# Variation In Mathematics Types And Examples

A variation is a relation between a set of values of variables, and under certain conditions, the values changes. In a variation, there are a set of values that are constant i.e. doesn’t change.

**Types of Variation**

There are four types of variation in mathematics and they are

- Direct Variation
- Inverse Variation
- Joint Variation
- Partial variation

### Direct Variation

In this type, the set of variables behaves the same way i.e. if one variable increases, the other will increase as well and vice-versa. Therefore, if A varies directly as B, it can be represented as A α B

Mathematically, it would then be expressed as A = KB

K is now the proportionality constant and it won’t change regardless of the condition of A and B.

### Inverse Variation

In this type, the variables vary disproportionately i.e. if one increases, the other decreases and vice-versa. Therefore, if A varies inversely as B, it can be represented as A α 1/B

Mathematically, it would be expressed as A = K/B

K is the proportionality constant.

### Joint Variation

This describes a situation where one variable depends on two or more other variables and varies directly as each of them. For instance, x varies jointly as a and b. This can be written as

X α ab

X = kab

### Partial variation

Partial variation exists when a quantity is partly constant and partly varies with another quantity. This type always leads to a simultaneous equation before you can get the two constant values.

For example, A is partly constant and partly varies as B can be written as

A = C + KB

C and K are the proportionality constant.

**Examples on Variation**

No 1: If x varies directly as y, and x = 6 when y = 2, what is the equation that describes this direct variation and find x when y = 5

Solution

X α y

X = ky

6 = k*2

K = 6/2 = 3

To write the equation,

X = 3y

To find x when y is 5

X = 3*5 = 15

No 2: If x varies inversely as y, and x = 9 when y = 3, write the equation for the variation and find y when x = 12

Solution

X α 1/y

X = k/y

K = xy

K = 9*3 = 27

The equation,

X = 27/y

To find y when x = 12

X = 27/y

12 = 27/y

Y = 27/12 = 9/4

No 3: If a varies jointly as b and c and a = 12, when b = 6 and c = 8, find a when b = 2 and y = 6.

Solution

A α bc

A = kbc

12 = k *6*8

K = 12/48 = 1/4

A = 1/4bc

To find a when b = 2 and y = 6

A = ¼ * 2 * 6 = 3

A = 3

No 4: The cost of car service is partly constant and partly varies with the time it takes to do the work. It costs ₦3,500 for a 10 hours service and ₦2,900 for a 4 hours service.

(a)Find the formula connecting cost, ₦C with time T hours.

(b) Hence find the cost of a 14 hours service.

Solution

₦C = K1 + TK2

K1 and K2 are the proportionality constant

For the first condition

₦3,500 = K1 + 10K2 ……… (i)

₦2,900 = K1 + 4K2 ……….. (ii)

This is now simultaneous equation

Now, subtract equ (ii) from equ (i)

₦3,500 – ₦2,900 = K1 – K1 +10k2 – 4K2

₦600 = 6K2

K2 = ₦600/6 = 100

Substitute k2 = ₦100 into equ (i)

₦3500 = K1 + 10*100

₦3500 = k1 + ₦1000

K1 = ₦3500 – ₦1000 = 2500

The formula connecting the ₦C and T

₦C = 2500 + 1000T

Hence find the cost of a 14 hours service

₦C = ₦2500 + 1000*14 = ₦2500 + ₦14000 = ₦16500

The cost will be ₦16500