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- END UNIT 1 ASSIGNMENTOpened: Saturday, 27 May 2023, 12:00 PMDue: Monday, 29 May 2023, 12:00 AM
UNIT 6 :Application of direct proportions
6.1 Concept of direct proportion
Let us do the activity below. We will then explain the concept of direct
proportion.
Activity 6.1
Have your five books or counters. Record your counters as shown.
Have two of you put their counters together. Record the number of your
counters.Carry on for up to four of you and fill in the blanks accordingly.
What have you observed? Explain your observations.
Tip:
It is clear that pupils increase from 1 to 2. In the same way, counters
increase from 5 to 10.
We note: 2/1= 10/5 , increase in the same way.
We can say 2 pupils have 10 counters. When we decrease the pupils from
2 to 1, the counters reduce from 10 to 5.
We note 1/2= 5/10, decrease in the same way.
Activity 6.2
Materials: water, 1/2
litre bottles, 1 litre bottles, similar cups.
• Pour water into the 1/2 litre bottle and the 1 litre bottle.
• Pour water from the 1/2 litre bottle into the cups. How many cups are filled?
• Pour water from the 1 litre bottle into the cups. How many cups are filled?Now fill in the table below.
• Divide: 1 litre ÷ 1/2 litre and their respective number of cups. What do
you notice? Explain your findings.
Example 6.1
(a) On a scale drawing, 1 cm represents 10 km of road. What length of
road is represented by 3 cm?
1 cm rep 10 km
3 cm rep __?_
It is clear that 3 cm will represent a longer length of road.
Thus, 3 cm represents 3 × 10/1 km = 30 km
(b) I take 30 minutes to walk to school. How much time do I need to:
(i) walk to school and back home?
(ii) walk to school and back home for 5 days?
Solution
The distance to school from home is fixed. I take 30 minutes one way.
(i) To walk to school and back home is two way.
I take 30 minutes × 2 = 60 minutes = 1 hour
(ii) In 1 day, walking to and from school, I take 60 minutes.
In 5 days, I take 60 minutes × 5 = 300 minutes
or 1 h × 5 = 5 hours.
Tip: Rule for direct proportion
(i) When one quantity increases, the second quantity increases in a
similar way.
(ii) When one quantity decreases, the second quantity decreases in asimilar way.
Practice Activity 6.1
Work out the following1. Fill in the table below.
2. Study the table below. Fill in the missing numbers. Justify your
answers
3. I have six water tanks. Each tank holds 1 500 litres of water. How
many litres of water can my tanks hold? Explain your answer.
4. It takes 2 minutes to walk round the school field once. How long does
it take to walk round the field 7 times? Discuss your answer.
5. We are twenty pupils. Each of us is 10 years old. What is the total ofour ages
6.2 Ratios and direct proportion
In direct proportions, we compare quantities of different items. For example,
1 boy has 2 books. A boy and books are different items. In this case, 1:2 is
the ratio of boy to books.
Let us study the activity below.
Activity 6.3
• Give three books to each volunteer.
• Now, have 2 volunteers put their books together. How many books
do they have?
• Have four volunteers put their books together. How many books do
they have?
• Divide: number of books
number of volunteers. This is the book to volunteer ratio.
• Count the number of boys and girls in your group. What ratio is it?
• Explain your findings.
• Discuss situations where ratios are used in daily life.
Example 6.2
Three children had nine sweets.
(a) What is the ratio of child to sweets?(b) How many sweets are needed for 4 children?
Solution
(a) Ratio = number of children
number of sweets
= 3 children
9 sweets
= 1 child to 3 sweets or 1 : 3
(b) 1 child gets 3 sweets.
4 children get 3 × 4 sweets= 12 sweets
Note: From Example 6.2(a);
Practice Activity 6.2
Work out the following.
1. Express each of the ratios in its simplest form.
(a) 8:24 (b) 21:42 (c) 9:27 (d) 16:12
(e) 18:8 (f) 24:16 (g) 8:50
2. A minibus carries 28 passengers. A bus carries 64 passengers. What
is the ratio of passengers carried by minibus to bus? Present your
answer.
