TOPIC 1 RATES AND VARIATIONS – MATHEMATICS FORM TWO
Introduction
Understanding rates and variations is essential for grasping how quantities change in relation to each other. Some variables are interdependent, meaning a change in one affects the others.
Many real-life situations involve such relationships. In this chapter, you will describe the concepts of rates and variations, explain the types of variations, and solve problems on rates and variations.
The competencies developed will enable you to solve real-life challenges such as finding shopping deals, assessing fuel efficiency, or predicting data trends in fields like economics, science, engineering, and many other applications.
Think
Comparing quantities and establishing relationships without the concept of rates and variations
RATES
Rates show how one quantity is related to another quantity either increasing or decreasing its quantity. For example, speed of moving object is the ratio between the distance covered and time taken.
Engage in Activity 1.1 to explore the concept of rates in real life.
Activity 1: Demonstrating rates in real life
Measure and compare the time it takes for people to walk, run or complete any other similar activity.
Determine different times of completing the activity using the concept of rates and give your reasons.
Suggest other real life examples of rates by applying the same reasoning and provide explanations.
Example 1
If a car travels 150 kilometres in 3 hours. What is its speed?
Solution
Given distance =150 km and time 3 hrs. But, speed is the rate of change of distance with respect to time. That is,
speed = distance time
Thus, speed =150 km3hrs
= 50 km per hour
Therefore, the rate of travel of the car is 50 km in every hour.
Example 2
If a water tank fills up with 200 litres of water in 5 minutes, what is the rate of flow of water?
Solution
Given the volume of water is 200 L and the time the tank takes to be filled is 5 minutes.
Rate of flow = Volume Time =200 L5 min=40 L/min
Therefore, water enters the tank at the rate of 40 litres per minute.
Example 3
A student had two plant seedlings. She measured the rate at which the seedlings were growing. Seedling A grew 5 cm in 10 days and seedling B grew 8 cm in 12 days. Which seedling was growing more quickly?
Solution
The rates of growth of the two seedlings is computed as follows:
Rate of growth of seedling A
=5 cm10 days =0.5 cm per day
Rate of growth of seedling B
=8 cm12 days =0.67 cm per day
The growth rate of seedling B is higher than that of seedling A. Therefore, seedling B was growing more quickly than seedling A.
Example 4
Two pipes, A and B are used to fill a water tank. Pipe A can fill the tank in 6 hours, while pipe B can fill the same tank in 4 hours. If both pipes are opened at the same time, how long will it take to fill the tank?
Solution
Pipe A fills the tank in 6 hours, so it fills 16 of the tank in 1 hour.
Pipe B fills the tank in 4 hours, so it fills 14 of the tank in 1 hour.
Combined rate = Rate of A+ Rate of B
Combined rate =16+14
Combined rate =2+312=512
Therefore, both pipes together fill 512 of the tank in 1 hour.
If 512 of the tank is filled in 1 hour, then the time taken to whole tank 1212 will be;
Time = Whole tank Combined rate Time =1212512=1212×125=125 hours 125 hours =2 and 25 hours But 25 of an hour =25×60 minutes =24 minutes
Therefore, it will take 2 hours and 24 minutes to fill the tank when pipes A and B are opened at the same time.
EXERCISE 1
1. What rate in metres per second is equivalent to a speed of 45 kilometres per hour?
2. A car covers a distance of 200 kilometres in 60 minutes. What is the speed of the car in metres per second?
3. A water tap takes 10 minutes to fill a 500 litres tank. Find the rate of flow of water in litres per second.
4. If premium petrol costs Tshs 2,500 per litre, how many litres can be purchased for Tshs 10,000?
5. Find the rate in kilometres per hour if a racing car travels 11512 kilometres in 35 minutes.
6. Mwajuma walks 6 kilometres in 1 hour and John walks the same distance in 2 hours. What are their walking speeds? Who walks faster, and by how much?
7. A garden hose fills a 20 litre bucket in 4 minutes. What is the rate of flow in litres per minute?
8. A car travels 400 km using 25 litres of fuel. What is the car’s fuel efficiency in kilometres per litre?
9. Mr. Magoda’s salary increased from Tshs 800,000 to Tshs 1,000,000 in a year. What is the rate of change of his salary?
