RATIO AND PROPORTION

Subject :

Mathematics

Topic :

RATIO AND PROPORTION

Class :

Basic 6 / Primary 6

 

Term :

 

Second Term

Week :

 

Week 2

Instructional Materials :

• Pictures
• Wall Posters
• Related Online Videos
• Role Playing

 

Reference Materials

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

Previous Knowledge :

The pupils have previous knowledge of money in their previous classes

 

 

Behavioural Objectives :  At the end of the lesson, the pupils should be able to

1. Understand the concept of ratio and how it is used to compare quantities.
2. Be able to express ratios in different forms (e.g. 1:2, 1/2, 1 to 2).
3. Understand the concept of proportion and how it is used to compare ratios.
4. Be able to set up and solve proportions using cross products.
5. Understand how to use ratios and proportions to solve real-world problems, including scaling and unit conversion.
6. Understand the difference between direct and inverse variation, and be able to use these concepts to solve problems.

 

Content :

Direct proportion 

In mathematics, a ratio is a way to compare two or more quantities. It is usually expressed as a fraction or a pair of numbers separated by a colon (e.g. 1:2, 3/4). For example, if you have three red balls and six green balls, the ratio of red balls to green balls is 3:6 or 1/2.

A proportion is an equation that states that two ratios are equal. For example, if the ratio of red balls to green balls is 3:6, and the ratio of apples to bananas is 6:9, then we can say that these ratios are proportional because 3/6 = 6/9. Proportions can be used to solve real-world problems, such as scaling, unit conversion, and finding missing values in a ratio.

In general, ratios and proportions are useful tools for comparing quantities and finding relationships between different values. They are often used in mathematics and other fields, such as science, engineering, and economics.

 

In mathematics, a direct proportion is a relationship between two variables in which one of the variables is directly proportional to the other. This means that the value of one variable increases or decreases in direct proportion to the value of the other variable.

For example, consider the following statements:

1. The number of hours worked is directly proportional to the amount of money earned. This means that if you work more hours, you will earn more money.
2. The volume of a gas is directly proportional to its temperature, provided that the pressure remains constant. This means that if you increase the temperature of a gas, its volume will increase.
3. The length of a wire is directly proportional to its resistance, provided that the wire’s cross-sectional area and temperature remain constant. This means that if you increase the length of a wire, its resistance will also increase.
4. The weight of an object is directly proportional to its mass. This means that if you increase the mass of an object, its weight will also increase.
5. The brightness of a light bulb is directly proportional to the electric current passing through it. This means that if you increase the current, the brightness of the light bulb will also increase.

 

 

 

Examples involving direct proportion:

1. If 3 workers can paint a house in 6 hours, how long would it take 1 worker to paint the house? Answer: 18 hours (since 1/3 of the work can be done in 1/6 of the time).
2. If 2 meters of fabric cost $4, how much would 6 meters of fabric cost? Answer: $12 (since 3 times the length of fabric costs 3 times the price).
3. If a car can travel 100 miles on 5 gallons of gas, how far can it travel on 15 gallons of gas? Answer: 300 miles (since 3 times the amount of gas allows 3 times the distance to be traveled).
4. If a box of apples weighs 10 kilograms and contains 5 apples, how much does each apple weigh? Answer: 2 kilograms (since the total weight is divided equally among the apples).
5. If a cup of coffee costs $2 and contains 8 ounces of coffee, how much does each ounce cost? Answer: $0.25 (since the total cost is divided equally among the ounces).

 

 

Inverse or Indirect proportion 

In mathematics, an indirect or inverse proportion is a relationship between two variables in which one of the variables is inversely proportional to the other. This means that the value of one variable decreases or increases inversely with the value of the other variable.

For example, consider the following statements:

1. The time it takes to travel a certain distance is inversely proportional to the speed at which you travel. This means that if you increase your speed, it will take you less time to travel the same distance.
2. The intensity of a sound is inversely proportional to the distance from the source. This means that the closer you are to the source of the sound, the louder it will seem.
3. The resistance of a wire is inversely proportional to its cross-sectional area, provided that the wire’s length and temperature remain constant. This means that if you increase the cross-sectional area of a wire, its resistance will decrease.
4. The strength of a gravitational field is inversely proportional to the distance from the center of the object creating the field. This means that the closer you are to the object, the stronger the gravitational field will be.
5. The rate of cooling of an object is inversely proportional to the thickness of the material it is made of. This means that if you increase the thickness of the material, the rate of cooling will decrease.

