Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Apply percentages in diverse contexts (discounts, interest).
• Solve problems with rates, ratios, and proportions.
• Use direct and indirect proportion for scaling and resource allocation.
• Interpret numbers in tables, charts, and graphs.
• Apply rounding and estimation strategies effectively.

Lesson summary

This week, we delve deeper into numbers and calculations, building upon what you learned in previous grades and expanding your understanding of their practical applications. In Mathematical Literacy, being comfortable and confident with numbers isn’t just about doing calculations; it's about making informed decisions in everyday life, from managing your personal finances to understanding statistical data presented in the news. This topic is crucial for understanding aspects of South African society and participating effectively in the economy. Imagine budgeting your salary, interpreting election results, or comparing the cost of different cellular data plans.

Lesson notes

2.1 Percentages: More Than Just a Sign Percentages are a way of expressing a number as a fraction of

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0. The word "percent" means "out of one hundred". It is denoted by the symbol %.

Calculating a Percentage: To find x% of a number, multiply the number by x/

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0. Example 1: A shop in Durban is having a 20% off sale on all clothing. A shirt originally costs R

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0. How much will the shirt cost after the discount? Discount amount = 20/100 R150 = R30 Sale price = R150 - R30 = R120 Therefore, the shirt will cost R120 after the discount.

Percentage Increase/Decrease: Percentage Increase: ((New Value - Original Value) / Original Value) 100% Percentage Decrease: ((Original Value - New Value) / Original Value) 100% Example 2: The price of petrol increased from R18.50 per liter to R20.00 per liter. What is the percentage increase? Increase = R20.00 - R18.50 = R1.50 Percentage increase = (R1.50 / R18.50) 100% ≈ 8.11% The price of petrol increased by approximately 8.11%. 2.2 Ratios, Rates and Proportions Ratio: A ratio compares two or more quantities.

It can be written as a:b or a/b.

Rate: A rate compares two quantities with different units. For example, kilometers per hour (km/h).

Proportion: A proportion states that two ratios are equal.

Example 3: A recipe for pap requires a ratio of 1:3 of maize meal to water. How much water is needed if you use 2 cups of maize meal?

Ratio: Maize meal : Water = 1:3 If maize meal = 2 cups, then water = 2 3 = 6 cups.

Therefore, you need 6 cups of water.

Example 4: Exchange Rates The current exchange rate is R19.50 to 1 USD. How much would it cost in Rands to buy something that costs $50? Cost in Rands = $50 R19.50/USD = R975 Therefore, it would cost R975. 2.3 Direct and Indirect Proportion Direct Proportion: As one quantity increases, the other quantity increases proportionally. If y is directly proportional to x, then y = kx, where k is a constant.

Indirect Proportion (Inverse Proportion): As one quantity increases, the other quantity decreases proportionally. If y is inversely proportional to x, then y = k/x, where k is a constant.

Example 5: Direct Proportion If 5 workers can build a wall in 8 hours, how long will it take 10 workers, assuming they work at the same rate? Since the work is completed faster with more workers, but each worker individually does less work, this is actually an inverse proportion problem (see Example 6). It's important to read questions carefully!

Example 6: Indirect Proportion If 5 workers can build a wall in 8 hours, how long will it take 10 workers, assuming they work at the same rate? Let 'h' be the number of hours it takes 10 workers. Since the amount of work is constant, the product of the number of workers and the time taken is constant.

Therefore, 5 8 = 10 * h 40 = 10h h = 4 hours Therefore, it will take 10 workers 4 hours to build the wall. 2.4 Rounding and Estimation Rounding is approximating a number to a specific place value. Estimation is approximating the answer to a calculation. These are important for checking the reasonableness of your answers and quickly approximating solutions.

Example 7: Estimate the total cost of the following grocery items: Bread: R12.99 Milk: R15.50 Eggs: R32.25 Estimate by rounding each to the nearest Rand: Bread: R13 Milk: R16 Eggs: R32 Estimated Total = R13 + R16 + R32 = R61 2.5 Large Numbers and Scientific Notation Scientific notation is a way of expressing very large or very small numbers in a compact form. A number in scientific notation is written as a × 10 n , where 1 ≤ a 7 Guided Practice (With Solutions)

Question 1: A cellphone originally costing R3500 is on sale for 15% off. What is the sale price?

Solution: Discount = 15% of R3500 = (15/100) R3500 = R525 Sale Price = R3500 - R525 = R2975 Therefore, the sale price is R

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5. This demonstrates a direct application of percentage calculations in a retail context.

Question 2: A recipe requires a ratio of flour to sugar of 2:

1. If you want to use 300g of flour, how much sugar do you need?

Solution: Flour : Sugar = 2:1 Let 'x' be the amount of sugar needed. 2/1 = 300/x 2x = 300 x = 150g Therefore, you need 150g of sugar. This question reinforces understanding of ratios and proportions in a practical cooking setting.

Question 3: If 3 taps can fill a tank in 12 hours, how long will it take 4 taps to fill the same tank (assuming all taps have the same flow rate)?

Solution: This is an example of indirect proportion. The more taps, the less time it takes. Total water needed = 3 taps 12 hours = 36 tap-hours Let 'h' be the number of hours it takes 4 taps. 4 taps h hours = 36 tap-hours h = 36/4 = 9 hours Therefore, it will take 4 taps 9 hours to fill the tank. This demonstrates indirect proportion applied to a resource management scenario.

Question 4: The number of tourists visiting Kruger National Park increased from 1,200,000 in 2022 to 1,350,000 in

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3. Calculate the percentage increase.