10. The temperature in Mrs. Kidunula’s room increased from 15∘C at 8 am to 25∘C at 2 pm . What was the average rate of temperature change per hour?
11. A car accelerates from 0 to 60 km/h in 10 seconds. What is the rate of change of its speed in km/h per second?
12. Mr. Kilenzi’s heartbeat increased from 60 beats per minute to 120 beats per minute during exercise over 3 minutes. What was the average rate of change in his heartbeats?
13. A family uses 80 GB of internet data in 30 days. What is the average daily rate of data usage in GB per day?
14. Pipe X can fill a tank in 8 hours, while Pipe Y can fill it in 5 hours. How long will it take to fill the tank if both pipes are opened simultaneously?
15. Pump A can fill a pool in 5 hours, Pump B in 4 hours, and Pump C in 10 hours. How long will it take to fill the pool if all three pumps are operated at the same time?
16. Study the following figure which shows different altitudes (in thousands feet) attained by a plane and then answer the questions that follow.
(a) During the first 4 minutes of flight, the plane took off to the altitude shown by point A . Estimate the rate of change in altitude.
(b) Between points E and F , the plane is descending. Estimate the rate of change of altitude during this time.
(c) Is the rate of change in (b) positive or negative? Give reasons for your answer.
(d) Find the rate of change in altitude between points C and D . Is this rate greater or less than the rate between points D and E ? How does the graph visually show that the two rates are different?
(e) Can the rate of change between any two points be 0 ? Briefly explain and give an example from the figure.
17. Ana and Lina can weed a garden in 2 hours. Working alone, Ana can do the same work in 3 hours. How long would Lina weed the garden alone?
Exchange Rates
In any country, people expect to do transactions in in the currency of of their own country. When money from country A is to be used in country, it is necessary to exchange the currency of country A to the currency of country B. Various currencies in the world are linked together by exchange rates. This enable smooth transfer of money and payments to take place between countries.
Engage in Activity 1.2 to explore more about the concept of currency exchange in reallife situations.
Activity 2: Performing currency exchange
1. Choose a suitable computer Currency Conversion Application.
2. Analyse current exchange rates between the Tanzanian currency and other currencies of your choice using the computer application.
3. Compare the app’s rates with market rates and evaluate its accuracy for financial decisions, especially for travel.Y
EXERCISE 2
Use any computer Currency conversion application to answer questions 1-10.
1. How much is Tshs 20,600 worth in Indian Rupees?
2. Convert 1 New Zealand Dollar into Saudi Arabia Riyal.
3. How much is Tshs 500,000 worth in Euros?
4. How many Yen are equivalent to Tsh 1?
5. How many Tanzanian shillings can a visitor from Kenya get for exchanging 930 Kenyan shillings?
6. Find the amount in Tanzanian shillings of each of the following:
(a) 30,000 Euros
(b) 4,200 Pula
(c) 640 Rands
(d) 12,000 Riyal
7. Exchange Tshs 300,000 into the following currencies:
(a) Mozambican meticals
(b) Malawian kwachas
(c) Swiss francs
(d) Indian rupees
8. How much is Tshs 6,000,000 worth in Pounds Sterling?
9. Mwanaisha bought story books for 200 AUD (Australian Dollars). How much did she spend in Tanzanian shillings?
10. Mr. Utaligolo wants to exchange 1,000 USD for EUR. If the exchange fee is 2%, how much will he receive after charging the fee?
11. Mrs. Uwemba is shopping in a country where 1USD=205 Local Currency (LC). If she spent 205,000 LC, how much did she spend in USD?
12. Ayota Stationery wants to buy items from an online store, which cost 2500 NOK (Norwegian Kroner). If the exchange rate is 1NOK=9 Local Currency, how much does the stationery pay in Local Currency?
VARIATIONS
variation is a relationship where a change in one quantity leads to a proportional change in the other. It allows for the exploration of connections between two or more quantities. The four basic types of variations are direct, inverse, joint, and combined.
Direct variations
Direct variation is a relationship between two variables where one variable is a constant multiple of the other. In other words, if one variable increases or decreases, the other variable changes proportionally in the same direction.
This type of variation is useful for understanding how changes in one variable affect another in a directly proportional manner.
Engage in Activity 1.3 to explore the application of direct variation in real life.
Activity 3: Exploring direct variation in real life
Learn about direct variation from books or from the internet.
