What is Variation?
If the value of a quantity can change according to different situations, it is termed a variable. If a change in the value of variable results in a change in the value of a related variable, their relationship is termed a variation.
Direct Variation
The variables in a direct variation are said to be directly proportional. If the value of one variable increases, the value of the other variable also increases, and vice versa. Assuming k to be constant, a direct variation can be expressed as:
y = kx
Examples:
- More construction, more materials
- Less money, less shopping
- More students, more teachers
- Less workers, less work
Inverse Variation
The variables in an inverse variation are said to be inversely proportional. If the value of one variable increases, the value of the other variable decreases, and vice versa. Assuming k to be constant, an inverse variation can be expressed as:
xy = k or y = k/x
Examples:
- Less speed; more time taken
- More speed; less time taken
- More time; less workers required
- Less time; more workers required
Examples
1. If the sales tax on a Rs 60 purchase is Rs. 4, what would it be on a Rs 300 purchase?
Solution:
Here, the sales tax increases if the cost of purchase increases. Therefore, this is a direct variation.
Sales tax on Rs 60 purchase = Rs 4
Sales tax on Rs 1 purchase =rs 4/60 = rs 1/15
Sales tax on Rs 300 purchase = (1/15) x 300 = Rs 20
Therefore, the sales tax on Rs 300 purchases is Rs 20.
2. If 15 men can complete a piece of work in 50 days. How many days will be required to complete the work if 30 men work together?
Solution:
Here, the number of days required to complete the work decreases if the number of men increases. Hence, this is an inverse variation.
15 men can complete a piece of work in 50 days.
1 man can complete the work in 50 x 15 = 750 days
30 men can complete the work in 750/30 = 25 days
Therefore, 30 men can complete the work in 25 days.
3. If y varies directly as x, and y = 25 when x = 75, find x when y = 80.
Solution:
First, write an equation of direct variation:
y=kx
Where k is a constant.
Substituting the values of x and y,
y = kx
25 = 75k
k = 25/75
k = 1/3
Therefore, an equation of the direct variation is y = ⅓ x
To find x when y = 80, substitute in y = ⅓ x
y = ⅓ x
80 = ⅓ x
x = 80 x 3
x = 240
Therefore, when y =80, x = 240
4. If y is inversely proportional to x, and y = 8 when x = 20, find x when y = 16.
Solution:
First, write an equation of inverse variation:
xy = k
Where k is a constant.
Substituting the values of x and y,
xy = k
20 x 8 = k
k = 160
Therefore, an equation of the direct variation is xy = 160
To find x when y = 16, substitute in xy = 160.
xy = 160
x(16) = 160
x = 160/16x = 10
Therefore, when y = 16, x = 10
5. A man driving at a speed of 60 km/h covers a distance in 40 minutes.
i) How long will he take to cover the same distance as a speed of 50 km/hr.
ii) What will be his speed if he covers the same distance in 60 minutes?
Solution:
i) Speed (Km/hr) 60 50
Time (min) 40 ?
Let the time taken be x minutes. As speed decreases, the time is taken to travel a given distance increases, hence, this is an inverse variation. Therefore,
60 x 40 = 50 x x
x = (60 x 40)/50
x = 48
The man takes 48 min to travel if he drives at a speed of 50 km/hr.
ii) Speed (Km/hr) 60 ?
Time (min) 40 60
Let speed be x km/hr. Since speed decreases as time increases, hence, this is an inverse variation. Therefore,
60 x 40 = x x 60
x = (60 x 40)/60
x = 40
The man drives at a speed of 40 km/hr.
Did You Know?
- A variation where one quantity varies directly as the product of two or more quantities is called a joint variation. For example: if x varies directly as y and the square of z, then, x = kyz2, where, k is a constant. It can be said that z varies jointly as y and z2.
- Sums on direct and inverse variation can be solved using the unitary method or proportion.
It states that the force between two bodies varies jointly as their masses m1 and m2 and inversely to the square of the distance between their centers.
- k is a non-zero constant in both direct and inverse variations.
- Simple interest vs time, density vs mass, and force vs acceleration are some examples of direct variation.