Mathematical Literacy Level 3 Module 4

Mathematical Literacy NQF 3 Downloaded from hlayiso.com Maths Lit NQF Level 3

Solve rate, ratio and proportion problems in a workplace context Module 4 Downloaded from hlayiso.com Maths Lit NQF Level 3

Overview 4.1 Performing workplace related calculations involving ratios 4.2 Perform workplace related calculations involving proportions 4.3 Perform workplace related calculations involving rates 4.4 Determine the benefits of buying in bulk and buying different sizes to select an appropriate option 4.5 Solve problems using percentages Downloaded from hlayiso.com Maths Lit NQF Level 3

Performing workplace related calculations Unit 4.1 involving ratios Ratio A ratio is a comparison of two or more numbers. We write the numbers with a colon between them. Example: • If there are 8 students who drive by bus and 12 students who travel by taxi in one class, then we say we have a ratio of 8 to 12. We write this as 8 : 12. A ratio of 8 : 12 is not the same as a ratio of 12 : 8 Downloaded from hlayiso.com Maths Lit NQF Level 3

Simplifying and comparing ratios Unit 4.1 • To simplify ratios, check whether both numbers can be divided by a common factor. • Whatever you do to one number in the ratio, you must do to the other number. Equivalent ratios • Different ratios that simplify to the same ratio are called equivalent ratios. • Equivalent fractions can be written as equivalent ratios. Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 4.2 page 59 Unit 4.1 1. Write the following ratios in simplest form: a) 12 : 20 b) 20 : 40 c) 2 : 100 2. a) Are the ratios 2 : 5 and 12 : 16 equal? b) Is the ratio of 1 to 20 equal to the ratio 5 : 100? Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 4.2 page 59 continued ... Unit 4.1 1. a) 12 : 20 = 3 : 5 Divide both sides by 4 b) 20 : 40 = 1 : 2 Divide both sides by 20 c) 2 : 100 = 1 : 50 Divide both sides by 2 2. a) 12 : 16 = 3 : 4 The ratios are not equal b) 5 : 100 = 1 : 20 Divide both sides by 5 Yes, they are equal Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 4.4 page 60 Unit 4.1 1. How many millilitres of tint and peroxide will a hairdresser need to make a 250 ml mixture if the tint and peroxide is mixed in a ratio 2 : 3? • There are 2 + 3 = 5 parts altogether. • 250 ÷ 5 = 50 ml • Multiply to find each part: 2 × 50 ml = 100 ml of tint 3 × 50 ml = 150 ml of peroxide Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 4.4 page 60 Unit 4.1 There are 2 + 3 = 5 parts altogether. 250 ÷ 5 = 50 ml Multiply to find each part: 2 × 50 ml = 100 ml of tint 3 × 50 ml = 150 ml of peroxide Downloaded from hlayiso.com Maths Lit NQF Level 3

Different forms of a ratio Unit 4.1 • Ratios are written without units to show that the units on both sides are exactly the same. • A scale ratio on a map refer to measurement units. So we can write in the correct units to get a different form of the ratio. Example 4.3 page 59 A map is drawn to a scale of 1 : 1 000. Write this in measurement units. • We can use cm. • So 1 cm on the map represents 1 000 cm on the ground. • Convert 1 000 cm to metres. • 1 000 cm ÷ 100 = 10 m. So 1 cm represents 10 m. Downloaded from hlayiso.com Maths Lit NQF Level 3

Share an amount in a given ratio Unit 4.1 • When you share an amount in a given ratio, calculate how many parts the amount is divided into, and the amount of each part. Example: Share 1 500 bricks in the ratio 1 : 2. 1. Add the number of parts of the ratio together: 3 parts 2. Divide the total amount by the number of parts: 1 500 ÷ 3 = 500 3. Multiply the answer by the ratio parts: 500 x 1 = 500 and 500 x 2 = 1 000 So 1 500 bricks are shared in the ratio 500 : 1 000 Downloaded from hlayiso.com Maths Lit NQF Level 3

Find missing numbers in a ratio Unit 4.1 • When we are not given the total amount, the method is different. Example: • We need to divide tiles of two types in the ratio 2 : 5 for decorative paving. We have 300 of tile type A. • How many of tile type B do we need? Downloaded from hlayiso.com Maths Lit NQF Level 3