Find real-life examples that illustrate direct variations.
Demonstrate mathematically how variables in the real-life scenarios relate and use the relationship to solve related problems.
If y is directly proportional to x, it can be written as y∝x, where ∝ is a symbol of proportionality. The corresponding mathematical equation connecting x and y is formed by introducing a proportionality constant k, and replaces ∝ with an equal sign to get, y=kx.
For instance, if y varies directly as the square of x, then y∝x2 and the corresponding equation is y=kx2, where k is the constant of proportionality.
Graph of the relation y∝x
From Figure 1.1, it can be observed that an increase (or decrease) in the quantity x results in a proportional increase (or decrease) in the quantity y, and vice versa.
Example 5
Example 6
Example 7
EXERCISE 3
If x varies directly as y and x=16 when y=10, find the value of y when x=20.
The surface area of a circular object varies directly as the square of its radius. If its surface area is 78.5 cm2, when the radius is 5 cm, find the surface area when the radius is 7 cm.
If x varies directly as y and x=30 when y=40, find the value of x when y=16.
A mason can build 100 metres of fence in 20 hours. How long will it take 5 masons to build 875 metres?
If x varies directly as 2y+7 and x=5 when y=4, find y when x=6.
If 8 men can assemble 16 machines in 12 days, how long will it take 15 men to assemble 100 machines?
If y varies directly as the square root of x, and y=12 when x=4, find y when x=9.
If y varies directly as x and y=8 when x=3, find y when x=18.
Two variables x and y that vary directly have corresponding values as shown in the following table.
(a) Find the rule connecting x and y.
(b) Fill in the missing values.
(c) Draw a graph which shows that y∝x for k=1.
Study the following table and answer the questions that follow.
(a) Do earnings vary directly as the number of hours worked?
(b) If yes, calculate the constant of proportionality and find the equation that describes the relationship.
The volume of a sphere varies as the cube of its radius. Three solid spheres of diameters 32 m,2 m, and 52 m are melted and combined to form a new solid sphere. Find the diameter of the new sphere.
When observing two buildings simultaneously, the length of each building’s shadow varies directly with its height. If a 5 floor building has a shadow of length 20 m, how many floors would form a shadow of length 32 m?
The resistance of a wire varies as the square of the diameter of its cross-section. Find the percentage change in the resistance when the diameter is (a) doubled (b) reduced by 20%.
Two variables A and x are related by the formula A=axn. The following set of data was generated based on this formula.
(a) Find the values of a and n.
(b) Find the value of A when x=5.
A precious stone worth Tshs 15,600,000 is accidentally dropped and broken into three pieces. The weights of the pieces are in the proportions of 2:3:5, respectively. If the value varies directly as the cube of its weight, calculate the value of the remaining stone in percentage.
Inverse variations
A relationship between two or more variables is said to be an inverse variation if the value of one variable increases while the other value decreases, or vice versa. Engage in Activity 1.4 to explore further inverse variations.
Activity 4: Identifying inverse variations in daily life
Identify three real-life activities where increasing one variable decreases the other variable.
Perform one of the activities to experience how changes of the variables are related in the activity.
Find out through reading books and browsing the internet how to express these inverse relationships mathematically.
Use the mathematical expression to explain how changes in one variable affect the other, and present your findings.
Quantities with an inverse variation relationship are said to be inversely proportional to each other. In this case, the quantities vary inversely or in inverse proportion. Inverse proportion is sometimes referred to as indirect proportion.
For example, the number of men employed to cultivate a farm and time taken to complete the work are inversely related. Likewise, the time to travel to a certain place and the speed are inversely related.
Example 8
Example 9
Example 10
EXERCISE 4
Joint and combined variations
In some activities, one variable can depend on several other variables to operate effectively. Such relationships are described by joint and combined variations. Engage in Activity 4.5 to explore more about joint and combined variations in real-life activities.
Activity 5: Discovering joint and combined variations in daily life
Explore the concepts of joint and combined variations using books and online resources.
Find and describe real-life examples of such variations using daily practices such as formulas.
Share your final observations and discuss the examples you discovered.
Example 12
Example 13
Combined variations
Example 14
Example 15
Example 16
Example 17
Exercise 5
Chapter summary
TOPIC 1 RATES AND VARIATIONS – MATHEMATICS FORM TWO
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