Chapter Two: Variations – Additional Mathematics Form Two
Introduction
Some quantities relate to one another. The relationship in which a change in one quantity results in a proportional change in the other quantity is called variation. In this chapter, you will learn about direct, inverse, and joint variations.
The competencies developed will help you to relate quantities that require changes in response to other quantities. For instance, changing the quantity of food in relation to the change in the number of people; speed of a car in relation to distance covered; demand for a commodity in relation to its price; population of a town in relation to social services; number of workers in response to the change in the number of tasks to be accomplished, among many other real life applications.
Direct Variations
Some quantities in real life relate in such a way that they change together at the same rate. For instance, change in the quantity of food relates directly to the change in the number of people. This type of relationship is called direct variation. The quantities with direct variation relationship are said to be directly proportional.
Generally, if quantities A and B vary directly, then the change in quantity A causes a proportional change in quantity B. The symbol for proportionality is ∝, hence, the direct variation of two quantities A and B is written as A ∝ B.
If A ∝ B, then A = kB, where k is a proportionality constant.
Activity 2.1: Illustrating variations of two quantities
Individually or in a group, perform the following tasks:
- Prepare a rectangular piece of paper with dimension of your choice.
- Use a ruler to measure and record the length and width of the rectangular piece of paper constructed in task 1.
- Find the area of the rectangular piece of paper in task 1.
- Divide the rectangular piece of paper in task 1 into five rectangular pieces of paper of different widths having the same length.
- Find and record the area of each rectangular piece of paper in task 4.
- Draw the graph of width against area using the records in task 5.
- Use the graph from task 6 to find the relationship between width and area.
- Comment on the relationship obtained in task 7.
- List at least five examples of quantities you have encountered in your daily life that have similar relationship to the one found in task 7.
- Share your results with other students through discussion for more input.
Key Point: If A₁ and B₁ are first pair of values of A and B, respectively, then second pair of values A₂ and B₂ can be computed using the following relation:
A₁/B₁ = A₂/B₂ = k
Example 1
Given that x varies directly as y. If x is 20 when y is 45, find:
(a) The value of y when x = 100
(b) The value of x when y = 70
Solution
(a) Given that x varies directly as y. It implies that,
x ∝ y
Introduce the constant of proportionality k to obtain,
x = ky
Making k the subject of the formula gives,
k = x/y
Substitute x = 20 and y = 45 to obtain,
k = 20/45 = 4/9
When x = 100, then
4/9 = 100/y
4y = 900
y = 900/4 = 225
Therefore, the value of y is 225.
(b) When y = 70, it implies that,
4/9 = x/70
9x = 280
x = 280/9
Therefore, the value of x is 280/9.
Example 2
If y varies directly as the square of x, and y = 8 when x = 2, find:
(a) The values of x when y = 20
(b) The value of y when x = 1
Solution
(a) Given y varies directly as x². It implies that,
y ∝ x²
y = kx²
k = y/x²
But y = 8 when x = 2, substitute the values to obtain,
k = 8/2² = 8/4 = 2
When y = 20, then
2 = 20/x²
2x² = 20
x² = 10
x = ±√10
Therefore, the value of x = √10 or x = -√10.
(b) When x = 1, it implies that,
2 = y/1²
y = 2
Therefore, the value of y is 2.
Exercise 1
- If y varies directly as x and y = 10 when x = 15, find the value of y when x = 7.
- Given that y varies directly as three more than the square of x and that y = 24 when x = 3. Find:
- The equation relating x and y
- The value of x which corresponds to y = 168
- The value of y which corresponds to x = 7
- If A varies directly as B and the value of A = 15 when the value of B is 25:
- Express B in terms of A
- Find the value of A when the value of B = 60
- Find the value of B when the value of A = 18
- If the cube root of x varies directly as the square of y and x = 216 when y = 2, find:
- The equation connecting x and y
- The value of y when x = 1728
- The value of x when y = 4
Inverse Variations
Inverse variation is the relationship of quantities in which the increase in one quantity causes the decrease of the other quantity and vice versa. The quantities with an inverse variation relationship are said to be inversely proportional to each other.
For instance, if a person drives from town A to town B, the more the speed of the car, the less time it takes to travel between the two towns, thus speed of the car and time of travel are said to be inversely proportional.
If p has an inverse variation relationship with q, then p ∝ 1/q
The equation relating p and q is: p = k/q, where k is a constant of proportionality.
Thus, k = pq
Activity 2: Determining inverse variations of two quantities
In a group, perform the following tasks:
- Find at least 20 text books.
- One student in the group should arrange the text books in task 1 in a shelf and another student record the time taken to complete the task.
- Repeat task 2 using two, three, and four students in a group.
- Use the records in task 3 to draw the graph of number of students against the reciprocals of time.
- Use the graph from task 4 to find the relationship between the number of students and time.