3. The weight of Pierre’s mathematics book is 420 g. The weight of his
dictionary is 560 g. What is the ratio of his mathematics book to his
dictionary?
4. Dusabimana has 360 oranges and 120 mangoes. Find the ratio of her
mangoes to her oranges. Discuss your answer.
5. In a game park, there are 120 giraffes and 360 antelopes. Find the
ratio of giraffe to antelope in its simplest form.
6. The ratio of mass of maize to rice was 3:5. The total mass of maize and
rice was 96 kg. Find the mass of rice and maize. Explain your steps to
answer.
7. A class has 56 pupils. There are 14 boys in the class. Find the ratio of
boys to girls in the class.
8. The mass of a pupil’s book is 300 g. The mass of a teacher’s book is
900 g. Find the ratio of the masses of teacher’s book to pupil’s book.
Present your results.
9. Observe and find the ratio of items in your home or school. For example,
the ratio of
(a) number of teachers to pupils in your school.
(b) boys to girls in your class.
(c) number of cups to plates at home.Discuss your results.
6.3 Problems involving direct proportion
Activity 6.4
Solve the problems below:
• A family uses 90 litres of water every day from their tank.
(a) How many days will it take the family to use 2 700 litres of
water from the tank?
(b) How many litres of water from the tank are used in 20 days?
Explain your steps.(c) Discuss examples where you use direct proportions
Example 6.3
In a peace rally, 3 speakers talk to people in 7 districts.
How many speakers are needed for 42 districts? Discuss the importance
of peace in our country.
Solution
3 speakers talk to 7 districts.
We can write this as 3/7 speakers per district.
Then 42 districts require 42 × 3/7 speakers
= (6 × 3) speakers
= 18 speakers18 speakers can talk about peace in 42 districts.
Practice Activity 6.3
1. The weight of eight copies of P5 mathematics book is 480 g. What is
the weight of one copy?
2. A car travels 12 km on one litre of petrol. How many kilometres will
the car travel on three litres?
3. In transport business, three minibuses carry 54 passengers. How
many passengers will 8 such minibuses carry? Discuss the importance
of transport business.
4. A car uses 3 litres of petrol to travel 72 kilometres. How much petrol
does it use in a journey of 648 kilometres? Explain your answer.
5. A house cleaner uses 8 litres of water everyday to clean a house. How
many days will 64 litres of water last for his work?
6. A mother feeds her baby with 4 glasses of milk every day. This is in
order to keep the baby healthy. How many days will the baby take to
drink 132 glasses of milk? Discuss your steps to answer.
7. The weight of 15 boys is 300 kg. The boys have the same weight.
Calculate the weight of one boy? Present your answer.
8. One aircraft carries 100 passengers. How many passengers are carried
by 6 such aircrafts? Explain your answer.
9. Three tractors can dig 10 acres of land in a day during farming season.
How many tractors are needed to dig 30 acres in a day during theseason? Justify your answer. Explain importance of farming.
Activity 6.5
(a) In a certain school, there are 700 pupils. The ratio of boys to girls is
3:4. Find the number of boys and girls in the school.
(b) The ratio of boys to girls was 3:5 in a group. 24 girls left the group
and 24 boys joined the group. The ratio of boys to girls became 5:3.
How many boys and girls were in the original group? Explain thesteps used.
Example 6.4
1. In a certain town, the ratio of adults to children is 4:5. The number
of adults is 400.
(a) Calculate the number of children in the town.
(b) Every Monday to Friday, 100 adults go for their jobs away from
the town. Similarly, 200 children go to their schools away from
the town. Find the total number of adults and children in the
town on Tuesday at 10 a.m.
2. The ratio of girls to boys was 5:3 originally. Later 24 boys joined the
group and 24 girls left the group. The ratio of girls to boys became
3:5.
(a) How many boys and girls were in the original group?
