Ratio and Proportion

Ratio and Proportion

Introduction
In real life, ratios and proportions are used on a daily basis; for example, cooks us them when following recipes. They are also used in business when dealing with money, to figure out how much is earned when more products are sold. In this topic, you will understand ratios and proportions and be able to use them in a range of contexts.

5.1 Understanding and Applying Equivalent Ratios
Activity 5.1 (a) (Work in groups)
Study your environment and identify situations where ratios are applicable in real
life. You can present a photograph, where necessary.
Activity 5.1 (b) (Work in groups)
you are required to read and interpret your class timetable properly and then
answer the questions below
(a) How many periods are given to;
iii) all vocational subjects
i) all science subjects ii) all arts subjects
(b) Write the ratios of the;
i) number of periods of science subjects to that of the vocational subjects
ii) number of periods of all subjects to that of the arts subjects
(c) Write two ratios which are equivalent to each of the ratios given above.
(d) Present your work to the rest of the class.

Activity 5.1 (c) (Work in groups)
(a) Count and record the number of books and pens in your group.
(b) Write the ratio of the number of books to that of the pens.
(c) Make drawings to show any three ratios which are equivalent to the ratio in (b .
Activity 5.1 (d) (Work in groups)
(a) Read the following statement.
In a mathematics quiz, Sarah was asked whether 24:30 is equivalent to 4:5.
(b) If you were Sarah, what would be your response?
(c) Make drawings tojustify your response.

activity 511 (d) (Work in groups)
(a) Read the following statement.
In a mathematics quiz, Sarah was asked whether 24:30 is equivalent to 4:5.
(b) If you were Sarah, what would be your response?
(c) Make drawings to justify your response.
Example 5.1
Kule and his friend won the District Athletics Championship and they were awarded UGX 180,000 and UGX 120,000, respectively. They decided to jointly invest all their money in a chapatti-making business in their local community. Their business made a profit of UGX 600,000 in the first month. How should they divide up the profits fairly in line with how much each one of them invested?

. Project Work: (Work in groups)
Photo albums are a good way to store, organise and proudly show the Photographs that you value most. Use materials in your environment to design a photo album that will be kept in the living room, to serve as a reminder of the unforgettable times of your childhood.

5.2 Understanding and Applying Direct and Inverse
Proportional Reasoning
Direct and inverse proportions are used to show how quantities are related to eachother. The symbol used to denote proportionality is “a”.
Understanding direct proportion
Activity 5.2(a) (Work in groups)
Suggested materials:

• a thread
  Instructions:
  . a ruler
• pencil / pen
  (a) Draw any four squares of different dimensions.
  (b) Write down the dimensions of all the squares drawn.
  (c) Measure and record the distance around each square.
  (d) What conclusion do you make about the results in (b) and (c)?
  (e) Determine the distance around a square of side length 14 m.
  (f) Explain your results to the rest of the class.

Applying direct proportion
Activity 5.2(b) (Work in groups)
Naduk is a baker in Moroto town. She has been baking for the last 10 years. She knows that 8 pancakes require 400 grams of baking flour.
(a) How many pancakes will 750 grams of baking flour bake?
(b) Discuss the relationship between the number of pancakes and the quantity of bakina flour.

Understanding and applying inverse proportion
Activity 5.2(c) (Work in groups)
(a) Explore your classroom environment and identify items you can distribute amongst members in your group.
(b) Collect some of those items.
(c) Give all the items to one group member to count and record their number.
(d) Distribute the items to 2 group members. Record the number of items that each member gets.
(e) Collect the items and re-distribute them to more members in the group.
(f) Compare the number of items received by the first 2 group members to the number of items received by the other members in the group.
(g) Explain your observations to the class.