- Comment on the relationship obtained in task 5.
- List at least five examples of quantities you have encountered in daily life that have similar relationship to the one found in task 5.
- Share your results with other students through discussion for more input.
Key Point: If p₁ and q₁ are first set of values of p and q, respectively, then second set of values p₂ and q₂ can be computed using:
p₁q₁ = p₂q₂ = k
Example 6
If y varies inversely as x and y = 60 when x = 1/12, find:
(a) The constant of variation
(b) The value of x when y = 1/2
(c) The value of y when x = 100
Solution
(a) Given y varies inversely as x, it implies that,
y ∝ 1/x
y = k/x
k = xy
But y = 60 when x = 1/12, substitute the values to obtain,
k = 60 × 1/12 = 5
Therefore, k = 5.
(b) When y = 1/2, then
5 = (1/2)x
x = 10
Therefore, the value of x is 10.
(c) When x = 100, then
5 = 100y
y = 5/100 = 1/20
Therefore, the value of y is 1/20.
Exercise 2
- If y varies inversely as x and that the constant of variation is 2, find:
- The value of y when x = 6
- The value of x when y = 0.7
- If the cube root of x varies inversely as the square of y and x = 27 when y = 3, find:
- The constant of variation
- The value of x when y = 7
- The value of y when x = 125
- Given that y is indirect proportional to the square of x and y = 1.25 when x = 2. Find:
- A formula giving y in terms of x
- The value of y when x = 1/4
- The value of x when y = 0.2
Joint Variations
A joint variation is the relationship among three or more quantities in which one quantity varies directly or inversely with two or more quantities. The quantities with a joint variation relationship are said to be jointly proportional.
For instance, if a quantity A varies directly with quantities B and C, then quantity A varies directly with both the quantities B and C. That is, A ∝ BC.
If A ∝ BC, then A = kBC, where k is a constant of proportionality.
If p ∝ q/r, then p = kq/r, where k is a constant of proportionality.
Example 11
If h varies jointly as l and m, such that h = 10 when l = 4 and m = 5:
(a) Find the value of the constant of proportionality
(b) Express m in terms of h and l
(c) Find the value of m when l = 20 and h = 30
(d) Find the value of h when m = 25 and l = 8
Solution
(a) Given h ∝ lm
h = klm
k = h/(lm)
But h = 10, l = 4, and m = 5, substitute the values to obtain,
k = 10/(4×5) = 10/20 = 1/2
Therefore, the constant of proportionality, k is 1/2.
(b) From h = klm, make m the subject of the formula to obtain,
m = h/(kl)
Substituting k = 1/2 gives,
m = h/((1/2)l) = 2h/l
Therefore, m = 2h/l.
(c) From m = 2h/l. If h = 30 and l = 20, it implies that,
m = (2×30)/20 = 60/20 = 3
Therefore, the value of m is 3.
(d) From h = klm. If l = 8 and m = 25, it implies that,
h = (1/2)×8×25 = 4×25 = 100
Therefore, the value of h is 100.
Exercise 3
- If y varies jointly as x and z such that x = 2 when z = 3 and y = 12, find the value of y when x = 7 and z = 4.
- If y varies jointly with x and z², such that y = 2 when x = 4 and z = 2, find the value of x when y = -1/2 and z = 1/2.
- Given that y varies jointly with x and the square root of z. If y = 2 when x = 1/8 and z = 1/4:
- Write the formula of y in terms of x and z
- Find the value of y when x = 3/8 and z = 1/9
- If 2 tractors can plough 6 acres of land in 4 hours. How many tractors with similar ability are needed to plough 8 acres of land in 8 hours?
Chapter Summary
- Two quantities are said to be directly proportional if a change in one quantity causes a proportional change (increase or decrease) in the other quantity.
- Inverse variation of two quantities is a relationship where the increase of one quantity causes a proportional decrease of the other quantity and vice versa.
- Joint variation is a relationship among three or more quantities in which one quantity varies directly or inversely with two or more quantities.
Revision Exercise 2
- Suppose y varies directly as x and y = 0.12 when x = 0.2.
- Find the value of x when y = 0.30
- Show that the value of y when x = 0.8 is 0.48
- A quantity p varies inversely as the square root of w and p = 1/3 when w = 27.
- Express p in terms of w
- Find the value of w when p = √12
- If V varies jointly as h and the square of r, and V = 45π when r = 3 and h = 5. Find the value of r when V = 175π and h = 7.
- The temperature T of water in a lake varies inversely to the water’s depth D. If at the depth of 60 m the temperature of water is 20°C, find:
- An equation relating temperature and depth
- The temperature of water at 250m
- If x varies directly as y and inversely as z, such that x = 5 when y = 3 and z = 2.
- Express y in terms of other variables
- Find z when x = 4 and y = 6