sets of values of the variables B and A, A1 corresponding to B1 and A2 corresponding to B2, then, since B α 1/A, we can write down:
A1 B1 = A2 B2
Or A1/A2 = B2/B1

Real-life examples of of inverse proportion

A few instances of Inverse Proportion are:

a. More the number of people, less is the time needed to complete a given work.

b. Time taken to travel a distance decreases proportionally as speed increases (if distance is constant).

Q. If 10 men can build a wall in 20 days, then in how many days will 25 men build it?

Explanation:

Special Cases of direct proportion

A quantity B can also be inversely proportional to some power of A.

That is, B α $\frac{1}{A^n}$
or B $A^n$ = k (i.e. constant)

Inversely Related

If B is inversely related to or varies inversely with A (not inversely proportional), then as A increases (or decreases), B decreases (or increase) but not in the same proportion.

B = $\frac{k_1}{A}$ + $k_2$
(Where $k_1$ and $k_2$ are constants.)

Joint Variation

A variable may be proportional to more than one variables at the same time.

Let us see a few cases:

Case 1

If there are three quantities A, B and C such that:
A α B (if C is constant) and
A α C (if B is constant)

Then, A α BC (i.e. A varies jointly with B and C )
Or A = kBC (where k is the constant of proportionality)

Case 2

If there are three quantities A, B and C such that: A α B and A α 1/C

Then, A α B/C Or A = kB/C Or AC/B = k (where k is the constant of proportionality)