Exercise 5.3
1) The number of learners who feed on a certain amount of food is inversely proportional to the number of days for which the food is available. If 300 learners can be fed for 15 days, how many days will it take to feed 250 learners? 2) A loaf of bread weighing 0.5 kg costs UGX 2,500. What is the weight of a loaf
which could be sold for UGX 2,000? 3) The average speed ofa bus is inversely proportional to the time it spends on a journey. If 4 hours are spent on a journey while driving at 64 km/hour, how long would the same journey take a bus moving at an average speed of 80 km/hour? 4) The number of men working at the same rate is indirectly proportional to the working hours. If 8 men need 18 working hours, how long will it take 9 men working at the same rate to complete the same work? 5) If y is inversely proportional to x, find an equation connecting y and x for which y = 4 when x = 2. Hence, find; i) y when x = 6
ii) x when y = 24

Ratio and Proportion
5.3 Understanding and Applying Ratio, Proportion and Scale
Understanding and applying ratios
Activity 5.3(a) (Work in groups)
(a) Explore your classroom environment.
(b) Identify some items and classify them into two groups.
(c) Count and record the items in each group.
(d) Compare the number of items in one group to that of the other group.
(e) Share with other groups in your class.
Activity 5.3(b) (Work in groups)
Hajarah is a local poultry farmer who deals in broilers. She mixes her chicken feed
locally from home. In a particular month, all her birds were underweight, so she
sought advise from a local agricultural extension worker, who advised her to mix
J concentrate feed and maize brand in the ratio of 2:3. She came back home but did
not know how to interpret the ratio.
(a) Explain to Hajarah what the ratio means.
(b) If she had stocked 100 kg of maize brand, how much concentrate feed does
she need to make the recommended mixture?

3)A cake recipe calls for IX cups of milk and 17 cups of flour. How many cups of milk and flour are required to make 5 similar cakes?
4)Last week, Elopu answered 48 out of 60 questions correctly in a test. This week, he answered 32 out of 48 questions correctly. In which test did Elopu have better results? Explain your answer.
5)80 similar books weigh 160 kg. Find the weight of 25 similar books.
6)Divide UGX 60,000 in the ratio 2:1 between Ngabirano and Agabirira.
7)Ssembuya and Achieng save coins in the ratio of 4:3. If Ssembuya has 60
coins in his savings, how many coins does Achieng have?
8)Atugonza runs 300 m in 40 seconds. Isabirye runs 200 m in 30 seconds.
Who is faster than the other for the short races? Understanding and applying proportion A proportion is a comparative relation between things, sizes, quantities, and numbers, among others. A proportion is a statement that of equality between two or more ratios.

Activity 5.3(c) (Work in groups)
o

Example 5.6
A shoe company pays UGX 2,500 per hour to casual labourers in the assembly sector.
How much will a casual labourer earn after 5 days, working for 8 hours each day?
i Solution: For 1 hour, the casual labourer earns UGX 2,500.
For 8 hours or 1 day, the casual labourer will earn UGX 2,500 x 8 = UGX 20,000.
Therefore, for 5 days, the casual labourer will earn UGX 20,000 x 5 = UGX 100,000.

Understanding and applying a scale
When engineers make designs for buildings, they make accurate diagrams of those buildings. All the diagrams are made to scale. This means that the length of every part on the design is similar to the one that would be on real building but just divided by an amount. The ratio of the length on the design to the length on the real building is called a scale. For example, the scale of 1 cm on the design to represent 100 cm on the real building is written as the ratio 1:100. This implies that the drawing and the real
building are proportional. From this discussion, you observe that a scale is a ratio giving the linear relationship between a representative diagram or drawing and the actual item itself.

Activity 5.3(d) (Work in groups)
Suggested material: map extracts
Instructions:
(a) Write down scales of different map extracts in the school library.
(b) Identify some features on the map extracts and determine their actual distances on the ground.

Example 5.7
A map of a town is drawn to a scale of
(a) The main street is drawn 60 cm long on the map. How long is the real street (in km)?
(b) The distance from a post offce to a police station is 250 m. How long will it be drawn on the map (in mm)?