Share an amount in a given ratio Unit 4.1 Answer: We need to find the unknown part of the ratio. 2 : 5 = 300 : ? 1. Add the number of parts of the ratio together: 2 + 5 = 7 parts 2. Find how many there are in one part: 300 ÷ 2 = 150 3. Multiply by the ratio part needed: 150 x 5 = 750 We need 750 tiles of type B. Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 4.5 page 60 Unit 4.1 Cement must be mixed with sand in the ratio 1 : 3. Water is also added. There are 12 wheelbarrows of sand left. How many wheelbarrows of cement must it be mixed with? Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 4.5 page 60 continued ... Unit 4.1 • We need to find the unknown part of the ratio. 1 : 3 = ? : 12 1. Add the number of parts of the ratio together: 1 + 3 = 4 parts 2. Find how many there are in one part: 12 ÷ 3 = 4 wheelbarrows 3. Multiply by the ratio part needed: 4×1=4 We need 4 wheelbarrows of cement. Downloaded from hlayiso.com Maths Lit NQF Level 3

Exercise 4.1 Unit 4.1 Complete Exercise 4.1 on page 61 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 3

Perform workplace related calculations involving proportions Unit 4.2 Proportion • A proportion is an equation with a ratio on each side. • It is a statement that two ratios are equal (or “in proportion”) Examples: • Chefs often need to make more, or less, of a recipe, using proportion. • Cleaners may dilute a cleaning fluid in water in a 1 : 8 ratio and use proportion to make up the correct volume needed. • The number of builders needed to complete a job is indirectly proportional to the length of time the job takes. Downloaded from hlayiso.com Maths Lit NQF Level 3

Direct proportion Unit 4.2 Direct proportion If two quantities are in direct proportion, then as one value gets bigger, the other also gets bigger. If one value gets smaller, so does the other value. Example: • Joseph earns R1 500 per week when working 5 hours per day; he earns R3 000 if he works 10 hours a day. • His salary increases in proportion to his number of working hours per day. So his salary is directly proportional to the number of working hours. 1500 10 • We can write: = 3000 20 If you cross multiply, the products are equal. 30 000 = 30 000 Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 4.6 page 62 Unit 4.2 13 cardboard packaging boxes cost R208. How much will 6 boxes cost? Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 4.6 page 62 continued ... Unit 4.2 • Fewer boxes will cost less, so the quantities are in direct proportion. 13 boxes R208 6 boxes unknown cost • Let's use x to show the unknown value and write a proportion. 13 208 = 6 x 13x = 1 248 1248 x = 13 x = 96 6 boxes will cost R96. Downloaded from hlayiso.com Maths Lit NQF Level 3

Inverse or indirect proportion Unit 4.2 Inverse proportion If two values are indirectly proportional to each other, as one value gets bigger, the other gets smaller. If one value gets smaller, the other gets bigger. Example: • Ntsako paints a wall in 3 days. If Vusi worked with him at the same rate the job would take 1½ days. So the more people involved, the quicker the job gets done. • The time taken is indirectly proportional to the number of people. • The ratio of number of people is 1 : 2. • The ratio of days spent is 3 : 1½ • The products are equal. 1 × 3 = 2 × 1½ Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 4.7 page 63 Unit 4.2 1. A farmer ploughs his land in 6 days using 5 tractors. How many days will it take to plough the same area with 3 tractors? • Write a proportion using x as the unknown: 5 tractors x days = 3 tractors 6 days • Note: the second fraction is inverted. • Solve the equation by cross-multiplying: 5×6=3×x x = 30 ÷ 3 = 10 days If he uses 3 tractors, it will take 10 days. Downloaded from hlayiso.com Maths Lit NQF Level 3

Exercise 4.2 Unit 4.2 Complete Exercise 4.2 on page 63 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 3

