FRACTIONS, PERCENTAGES, RATIO AND PROPORTION

FIRST TERM 

LEARNING NOTES

CLASS: JSS 2 (BASIC 8)

SCHEME OF WORK WITH LESSON NOTES 

Subject: 

MATHEMATICS

Term:

FIRST TERM 

Week:

WEEK 5

Class:

JSS 2 (BASIC 8)

Previous lesson: 

The pupils have previous knowledge of

 ESTIMATION AND APPROXIMATION OF NUMBERS

that was taught as a topic during the last lesson.

Topic :

FRACTIONS, PERCENTAGES, RATIO AND PROPORTION

Behavioural objectives:

At the end of the lesson, the pupils should be able to

• convert fractions to decimals and decimals to fractions
• express fractions as decimals and percentages
• solve simple sums on ratio and proportion

Instructional Materials:

• Wall charts
• Pictures
• Related Online Video
• Flash Cards

Methods of Teaching:

• Class Discussion
• Group Discussion
• Asking Questions
• Explanation
• Role Modelling
• Role Delegation

Reference Materials:

• Scheme of Work
• Online Information
• Textbooks
• Workbooks
• 9 Year Basic Education Curriculum
• Workbooks

Content

FRACTIONS

TOPICS

1. Expressing fractions as decimals
2. Percentages: Percentages of quantities; expressing one quantity as a percentage of the other; percentage increase and decrease.
3. Ratios
4. Rates and Proportions
5. Word problems

EXPRESSING FRACTIONS AS DECIMALS

A fraction is a part of a whole number. It is used to express a part or fraction of a whole. The number that is up in a fraction is called the numerator while the number that is written down is the denominator. The line is called over and it functions as the division line. Given a figure like 4/5 , four is the numerator while the denominator is five. The fraction is read as four over five.

There are two methods of doing this conversion. There is a general method which can be used any time and on any type of vulgar fraction. There is also another method where the denominator of the fraction contains power/powers of ten.

In this second case, the given fraction can first be converted to an equivalent fraction.

Examples:

Convert the following common fractions to decimal fractions (decimal numbers).

25,34,144225

Solutions:

First: we can use the equivalent fractions method, before the general method.

◊ Write 25 as 25=2×25×2=410=0.4∴25=0.4

◊ Write 34 as 34=3×254×25=75100=0.75∴34=0.75

◊ Write 144225 as 144225=144×4225×4=5761000=0.576∴144225=0.576

Second: the general method (for all condition) is used when the denominator of the given fraction does not contain power(s) of 10. This is by dividing the numerator by the denominator mentally or via long-division previously learnt by students in their Primary School days.

Teacher demonstrates this approach to students as an alternative.

Conversion of Decimals to Fractions

Examples:

1. Convert 0.65 to a common fraction.

Solution:

To do this, we simply multiply the given decimal fraction by 100 and at the same time divide it by 100.

Write 0.65 as 0.65=0.65×100100. If we carefully notice the expression, we will see that what we are doing is just multiplying 0.65 by unity (1).  Because 100100=1.

⇒0.65×100100=65100=1320, (when further reduced to the lowest term).

∴0.65=1320

2. Convert 0.6 to a common fraction.

Solution:

To do this we simply multiply the given decimal fraction by 10 and at the same time divide it by 10.

Write 0.6 as 0.6=0.6×1010=610=35∴0.6=35

3. Convert 0.125 to a common fraction.

Solution:

To do this we multiply the given decimal fraction by 1000 and at the same time divide it by 1000 to have

0.125=0.125×10001000=1251000=25200=540=18

(when fully simplified to its lowest form).

CLASS ACTIVITY:

1. Change the following common fractions to decimal fractions.

915,119,223,173,123341

2. Change the following decimal fractions to the vulgar or common fractions.

(i) 0.56 (ii) 0.0015 (iii) 5.35 (iv) 0.222 (v) 1.98

PERCENTAGES

This is a number expressed as a fraction of 100. A percentage is a part of one hundred. The symbol for percentage is %. 1% means 1 out of 100. This can be written as a fraction 1100, 100 is the denominator.

Example

1. Express as percentage.

Solution:

There are two possible ways to answer the question.

METHOD 1:

Write 215 as 215×100=20015=1313%

∴215=1313% in percentage.

METHOD 2:

1. Write 215as 23153=235=23×205×20.

43100=1313100=1313% in percentage.

2. Express 65% as fraction.

Solution:

Write 65% as 65%=65100=1320

∴65%=1320

3. Express 24% as fraction.

Solution:

Write 24% as 24%=24100=625%

∴24%=625

Class activity

1. Express each of the following fractions as percentage.

58,1720,925,13,123341

2. Express each of the following percentages as fraction.

64%, 45%, 0.125%, 0.17%.

Percentage of Quantities

Examples: Find the value of the following

(i) 20% of 400 yams

(ii) 1212% of 2000km

Solution:

(i) 20% of 400 yams

=20100×400 yams

= 20 × 4

= 80 yams

(ii) 1212% of  2000km

=252×2000

= 25 × 1000

= 25 000km

Expressing One Quantity as a Percentage of the Other

Examples:

(i) What percentage of 214 hours is 45 minutes?

