Lesson Notes By Weeks and Term v5 - Grade 6

Lesson Video

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Performance objectives

• Define and represent ratios.
• Identify and calculate equivalent ratios.
• Understand and solve simple problems with rates.
• Convert between simple fractions and percentages.
• Solve basic percentage problems in real life.

Lesson summary

This week, we're diving into the fascinating world of ratios, rates, and percentages! These concepts are all around us, helping us understand proportions, comparisons, and changes in everyday life. From sharing sweets with friends to calculating discounts at your favourite shop, understanding ratios, rates, and percentages will make you a master of practical maths. Why is this important for South African learners? Imagine you're helping your family budget groceries or calculate the best deals on airtime. Knowing about percentages helps you make informed decisions. If you're saving up for a soccer ball or a new pair of school shoes, understanding ratios helps you plan your savings.

Lesson notes

2.1 Ratio A ratio is a way of comparing two or more quantities. It shows how much of one thing there is compared to another.

We can write ratios in three main ways: Using a colon: For example, 3:2 (read as "3 to 2")

As a fraction: For example, 3/2 Using words: For example, "3 to 2" The order of the numbers in a ratio is very important!

The ratio 3:2 is different from the ratio 2:

3. Example 1: In a class, there are 12 girls and 18 boys. What is the ratio of girls to boys?

Solution: The ratio of girls to boys is 12:

1

8. We can simplify this ratio by dividing both numbers by their highest common factor (HCF), which is

6. So, the simplified ratio is 2:

3. This means for every 2 girls, there are 3 boys.

Example 2: Thandi has 5 apples and Sipho has 3 apples. What is the ratio of Thandi's apples to Sipho's apples?

Solution: The ratio of Thandi's apples to Sipho's apples is 5:

3. Equivalent Ratios: Equivalent ratios are ratios that represent the same comparison, even though the numbers may be different. We can find equivalent ratios by multiplying or dividing both parts of the ratio by the same number.

Example 3: Find a ratio equivalent to 1:

4. Solution: We can multiply both parts of the ratio by

2. So, 1 x 2 = 2 and 4 x 2 =

8. Therefore, 2:8 is an equivalent ratio to 1:

4. We can also multiply by 3, 4, or any other number. 2.2 Rate A rate is a special type of ratio that compares two quantities with different units. It tells us how much of something occurs per unit of something else. Common examples include speed (kilometres per hour) and price (Rands per kilogram).

Example 1: A car travels 200 kilometers in 4 hours. What is the car's speed?

Solution: Speed is a rate that compares distance to time.

We can write this as a ratio: 200 km : 4 hours. To find the speed per hour, we divide both parts of the ratio by 4: (200 km / 4) : (4 hours / 4) which simplifies to 50 km : 1 hour.

Therefore, the car's speed is 50 kilometers per hour (50 km/h).

Example 2: Apples cost R15 for 3 kilograms. What is the price per kilogram?

Solution: The rate is R15 : 3 kg. To find the price per kilogram, we divide both sides by 3: (R15 / 3) : (3 kg / 3), which simplifies to R5 : 1 kg.

Therefore, the price is R5 per kilogram. 2.3 Percentage A percentage is a way of expressing a number as a fraction out of

1

0

0. The word "percent" means "out of one hundred." The symbol for percent is %. So, 50% means 50 out of 100, which is the same as the fraction 50/100 or the decimal 0.

5. Converting Fractions to Percentages: To convert a fraction to a percentage, we multiply the fraction by 100%.

Example 1: Express 1/4 as a percentage.

Solution: (1/4) x 100% = 25%.

Therefore, 1/4 is equal to 25%.

Example 2: Express 3/5 as a percentage.

Solution: (3/5) x 100% = 60%.

Therefore, 3/5 is equal to 60%.

Converting Percentages to Fractions: To convert a percentage to a fraction, we write the percentage as a fraction with a denominator of 100 and then simplify.

Example 3: Express 75% as a fraction.

Solution: 75% = 75/

1

0

0. We can simplify this fraction by dividing both the numerator and denominator by their highest common factor, which is

2

5. So, 75/100 = 3/

4. Finding a Percentage of a Number: To find a percentage of a number, we first convert the percentage to a fraction or decimal and then multiply.

Example 4: What is 20% of 50?

Solution: 20% = 20/100 = 1/5. (1/5) x 50 =

1

0. Therefore, 20% of 50 is

1

0. Example 5: A shop offers a 15% discount on a pair of shoes that cost R

2

0

0. How much is the discount?

Solution: 15% = 15/100 = 0.15. 0.15 x R200 = R

3

0. Therefore, the discount is R

3

0. Guided Practice (With Solutions)

Question 1: Write the ratio of squares to circles in the following diagram: [Imagine a diagram with 4 squares and 6 circles] Solution: Number of squares: 4 Number of circles: 6 Ratio of squares to circles: 4:6 Simplified ratio (dividing both by 2): 2:3 Answer: The ratio of squares to circles is 2:

3. Question 2: Complete the equivalent ratio: 3:5 = ____:10 Solution: We need to find what number multiplied by 5 gives us

1

0. That number is 2 (5 x 2 = 10). To keep the ratio equivalent, we also multiply the first number (3) by 2. 3 x 2 = 6 Answer: 3:5 = 6:10 Question 3: A baker uses 2 cups of flour for every 1 cup of sugar in a recipe. How many cups of flour are needed if the baker uses 3 cups of sugar?

Solution: The ratio of flour to sugar is 2:1 We need to find the equivalent ratio when the sugar is

3. So, the ratio becomes 2:1 = ?:

3. To get from 1 to 3, we multiply by 3 (1 x 3 = 3).

Therefore, we also multiply the flour amount by 3: 2 x 3 = 6 Answer: The baker needs 6 cups of flour.

Question 4: A train travels 150 km in 2 hours. What is its average speed in kilometers per hour (km/h)?

Solution: The rate is 150 km : 2 hours To find the speed per hour, we divide both sides by 2: (150 km / 2) : (2 hours / 2)

This simplifies to 75 km : 1 hour Answer: The train's average speed is 75 km/h.

Question 5: Express 60/100 as a percentage.