 

 

Examples involving indirect proportion:

1. If it takes 4 hours to travel 200 miles at a speed of 50 mph, how long would it take to travel the same distance at a speed of 25 mph? Answer: 8 hours (since half the speed results in twice the time).
2. If a sound seems twice as loud when you are half as far away from the source, how far away would you have to be to hear the sound at half the intensity? Answer: twice as far (since half the intensity requires twice the distance).
3. If a wire has a resistance of 8 ohms when its cross-sectional area is 4 square millimeters, what is its resistance when its cross-sectional area is 8 square millimeters? Answer: 4 ohms (since double the cross-sectional area results in half the resistance).
4. If an object weighs 10 kilograms on the surface of a planet with a gravitational field strength of 10 N/kg, what would its weight be on a planet with a gravitational field strength of 5 N/kg? Answer: 20 kilograms (since half the field strength results in twice the weight).
5. If it takes 10 hours for a 1-centimeter-thick piece of metal to cool from 100°C to 50°C, how long would it take for a 2-centimeter-thick piece of metal to cool from 100°C to 50°C under the same conditions? Answer: 20 hours (since double the thickness results in half the rate of cooling

 

1. What is the main difference between ratio and proportion in mathematics?

a) Ratio is a way to compare quantities, while proportion is a way to solve real-world problems. b) Proportion is a way to compare quantities, while ratio is a way to solve real-world problems. c) Ratio and proportion are the same thing. d) There is no difference between ratio and proportion.

2. Which of the following is NOT a way to express a ratio?

a) 3:6 b) 1 to 2 c) 1/2 d) 50%

3. Which of the following statements is NOT an example of a direct proportion?

a) The volume of a gas is directly proportional to its temperature, provided that the pressure remains constant. b) The weight of an object is directly proportional to its mass. c) The intensity of a sound is directly proportional to the distance from the source. d) The rate of cooling of an object is directly proportional to the thickness of the material it is made of.

4. If the ratio of red balls to green balls is 3:6, and the ratio of apples to bananas is 6:9, what can you conclude?

a) These ratios are proportional. b) These ratios are not proportional. c) There is not enough information to determine if these ratios are proportional. d) It is impossible for ratios to be proportional.

5. How is cross multiplication used in solving proportions?

a) To find the missing value in a ratio. b) To compare ratios. c) To solve real-world problems involving scaling and unit conversion. d) To determine if two ratios are proportional.

6. What is the main difference between direct and inverse variation?

a) In direct variation, the value of one variable increases or decreases directly with the value of the other variable. In inverse variation, the value of one variable decreases or increases inversely with the value of the other variable. b) In direct variation, the value of one variable increases or decreases inversely with the value of the other variable. In inverse variation, the value of one variable decreases or increases directly with the value of the other variable. c) In direct variation, the value of one variable decreases or increases with the value of the other variable. In inverse variation, the value of one variable increases or decreases with the value of the other variable. d) In direct variation, the value of one variable increases or decreases with the value of the other variable. In inverse variation, the value of one variable remains constant.

7. What is a unit rate?

a) A rate that compares two quantities with different units. b) A rate that compares two quantities with the same units. c) A rate that is expressed as a fraction. d) A rate that is expressed as a decimal.

8. What is an equivalent ratio?

a) A ratio that has the same value as another ratio. b) A ratio that has a different value than another ratio. c) A ratio that has a value between two other ratios. d) A ratio that is equal to 1.

9. How are ratios and proportions used in real-world problems?

a) To compare quantities. b) To solve problems involving scaling and unit conversion. c) To find missing values in a ratio. d) All of the above

 

Answers

1. a) Ratio is a way to compare quantities, while proportion is a way to solve real-world problems.
2. d) 50%
3. c) The intensity of a sound is directly proportional to the distance from the source.
4. a) These ratios are proportional.
5. c) To solve real-world problems involving scaling and unit conversion.
6. a) In direct variation, the value of one variable increases or decreases directly with the value of the other variable. In inverse variation, the value of one variable decreases or increases inversely with the value of the other variable.
7. b) A rate that compares two quantities with the same units.
8. a) A ratio that has the same value as another ratio.
9. d) All of the above.

