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.
In this article we would be looking at Variation in Mathematics Types and Examples.
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.
I hope you find this article helpful as well as interesting.
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
I hope you find this article helpful as well as interesting.
