Form Two Mathematics Notes: Rates and Variation

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  1. Chapter 1Rates and VariationRead below · Free
  2. Chapter 2CongruenceContinue in the app
  3. Chapter 3SimilarityContinue in the app
  4. Chapter 4AlgebraContinue in the app
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Lesson Objectives

By the end of this chapter, students should be able to:

1. Key Words and Concepts

Before we begin, let us define the core terms of this chapter.

Rate: A comparison of two quantities measured in different units. It shows how one quantity changes in relation to another.
Speed: The rate of distance covered per unit time: Speed = Distance ÷ Time.
Variation: A relationship in which one quantity changes as another quantity (or other quantities) changes.
Direct variation: A relationship where an increase in one quantity causes a proportional increase in the other: y = kx.
Inverse variation: A relationship where an increase in one quantity causes a proportional decrease in the other: y = k/x.
Joint variation: A relationship in which a quantity varies directly as the product of two or more quantities: y = kxz.
Combined variation: A relationship in which a quantity varies directly as one quantity and inversely as another: y = kx/z.
Constant of variation (k): The fixed number that links the varying quantities in a variation equation.
Proportional: Changing in the same ratio; for example when one quantity doubles, the other doubles too (direct).

2. Concept of Rate

A rate compares two quantities that are measured in different units. It shows how one quantity changes in relation to another.

Common Rates

Common Examples in Daily Life

ExampleRateUnit
Speed of a carDistance per timekm/h
Price of goodsCost per itemTsh/item
WagesMoney per hour or dayTsh/hour
Flow of waterVolume per timelitres/minute

Worked Examples

Example 1. A car travels 240 km in 4 hours. Find its speed.
Solution: Speed = Distance ÷ Time = 240 ÷ 4 = 60 km/h
Example 2. A worker earns Tsh 80,000 for 5 days. Find his rate of pay per day.
Solution: Rate = 80,000 ÷ 5 = Tsh 16,000 per day
Example 3. If 15 litres of petrol cost Tsh 45,000, find the price per litre.
Solution: Rate = 45,000 ÷ 15 = Tsh 3,000 per litre
Example 4. A pipe delivers 60 litres of water in 3 minutes. Find the rate of flow.
Solution: Rate = 60 ÷ 3 = 20 litres/minute

📐 Mathematics in Motion

The speed of light is about 300,000 km per second, one of the most basic rates in the universe. Light from the Sun takes about 8 minutes to reach Earth.

Practice question: A car travels 120 km in 2 hours. What is its speed?

Speed = 120 ÷ 2 = 60 km/h

3. Calculating Rates from Given Quantities

To calculate a rate, divide the first quantity by the second quantity. Rates always have units (such as km/h, Tsh/item or litres/min).

Rate = Quantity A ÷ Quantity B
Example 1. A cyclist covers 150 km in 5 hours. Find his speed.
Solution: 150 ÷ 5 = 30 km/h
Example 2. A pump fills 180 litres in 6 minutes. Find the rate of flow.
Solution: 180 ÷ 6 = 30 litres/minute
Example 3. A teacher earns Tsh 1,200,000 for 20 working days. Find the pay rate per day.
Solution: 1,200,000 ÷ 20 = Tsh 60,000 per day
Example 4. A car uses 25 litres of petrol to travel 200 km. Find the petrol consumption per kilometre.
Solution: 25 ÷ 200 = 0.125 litres/km

Practice question: A bus covers 480 km in 8 hours. Find its average speed.

Speed = 480 ÷ 8 = 60 km/h

🧠 Jog Your Brain — Check Your Understanding

Answer the questions. Click “View Answer” to reveal the correct response.

1. A rate compares two quantities measured in:

A. Different units

2. A car travels 200 km in 4 hours. Its speed is:

A. 50 km/h

3. Tsh 90,000 is earned in 6 days. The rate per day is:

A. Tsh 15,000

4. Density is calculated as:

A. Mass ÷ Volume

5. A pump fills 100 litres in 5 minutes. The rate of flow is:

A. 20 litres/minute

4. Direct Variation

Two quantities vary directly if an increase in one causes a proportional increase in the other (and a decrease causes a proportional decrease).

If y varies directly as x, then:

y ∝ x   or   y = kx, where k is the constant of variation and k = y ÷ x.
Example 1. y varies directly as x. When x = 4, y = 12. Find y when x = 7.
Solution: k = 12 ÷ 4 = 3, so y = 3 × 7 = 21
Example 2. The cost of rice varies directly with its mass. If 5 kg costs Tsh 15,000, find the cost of 8 kg.
Solution: k = 15,000 ÷ 5 = 3,000; Cost = 3,000 × 8 = Tsh 24,000
Example 3. At constant speed, distance varies directly with time. A car travels 90 km in 1.5 hours. How far will it travel in 4 hours?
Solution: k (speed) = 90 ÷ 1.5 = 60 km/h; Distance = 60 × 4 = 240 km

Practice question: y varies directly as x. When x = 5, y = 20. Find y when x = 9.

k = 20 ÷ 5 = 4; y = 4 × 9 = 36

5. Inverse Variation

Two quantities vary inversely if an increase in one causes a proportional decrease in the other.