(ii) A farmer found 24 out of 100 tubers of yam bad, what percentage of the yam is bad?

Solution

(i) 214 hours =94×601=9×15=135 minutes

Thus, 45 minutes as a percentage of 135 minutes

=45135×1001=13×100=1003=3313%

(ii) Total yams = 100

Bad yams = 24

Percentage of bad yams

=72100×100172%

Percentage Increase and Decrease

Percentage increase represents a rise or appreciation of a given number, amount or quantity while Percentage decrease represents a reduction or depreciation of a given number, amount or quantity. It is the opposite of percentage increase.

Example:

(i) Increase $210 by 30%

(ii) Find the percentage increase from 250 to 350

(iii) Reduce £650 by 10%

(iv) A tank containing kerosene is three-quarter full. If 20% of the kerosene is removed, find the amount of kerosene left in the tank.

Solution:

Method 1

(i) Amount = $210

Percentage increase = 30%

Total amount (30% of 210)

=30100×200=$63

Increasing $210 by $63 = $210 + $ 63 = $273

Therefore, total amount = $273

Method 2

Percentage increase = 30%

Total percentage = 100% + 30% = 130%

Amount to be increased = $210

Required amount =130100×210=13×21=$273

(ii) First, we subtract 250 from 350, then we express the difference over 250 and multiply by 100:

Thus, 350−250=100=100250×100=4×10=40%

(iii) 10% of £650=10100×650=1×65=£65

Decreasing £650 by £65 = £650 – £65

= £585

Therefore, the required amount = £585

(iv) Present level of kerosene = 34

Total decrease = 20%

Total percentage after decrease

= 100 – 20 = 80%

Level of kerosene left =80100×34=(20×3)100=35

Therefore the level of kerosene left in the tank is 35 of the tank.

Class Activity

1. Increase the following by 25%

(i) 420kg (ii) 16 000kg

2. A trader sells watermelon, tomatoes and onions. Their prices are £250, £1 450 and £4 500 respectively. If each of the prices were increased by 20%, find the price of water melon, tomatoes and an onion.
3. Reduce 75 litres of water by 15%
4. The price of a shirt was slashed from £1 500 to £1 250. Find the percentage decrease in the price of the shirt.

RATIO

Ratio compares two or more quantities or amounts of the same kind. It can be expressed as a fraction, e.g. 2:3=23. It can be expressed in its simplest or lowest term by cancelling, and it does not contain a fraction or decimal in its lowest term. In other words, ratio is a type of measure of the relative size of two or more quantities expressed as a proportion.

Example:

(i) If Honesty spent N224.00 on buying a story book, Sandra spent N70.00 Bobo drink. Find the ratio of the amount spent by Honesty to that of Sandra

(ii) Express 49.00 as a fraction and as ratio of 84.00

(iii) In a school of 720 students the ratio of boys to girls is 7:5. Find the number of boys and girls respectively.

Solution:

(i) Honesty N224.00, Sandra N 70.00

Then Amount spent by Honesty Amount spent by Sandra=22470=3210

∴ the ratio of the amount spent by Honesty to the amount spent by Sandra = 224:70 = 32:10.

Reducing it to its lowest term = 16:5.

(ii) As a fraction: N49N84=712

As a ratio: N49:N84 = 7:12

(iii) Ratio of boys to girls = 7:5

Total ratio = 7+5 = 12

Number of boys = 712 of 720=712×7201=7×60 = 420 boys

Number of girls 512 of 720=512×7201=5×60 = 300 girls

Class Activity

(1) 120,000 dollars is shared between 2 students who are on scholarship in Deeper Life High School in the ratio 36:28. How much does each student gets?

(2) If the ratio of a number to 4 is equal to the ratio 3:2, find the number.

RATE

74km/hr, $8 000/day, 9km/hr are all examples of rates.

(i) The first rate tells the distance gone in 1 hour.

(ii) The second rate tells how much money is made in 1 day.

(iii) The third rate tells the distance travelled in 1 hour.

Example:

A car travels 126km in 112h. Find its rate in km/h.

Solution:

In 112h, the car travels 126km

In 1 h, the car travels 126 ÷ 112h

= 12632

= 126 × 32

= 42 × 2

= 84km/h

PROPORTION

This is the relation of one part to another or to the whole with respect to quantity.

There are two types which is Direct and Inverse Proportion.

When a quantity increases in relation to another quantity it is Direct Proportion. On the other hand, an increase in a quantity in relation to a decrease in another quantity (or vice-versa) is called Indirect Proportion.

Example:

(i) A man gets #8 000 for 5 hours work. How much does he get for 14 hours?

(ii) A bag of rice feeds 15 students for 7 days. How long would the same bag feed 10 students?

Solution

(i) 5 hours = #8 000

1 hour = #8 000 ÷ 5

= #1 600

For 14 hours now,

14 × #1 600

= #22 400

(ii) 15 students = 7 days

1 student = 15 × 7

= 105 days

For 10 students we have,

= 105 ÷ 10

= 10510

= 1012 days.