Exercise 5.6
1) A plan ofa classroom is drawn to a scale of 1:20. If the real classroom is;
(a) 10 m long, how long is it on the plan?
(b) 6 m wide, how wide is it on the plan?
2) A plan of a compound is drawn to a scale of 1:25.
(a) A classroom block is drawn 800 mm long on the plan. How long is the
real classroom block?
(b) The headteacher’s offce is 3.5 m wide. How wide will it be on the plan?
3) A map is drawn to a scale of 1:30. On the ground, a school building is 24 m
long. How long (in cm) will it be on the map?
4) An engineer designed a model ofa new building using a scale of 1:50.
(a) If the model is;
120 cm long, how long is the real building?
i)
ii) 44 cm high, how high is the real building?
(b) If the real building is 20 m long, how long is the model?

o
5.4 Bivariate data, Scatter Graphs and Lines of Best Fit
“Bi” means two. Therefore, bivariate data are sets of data showing an association between
variables; for example, ice cream sales versus the average temperature of that day.
A scatter graph is a plot, using Cartesian coordinates, to display values for typically
two variables for a set of data. It is used to visualise the relationship between the two
variables plotted. Line of best fit refers to a line through a scatter graph that best
expresses the relationship between the two variables plotted.
Plotting a scatter graph
Plotting involves locating a point on a graph.
Activity 5.4(a) (Work in groups)
Suggested materials: a tape measure, a metre rule, a thread of length 50 cm
Instructions
(a) Measure the heights and the shoe sizes of all members in your group.
(b) Record your results.
(c) Plot the two variables on the same pair of axes.
(d) What can you conclude from this graph?

o
5.4 Bivariate data, Scatter Graphs and Lines of Best Fit
“Bi” means two. Therefore, bivariate data are sets of data showing an association between
variables; for example, ice cream sales versus the average temperature of that day.
A scatter graph is a plot, using Cartesian coordinates, to display values for typically
two variables for a set of data. It is used to visualise the relationship between the two
variables plotted. Line of best fit refers to a line through a scatter graph that best
expresses the relationship between the two variables plotted.
Plotting a scatter graph
Plotting involves locating a point on a graph.
Activity 5.4(a) (Work in groups)
Suggested materials: a tape measure, a metre rule, a thread of length 50 cm
Instructions
(a) Measure the heights and the shoe sizes of all members in your group.
(b) Record your results.
(c) Plot the two variables on the same pair of axes.
(d) What can you conclude from this graph?

• -9 iliie ui best tit tor a given bivariate
  data set on a scatter graph
  The line of best fit is the one, among the many possible lines, that gives the best
  relationship between the variables plotted. The numbers of points above and below
  the line are about equal.
  Activity 5.4(b) (Work in grouns)

O) Revision Questions:
1)” Can an 8-inch by 12-inch photograph be reduced to a 3-inch by 5-inch photograph? Explain your response.
2) The ratio of two numbers is 1:4 and their sum is 40. Find the numbers.
3) A learner who is 5 feet tall casts an 8-feet shadow At the same time, a tree casts a 40-feet shadow. How many feet is the tree tall?
4) On Masaka-Kampala highway, the speed limit is 110 km/hr. A motorist is driving at 70 miles/hr.
(a) Ifyou were the traffc offcer, would you have stopped the motorist for overspeeding?
(b) How far over or under the speed limit is the motorist travelling? (Use 1.6 km = 1 mile)
5) A road measuring 4.5 km is represented on a map by a length of 4.5 cm. Find the scale of the map.

Topic Summary
In this topic, you have learnt that:
1) Direct proportionality is a relation between two variables such that their ratio
is always the same value.
2) Inverse proportionality is a relation between two variable such that an increase
in one variable leads to a decrease in the other and vice versa.
3) The gradient of the line of best fit describes the relationship between the
variables, as illustrated below.

In (a), the relationship is positive, because as x increases, y also increases.
In (b), the relationship is negative, because as x increases, y decreases, and vice versa.
In (c), there is no relationship.