If y varies inversely as x, then:

y ∝ 1/x   or   y = k/x, where k = xy (a constant).
Example 1. y varies inversely as x. When x = 4, y = 6. Find y when x = 8.
Solution: k = 4 × 6 = 24; y = 24 ÷ 8 = 3
Example 2. 6 workers can build a wall in 12 days. How many days will 9 workers take (same work rate)?
Solution: More workers means fewer days, so this is inverse variation. k = 6 × 12 = 72; Days = 72 ÷ 9 = 8 days
Example 3. The time taken to travel a fixed distance varies inversely with speed. A journey takes 5 hours at 60 km/h. How long at 75 km/h?
Solution: k = 5 × 60 = 300; Time = 300 ÷ 75 = 4 hours

📐 Mathematics in Motion — Boyle's Law

Boyle's Law states that the pressure of a gas varies inversely with its volume at constant temperature. This means P × V = constant.

Practice question: y varies inversely as x. When x = 3, y = 10. Find y when x = 5.

k = 3 × 10 = 30; y = 30 ÷ 5 = 6

6. Joint Variation

A quantity varies jointly as two or more other quantities if it varies directly as their product.

If y varies jointly as x and z, then:

y ∝ xz   or   y = kxz, where k = y ÷ (xz).
Example 1. y varies jointly as x and z. y = 12 when x = 3 and z = 2. Find y when x = 4 and z = 5.
Solution: k = 12 ÷ (3 × 2) = 2; y = 2 × 4 × 5 = 40
Example 2. The volume V of a cylinder varies jointly as the square of the radius and the height. V = 100 when r = 2 and h = 5. Find V when r = 3 and h = 4. (Use V = kr²h.)
Solution: k = 100 ÷ (2² × 5) = 100 ÷ 20 = 5; V = 5 × 3² × 4 = 5 × 9 × 4 = 180

Practice question: y varies jointly as x and z. y = 16 when x = 2 and z = 4. Find y when x = 5 and z = 3.

k = 16 ÷ (2 × 4) = 2; y = 2 × 5 × 3 = 30

7. Combined (Direct and Inverse) Variation

A quantity may vary directly as one quantity and inversely as another at the same time.

If y varies directly as x and inversely as z, then:

y ∝ x/z   or   y = kx/z, where k = yz ÷ x.
Example 1. y varies directly as x and inversely as z. y = 24 when x = 4 and z = 2. Find y when x = 8 and z = 6.
Solution: k = (24 × 2) ÷ 4 = 12; y = 12 × 8 ÷ 6 = 16
Example 2. P varies directly as T and inversely as V. P = 10 when T = 200 and V = 40. Find P when T = 300 and V = 50.
Solution: k = (10 × 40) ÷ 200 = 2; P = 2 × 300 ÷ 50 = 12

Practice question: y varies directly as x and inversely as z. y = 6 when x = 3 and z = 2. Find y when x = 6 and z = 4.

k = (6 × 2) ÷ 3 = 4; y = 4 × 6 ÷ 4 = 6

8. Relating Variations to Graphs and Tables

VariationEquationGraph
Directy = kxA straight line passing through the origin; its slope is k.
Inversey = k/xA decreasing curve that approaches both axes but never touches them.

Table of Values

For y = 2x (direct variation):

x12345
y246810

For y = 20/x (inverse variation):

x124510
y2010542

Practice question: Describe the shape of the graph of y = 4x and of y = 16/x.

y = 4x is a straight line through the origin with slope 4. y = 16/x is a decreasing curve that approaches both axes but never touches them.

9. Applying Rates and Variation to Real-Life Problems

Rates and variation appear in business (unit prices), science (speed, density, pressure), agriculture (work rates) and everyday life (wages, water flow, fuel consumption).