(b) How many boys and girls were in the final group?
Solution
1. (a) Ratio of adult to children = 4:5 = 400 : children
Clearly number of children = 5 × 100 = 500
or 4/9× total population = 400
Total population = 9/4× 400 = 900, here adults are 400.
So children are 900 – 400 = 500.
(b) We know, there are 400 adults and 500 children.
On Tuesday, we have (400 – 100) adults = 300 adults and
(500 – 200) children = 300 children.
Total number of adults and children are 300 + 300 = 600.2. (a) Let the number of boys be b and girls be g in original group.
Practice Activity 6.4
1. A certain farmer has goats and chicken in her farm. The ratio of goats
to chickens is 3:5 in her farm. The total number of chickens and goats
are 320.
(a) How many chickens are there in her farm?
(b) Calculate the number of goats in her farm. Explain your steps.
(c) The farmer sold 20 goats and 80 chickens so as to get money for
school fees. Find the ratio of goats to chickens after selling her
animals. Why is it important to educate children?
2. In a church wedding, the ratio of children to adults was 3:4. The total
number of adults and children was 175. Later, 18 children and 5 adults
left the church. The ratio of children to adults became 3:5.
(a) How many children and adults were there initially?
(b) Find the number of children in the church after 18 of them left.
(c) Find the number of adults in the church after 5 of them left.
(d) Suppose, the ratio of men to women was 2:3 initially in the
church.
(i) Discuss how many women were present in the church?
(ii) Discuss how many men were present in the church?
3. In a shop, the ratio of number of shirts to trousers was 5:6. The
shopkeeper bought 10 more trousers and 10 more shirts. The new
ratio of shirts to trousers became 7:8.
(a) Calculate the original number of shirts.
(b) Calculate the original number of trousers.
(c) Calculate the new number of trousers.
(d) Calculate the new number of shirts. Explain importance of
selling.
4. During a sports day, the ratio of boys to girls was 5:6 in the morning.
At midday, 170 more boys and 180 more girls came. The ratio of boys
to girls became 7:8.
(a) Find the number of girls in the morning.
(b) Find the number of boys in the morning.
(c) Find the number of boys at midday.
(d) Find the number of girls at midday.Discuss your answers. Why are sports important?
Revision Activity 6
1. Study the following table. It shows the number of teachers to pupils
in a certain country. Fill in the missing numbers. State the ratioused and explain your observation
2. In a school, each pupil is given 5 exercise books. How many exercise
books will 30 pupils get from the school?
3. Abel’s farm has 20 mango trees and 100 coffee trees. Find the ratio
of mango to coffee trees on his farm. Why do we plant trees?
4. At breakfast 2 loaves of bread are served to 8 children. How many
loaves are needed for 64 children? Tell the importance of breakfast.
5. For good health, a pupil should drink 5 glasses of water every day.
Calculate how many glasses of water a pupil should drink in 10 days
for good health.
6. It takes 40 minutes to walk to the market. How much time do I need
to walk to the market and back home every day for one week?
7. In a game park, there are 120 lions and 240 antelopes. Find the ratio
of lions to antelopes in its simplest form?
8. A wagon travels 30 km in 1 1/2 hours. How many kilometres will it travel in a 1/2
hour? Explain the steps followed.
9. Dusabimana travelled from 6.00 a.m to 10.00 a.m on Monday to the
field. He was covering 60 km every hour. What was the total distance
of his journey? Present your answer.
10. In class activity, it takes a pupil 40 minutes to write a composition.
How many compositions should the pupil write in 160 minutes?
Discuss your steps to answer. Tell the importance of writingcomposition and keeping time.
Word list
Direct proportion Ratios Original group Final group
Increases Decreases Similar way
Task
Do the following.
(i) Read each word aloud to your friend.
(ii) Write the meaning of each of the words above. Discuss with your friend.
(iii) Write sentences using each of the words above. Read with your friend.(iv) Give daily life examples where you apply direct proportion.