Class Activity

1. It takes four people 3 days to dig a small field. How long would it take three people to do the same work?
2. A bag of corn can feed 100 chickens for 12 days. For how many days would the same quantity of corn feed 80 chickens?

Presentation

The topic is presented step by step

Step 1:

The class teacher revises the previous topics

Step 2.

He introduces the new topic

Step 3:

The class teacher allows the pupils to give their own examples and he corrects them when the needs arise

Evaluation

1. Express the following in decimal

25%,

34%,

14.42%

2.5%

2. Change the following decimals to fraction

(i) 0.0075

(ii) 0.0006

(iii) 0.55

3. A boy scored 15 out of 25 marks in a mathematics test. Calculate the score in percentage.
4. Change the following to ratios in their lowest term

(i) 30 minutes to 2 hours

(ii) 110cm to 250mm

(iii) #225 to #145

(iv) 105litres to 81 deciliters

5. 80 machines can produce 4 800 electric bulbs in 25 minutes. Express the proportion of machines. Electric bulbs and time in ratio.
6. Three children are to share 936 biscuits in the ratio 2:3:4. How many will each receive?

ASSIGNMENTS

1. Find the value of x and y in the following ratios

(i) 3:5 = x:20

(ii) 11:35 = 66: y

2. In a village of 375 children under the age of 10. If 68% of them have had malaria, how many children have escaped the disease?
3. Five buses need 3 trips to take 450 students to the stadium to watch a concert. How many trips would the 3 buses need to take the same number of students?
4. The ratio of girls to boys in a boarding school is 2:7. If there are 1 251 students in the school:

(i) How many of the students are girls?

(ii) What percentage of the students are boys?

5. An old carpet which now costs #1 000, was sold for #1 400 when new. Calculate the percentage decrease.
6. The cost of petrol is #65 per litre. What is the percentage increase if it is sold for #105 per litre by a dealer during fuel scarcity?
7. A packet of biscuits has a mass of 205g. if there are 28 biscuits in the packet, what is the approximate mass of 1 biscuit correct to 5 decimal places?
8. A woman’s pace is about 70cm long. She takes 2858 paces to walk from her home to the market. Find the distance from her home to the market correct to 4 significant figures.
9. The perimeter of a school compound is 1 616m. In the perimeter of the fence, there are 203 fence posts equally spaced. Find, approximating to 3 decimal places, the distance between two posts?
10. A nautical mile is 1.853km long. Calculate to:

(i) 2 dp

(ii) 2 s.f.

(iii) How many kilometres are there in 243 nautical miles?

[mediator_tech]

Simple Sums On Ratio And Proportion

1. In the ratio 4:5, the second part represents ___.
a) The larger quantity
b) The smaller quantity
c) The total quantity

2. If the ratio of boys to girls in a class is 3:5, and there are 24 girls, then the number of boys is ___.
a) 9
b) 15
c) 40

3. A recipe calls for 2 cups of sugar for every 3 cups of flour. If you want to make 6 cups of flour, how many cups of sugar do you need?
a) 2
b) 3
c) 4

4. If the ratio of apples to oranges is 2:3, and there are 10 apples, then the number of oranges is ___.
a) 6
b) 15
c) 12

5. If a map scale is 1 cm represents 10 km, then a distance of 50 km on the map would be represented as ___ cm.
a) 5
b) 10
c) 50

6. If 5 notebooks cost $15, then the cost of 8 notebooks is ___.
a) $18
b) $24
c) $20

7. A car travels 240 miles in 4 hours. At the same speed, how far will it travel in 6 hours?
a) 320 miles
b) 360 miles
c) 420 miles

8. If the ratio of red balls to blue balls is 3:2, and there are 25 red balls, then the number of blue balls is ___.
a) 15
b) 20
c) 30

9. If the ratio of the lengths of two sticks is 2:5, and the longer stick is 25 cm, then the length of the shorter stick is ___.
a) 5 cm
b) 10 cm
c) 12.5 cm

10. If a recipe calls for 1 teaspoon of salt for every 4 cups of flour, and you want to make 8 cups of flour, how many teaspoons of salt do you need?
a) 2
b) 3
c) 4

11. The ratio 5:7 can also be written as ___.
a) 7:5
b) 2:5
c) 10:14

12. If 4 meters of cloth cost $12, then the cost of 9 meters of the same cloth is ___.
a) $27
b) $18
c) $36

13. A recipe requires 2 cups of milk for 5 cups of flour. If you have 15 cups of flour, how many cups of milk do you need?
a) 6
b) 4
c) 8

14. If the ratio of students to teachers in a school is 25:1, and there are 500 students, then the number of teachers is ___.
a) 20
b) 25
c) 500

15. If the ratio of the prices of two items is 2:3, and the second item costs $15, then the cost of the first item is ___.
a) $10
b) $7.50
c) $20

Conclusion

The class teacher wraps up or concludes the lesson by giving out a short note to summarize the topic that he or she has just taught.

The class teacher also goes round to make sure that the notes are well copied or well written by the pupils.

He or she makes the necessary corrections when and where the needs arise.