Example 1. Fuel consumption. A car uses 6 litres of fuel for 90 km. How many litres are needed for 300 km?
Solution: Direct variation: k = 6 ÷ 90 = 1/15 litre per km; for 300 km, fuel = 300 ÷ 15 = 20 litres
Example 2. Work rate (inverse). If 5 men build a wall in 15 days, how many days will 10 men take?
Solution: k = 5 × 15 = 75; Days = 75 ÷ 10 = 7.5 days
Example 3. Direct cost. If 3 kg of rice cost Tsh 9,000, find the cost of 8 kg.
Solution: Rate = 9,000 ÷ 3 = 3,000 Tsh/kg; Cost = 3,000 × 8 = Tsh 24,000
Example 4. Combined variation. Pressure P varies directly as temperature T and inversely as volume V. P = 20 when T = 300 and V = 40. Find P when T = 600 and V = 30.
Solution: k = (20 × 40) ÷ 300 = 8/3; P = (8/3) × 600 ÷ 30 = 53.3 (to 1 decimal place)

✏️ Classroom Activity: Variation in My Community

In groups, find three real situations in your community (for example the price of sugar, bus fares, the time taken to dig a field with more people).

  • State whether each is direct, inverse, joint or combined variation.
  • Write the equation using k and invent a numerical problem.
  • Swap with another group and solve each other's problems.

📐 Mathematics in Motion

Every time you read a speedometer, work out a unit price in a shop or estimate how long a journey will take, you are using rates and variation.

10. Chapter Summary

  • A rate compares two quantities with different units.
  • Direct variation: y = kx, with k = y ÷ x. Quantities increase or decrease together.
  • Inverse variation: y = k/x, with k = xy. One increases as the other decreases.
  • Joint variation: y = kxz, with k = y ÷ (xz).
  • Combined variation: y = kx/z, with k = yz ÷ x.
  • Direct variation graphs are straight lines through the origin; inverse variation graphs are curves that never touch the axes.
  • These ideas are essential in science, business and everyday life.

Key Formulas

TypeFormula
RateQuantity A ÷ Quantity B (with appropriate units)
Direct variationy = kx (k = y ÷ x)
Inverse variationy = k/x (k = xy)
Joint variationy = kxz (k = y ÷ xz)
Combined variationy = kx/z (k = yz ÷ x)

📝 End-of-Chapter Assessment — Suggested Answers

Complete the questions, then click the buttons to check your answers.

Part A — Multiple Choice (10)

1. A car travels 150 km in 3 hours. Its speed is:

A. 50 km/h

2. y varies directly as x. When x = 3, y = 12. Find y when x = 5.

A. 20

3. y varies inversely as x. When x = 2, y = 12. Find y when x = 6.

A. 4

4. 4 workers finish a job in 10 days. How many days will 5 workers take?

A. 8 days

5. y varies jointly as x and z. y = 24 when x = 2 and z = 3. Find y when x = 3 and z = 5.

A. 60

6. y varies directly as x and inversely as z. y = 6 when x = 3 and z = 2. Find y when x = 6 and z = 4.

A. 6

7. The graph of y = kx is:

A. A straight line through the origin

8. Tsh 54,000 buys 18 litres of petrol. The price per litre is:

A. Tsh 3,000

9. Pressure P of a gas varies inversely with volume V. P = 20 when V = 6. Find P when V = 4.

A. 30

10. Which pair of quantities is compared in a rate?

A. Quantities in different units

Part B — Short Answer (10)

1. Define a rate and give two examples with their units.

A rate compares two quantities measured in different units. Examples: speed in km/h; price in Tsh/item (also wages in Tsh/hour, flow in litres/minute).

2. A worker earns Tsh 840,000 in 14 days. Find the pay rate per day.

840,000 ÷ 14 = Tsh 60,000 per day.

3. State what is meant by direct variation and write its equation.

Two quantities vary directly when an increase in one causes a proportional increase in the other. Equation: y = kx, where k = y ÷ x.

4. 4 kg of maize cost Tsh 10,000. Find the cost of 9 kg.

k = 10,000 ÷ 4 = 2,500; cost = 2,500 × 9 = Tsh 22,500.

5. 8 pipes fill a tank in 6 hours. How long will 12 pipes take?

Inverse variation: k = 8 × 6 = 48; time = 48 ÷ 12 = 4 hours.

6. y varies jointly as x and z. y = 18 when x = 3 and z = 2. Find y when x = 4 and z = 5.

k = 18 ÷ (3 × 2) = 3; y = 3 × 4 × 5 = 60.

7. y varies directly as x and inversely as z. y = 10 when x = 5 and z = 4. Find y when x = 9 and z = 6.

k = (10 × 4) ÷ 5 = 8; y = 8 × 9 ÷ 6 = 12.

8. A car uses 8 litres of fuel for 120 km. How many litres are needed for 450 km?

Fuel per km = 8 ÷ 120; for 450 km: 8 × 450 ÷ 120 = 30 litres.

9. Describe the graphs of direct and inverse variation.

Direct variation: a straight line through the origin. Inverse variation: a decreasing curve that approaches both axes but never touches them.

10. Give two real-life situations that show inverse variation.

Any two: more workers take fewer days for the same job; a higher speed takes less time for a fixed distance; gas pressure decreases as volume increases (Boyle's Law).

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