Lesson Notes By Weeks and Term v5 - Grade 6

Lesson Video

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Performance objectives

• Define and express ratios in their simplest form.
• Differentiate between ratio and rate.
• Calculate percentages of whole numbers.
• Express a ratio as a fraction and a percentage.
• Solve simple problems involving ratio and percentage.

Lesson summary

Ratio, rate, and percentage are fundamental mathematical concepts that help us understand and compare quantities in our everyday lives. Whether it's sharing sweets with friends, understanding the price of goods at the local spaza shop, or calculating discounts at a supermarket, these concepts are essential tools. In South Africa, understanding these concepts is especially important for making informed financial decisions and participating effectively in our economy. This week's focus will introduce the basic concepts of ratio, rate, and percentage, laying the groundwork for more advanced topics in the future.

Lesson notes

2.1 Ratio A ratio is a comparison of two or more quantities of the same kind. It shows how much of one thing there is compared to another.

Ratios can be written in several ways: Using the word "to": 3 to 5 Using a colon (:): 3:5 As a fraction: 3/5 The order in which the numbers are written matters! 3:5 is different from 5:

3. Simplifying Ratios: Just like fractions, ratios can be simplified. To simplify a ratio, divide each number in the ratio by their highest common factor (HCF).

Example 1: In a class of 30 learners, there are 12 boys and 18 girls. Write the ratio of boys to girls in its simplest form.

Ratio of boys to girls: 12:18 The HCF of 12 and 18 is

6. Divide both numbers by 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3 Simplified ratio: 2:3 This means for every 2 boys, there are 3 girls in the class.

Example 2: Thando has 20 apples and 15 oranges. What is the ratio of apples to oranges in its simplest form?

Ratio of apples to oranges: 20:15 The HCF of 20 and 15 is

5. Divide both numbers by 5: 20 ÷ 5 = 4 and 15 ÷ 5 = 3 Simplified ratio: 4:3 2.2 Rate A rate is a ratio that compares two quantities of different kinds. The units of the quantities must be included.

Common examples include: Speed: kilometers per hour (km/h)

Price: rands per kilogram (R/kg)

Wage: rands per hour (R/hour)

Example 1: A car travels 240 kilometers in 3 hours. What is the car's average speed? Speed = Distance / Time Speed = 240 km / 3 hours Speed = 80 km/h Example 2: The price of potatoes is R30 for 5 kg. What is the price per kilogram? Price per kilogram = Total price / Number of kilograms Price per kilogram = R30 / 5 kg Price per kilogram = R6/kg 2.3 Percentage A percentage is a way of expressing a number as a fraction of

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0. The word "percent" means "out of one hundred." The symbol for percent is %.

Converting a Fraction to a Percentage: To convert a fraction to a percentage, multiply the fraction by 100%.

Example 1: What is 1/4 as a percentage? (1/4) 100% = 25% Example 2: What is 3/5 as a percentage? (3/5) 100% = 60% Calculating a Percentage of a Number: To find a percentage of a number, convert the percentage to a decimal or fraction and then multiply by the number.

Example 1: What is 20% of 50?

Method 1: Convert 20% to a decimal: 20% = 0.20 0.20 50 = 10 Method 2: Convert 20% to a fraction: 20% = 20/100 = 1/5 (1/5) 50 = 10 Example 2: What is 75% of 80?

Method 1: Convert 75% to a decimal: 75% = 0.75 0.75 80 = 60 Method 2: Convert 75% to a fraction: 75% = 75/100 = 3/4 (3/4) 80 = 60 Expressing a Ratio as a Percentage: First express the ratio as a fraction, then convert the fraction to a percentage.

Example: Express the ratio 1:4 as a percentage.

Ratio 1:4 can be written as the fraction 1/4. (1/4) 100% = 25% Guided Practice (With Solutions)

Question 1: A fruit basket contains 8 apples and 12 bananas. What is the ratio of apples to bananas in its simplest form?

Solution: Ratio of apples to bananas: 8:12 The HCF of 8 and 12 is

4. Divide both numbers by 4: 8 ÷ 4 = 2 and 12 ÷ 4 = 3 Simplified ratio: 2:3

Commentary: This question reinforces the understanding and simplification of ratios. Identifying the HCF is key to simplifying correctly.

Question 2: A train travels 360 km in 4 hours. What is the average speed of the train?

Solution: Speed = Distance / Time Speed = 360 km / 4 hours Speed = 90 km/h

Commentary: This problem helps learners differentiate between ratio and rate and reinforces the calculation of rate using the correct formula. Remember to include the correct units (km/h).

Question 3: Calculate 30% of

9

0. Solution: Method 1: Convert 30% to a decimal: 30% = 0.30 0.30 90 = 27 Method 2: Convert 30% to a fraction: 30% = 30/100 = 3/10 (3/10) 90 = 27

Commentary: This question demonstrates how to find a percentage of a whole number. Students can choose the method (decimal or fraction) that they find easier.

Question 4: Express the ratio 3:10 as a percentage.

Solution: Ratio 3:10 can be written as the fraction 3/10. (3/10) 100% = 30%

Commentary: This question combines understanding of ratio and percentage. Convert the ratio to a fraction before calculating the percentage.

Question 5: At a shop, a shirt originally costing R120 is on sale with a 10% discount. How much is the discount in Rands?

Solution: Calculate 10% of R120: (10/100) R120 = R12 The discount is R

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2. Commentary: This practical problem introduces percentages in a real-life shopping context. It requires understanding how to calculate a percentage of a value to find a discount amount. Independent Practice (Questions Only)

Simplify the ratio 24:

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6. A recipe uses 2 cups of flour and 3 cups of water. What is the ratio of flour to water? A cyclist travels 60 km in 2 hours. What is the cyclist's average speed? The price of apples is R25 for 5 apples. What is the price per apple? Calculate 45% of

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0. What is 60% of 150?

Express the ratio 1:5 as a percentage.

Express the ratio 7:20 as a percentage. A store offers a 15% discount on a TV that originally costs R2000.