 

Examples

1. The ratio of apples to bananas is 3:5. This can also be expressed as 3/5 or 3 to 5.
2. The ratio of boys to girls in a classroom is 8:6. This can also be expressed as 8/6 or 8 to 6.
3. The ratio of cars to trucks on a highway is 4:1. This can also be expressed as 4/1 or 4 to 1.
4. The ratio of dogs to cats in a pet store is 3:2. This can also be expressed as 3/2 or 3 to 2.
5. The ratio of red balls to green balls is 5:7. This can also be expressed as 5/7 or 5 to 7.
6. The ratio of books to magazines in a library is 2:3. This can also be expressed as 2/3 or 2 to 3.
7. The ratio of cups to saucers in a cupboard is 3:4. This can also be expressed as 3/4 or 3 to 4.
8. The ratio of pencils to pens in a pencil case is 4:3. This can also be expressed as 4/3 or 4 to 3.
9. The ratio of cookies to cakes in a bakery is 5:6. This can also be expressed as 5/6 or 5 to 6.
10. The ratio of chairs to tables in a living room is 2:1. This can also be expressed as 2/1 or 2 to 1.

 

 

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. You and your friend split the cost of a $100 dinner evenly. How much does each of you pay?
2. You have a bag of M&Ms that contains 20 red M&Ms and 30 green M&Ms. What is the ratio of red to green M&Ms?
3. Your school has a total of 500 students, 300 of whom are boys. What is the ratio of boys to girls in your school?
4. You are making a cake and the recipe calls for 2 cups of sugar for every 3 cups of flour. If you want to make a cake with 6 cups of flour, how much sugar will you need?
5. You want to mix a solution that is 40% alcohol. If you have a bottle of 50% alcohol and a bottle of 80% alcohol, how much of each do you need to mix to get the desired solution?
6. You have a total of $100 to invest and you want to invest 60% of it in stocks and 40% in bonds. How much money will you invest in each type of investment?
7. You have a basket of fruit that contains 5 apples, 3 oranges, and 4 bananas. What is the ratio of apples to oranges to bananas?
8. You are planning a party and you want to have twice as many cups as plates. If you have 20 plates, how many cups do you need?

 

Solution

 

1. You and your friend split the cost of a $100 dinner evenly. How much does each of you pay?

Each of you will pay $100/2 = $<<100/2=50>>50.

2. You have a bag of M&Ms that contains 20 red M&Ms and 30 green M&Ms. What is the ratio of red to green M&Ms?

The ratio of red to green M&Ms is 20:30, or 20/30, or 20 to 30.

3. Your school has a total of 500 students, 300 of whom are boys. What is the ratio of boys to girls in your school?

The ratio of boys to girls in your school is 300:200, or 300/200, or 300 to 200.

4. You are making a cake and the recipe calls for 2 cups of sugar for every 3 cups of flour. If you want to make a cake with 6 cups of flour, how much sugar will you need?

To make a cake with 6 cups of flour, you will need 2 cups of sugar for every 3 cups of flour, or 2/3 * 6 cups of flour = 4 cups of sugar.

5. You want to mix a solution that is 40% alcohol. If you have a bottle of 50% alcohol and a bottle of 80% alcohol, how much of each do you need to mix to get the desired solution?

Let x be the amount of 50% alcohol and y be the amount of 80% alcohol. We can set up the proportion: 50/100 = x/(x+y) and 40/100 = y/(x+y). Cross multiplying gives us: 2x = 4y and y = 2x/5. Solving for x and y, we find that x = 10 and y = 8. Therefore, you will need to mix 10 mL of 50% alcohol and 8 mL of 80% alcohol to get the desired solution.

6. You have a total of $100 to invest and you want to invest 60% of it in stocks and 40% in bonds. How much money will you invest in each type of investment?

You will invest 60/100 * $100 = $<<60/100100=60>>60 in stocks and 40/100 * $100 = $<<40/100100=40>>40 in bonds.

7. You have a basket of fruit that contains 5 apples, 3 oranges, and 4 bananas. What is the ratio of apples to oranges to bananas?

The ratio of apples to oranges to bananas is 5:3:4, or 5/3/4, or 5 to 3 to 4.

8. You are planning a party and you want to have twice as many cups as plates. If you have 20 plates, how many cups do you need?

You will need 2 * 20 plates = <<2*20=40>>40 cups.

Conclusion :

The class teacher wraps up or conclude the lesson by giving out 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 does the necessary corrections when and where  the needs arise.