Perform workplace related calculations Unit 4.3 involving rates Rates A rate is a special ratio where the two terms are in different units. • “per” is clue that you are dealing with rate. It can be represented by /. e.g. R1,20 per egg can be written as R1,20/egg • To solve rate problems, we need a formula to link the quantities e.g. speed = distance time Examples of rates: • Speed in km per hour • cost per unit in shopping, in rands per kilogram • amount of medicine per millilitre • cost rates of water and electricity services in rands per unit. Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 4.8 page 65 Unit 4.3 1. a) Kaashief is a data capturer for a statistics organisation. He is able to input 20 pages per hour. How long will it take him to imput data of 73 pages? b) How many pages of data can he input in his 7,5 hour working day? Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 4.8 page 65 continued ... Unit 4.2 1. Set up a formula in words and write it in different ways: number of pages speed (pages/hr) = time in hours number of pages time = speed in pages/hr number of pages 72 a) Time = = speed in pages/hr 20 = 3,6 hours = 3 hours 36 minutes b) Number of pages = speed × time = 20 × 7,5 = 150 pages. Downloaded from hlayiso.com Maths Lit NQF Level 3

Exercise 4.3 Unit 4.3 Complete Exercise 4.3 on page 65 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 3

Determine the benefits of buying in bulk and buying different sizes to select an appropriate option Unit 4.4 • To find the best “deal” for a product, you must find the unit price for the product. Example 4.9 page 66 1. A shop advertises 4 kg of chicken priced at R66,80 or a different brand of 5 kg of chicken costing R88,95. Which is cheaper? 1. R66,80 ÷ 4 = 16,70/kg R88,95 ÷ 5 = R17,79/kg The 4 kg pack of chicken is cheaper per kilogram. Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 4.9 page 66 Unit 4.4 2. You can buy 750 ml of cooldrink for R12,95 or 550 ml for R10,85. Which is cheaper? Find the unit price (for 1 kg) for each package of chicken. R66,80 ÷ 4 kg = R16,70/kg R88,95 ÷ 5 kg = R17,79/kg The 4 kg package of chicken is cheaper. Downloaded from hlayiso.com Maths Lit NQF Level 3

Exercise 4.4 Unit 4.4 Complete Exercise 4.4 on page 67 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 3

Solve problems using percentages Unit 4.5 • Percentages are fractions with a denominator of 100. • The % symbol means 'out of 100'. • Business calculations often include percentage price increases and decreases and gross profit percentages. • Companies use percentages to compare and analyse data. Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 4.11 page 69 Unit 4.5 67 51 If you get 80 for one test and 60 for your next test, have your marks improved? Convert both fractions to percentages: Multiply by 100. 67 × 100 = 83,75% On a calculator, use 67 ÷ 80 % 80 51 × 100 = 85% On a calculator, use 51 ÷ 60 % 60 So the second test marks were better. Downloaded from hlayiso.com Maths Lit NQF Level 3

Calculate a percentage of a value Unit 4.5 • To find a percentage of a number, multiply the number by the percentage and divide by 100. Example: • 30% of R600 = 30 × R600 = R180 100 (On a calculator, use 600 × 30 %) Downloaded from hlayiso.com Maths Lit NQF Level 3

Decrease and increase a value by aUnitpercentage 4.5 • Percentage increases and decreases are calculated on the original value before the percentage change. • If you increase an amount by 30%, you need to find the value of 130% of the amount. Example 4.13 page 70 1. Increase 135 by 30%. Method 1: Method 2: • 130 × 135 = 175,5 • 100% = 135 100 1 130 135 • The increased value is • So 130% = × = 175,5 100 1 135 + 40,5 = 175,5. Downloaded from hlayiso.com Maths Lit NQF Level 3

Express a part of a whole as a percentage Unit 4.5 • This is the same as converting a fraction to a percentage. Example 4.14 page 71 Of the 18 000 000 children in South Africa, 4 000 000 live in KwaZulu-Natal. What percentage is this? 4 000 000 • 18 000 000 ×100% = 22,2% (rounded off to 1 decimal place) • When you know the part of the whole quantity and you know the percentage of the whole that it represents, you can calculate the whole quantity. Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 4.15 page 71 Unit 4.5 1. A computer costs R14 250 including VAT. Calculate the VAT-exclusive price. Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 4.15 page 71 Unit 4.5 Since the VAT percentage of 14% was added to the original price, the VAT-inclusive price is 114% of the original price. The original price is the unknown. We can set up the following proportion: 14 250 114 = original price 110 14 250 × 100 = 114 × original price 1 425 000 = 114 × original price Divide both sides by 114: 12 500 = original price So the original price was R12 500. Downloaded from hlayiso.com Maths Lit NQF Level 3

Exercise 4.5 Unit 4.5 Complete Exercise 4.5 on page 72 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 3

Module assessment Module 4 Complete Module assessment on page 74 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 3

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