The concept of ratio and ratios in their simplest form

MATHEMATICS

Grade 8

RATIO AND PROPORTION

MEASUREMENT

CONSTRUCTIONS

Module 15

THE CONCEPT RATIO AND RATIOS IN THEIR SIMPLEST FORM

ACTIVITY 1

Explaining the concept of ratio and expressing ratios in their simplest form

[LO 1.4, 1.5] 1. The [ : ] sign means that you can express two or more quantities (of the same kind) as a ratio, e.g., if you receive R10 and I receive R15, we could express the two amounts as a ratio: 10 : 15. Note: Units are not named when we deal with ratios. 2. Ratios can also be written as fractions and be simplified, 23 size 12{ { {2} over {3} } } {}1015 size 12{ { {"10"} over {"15"} } } {} e.g.: 10 : 15 = = 2 : 3 (The ratio is now expressed in its simplest form.) 3. How are the following ratios expressed in its simplest form? 1 14 size 12{ { {1} over {4} } } {} : 1 18 size 12{ { {1} over {8} } } {} ? Suggestion:Write mixed numbers as fractions and treat the problem as division of fractions. 4. Now write each of the following ratios in its simplest form: 4.1: 18 : 24 : 30 4.2: 3 12 size 12{ { {1} over {2} } } {}: 4 12 size 12{ { {1} over {2} } } {} 4.3: 70 min : 1 14 size 12{ { {1} over {4} } } {} h 4.4: 7,5 kg : 500 g 4.5: 30 m : 300 cm 5. Given: ratio 3 : 5. 5.1 What is the smaller amount if the bigger amount is R50? Suggestion: Always draw a table. In the first column, write: ratio and money. Fill in the given data. Calculate a. RATIO 3 5 MONEY (R) a 50

6. Job opportunities in Langa: Draw a table and calculate the following: How many men are there in Langa if the ratio in numbers between men and women is 3 : 7 , and the number of women is 2 520? RATIO

7. At present, education policy in South Africa requires the following in schools: one educator for every 35 learners. 7.1 How many educators may be appointed at the Morningstar Primary School if 315 learners have been enrolled? 7.2 What must the number of learners be if the school has 23 educators? 8. The Education Department attempts to ensure a proportional ratio between educators and learners in all schools: 8.1 How many teachers are in excess and are therefore paid by the school's management council if a particular school has 700 learners and 32 educators? 8.2 What does it cost the parents to pay the salaries of the extra teachers if a teacher earns R4 982,55 per month? ACTIVITY 2

Using ratios for easy comparison of particular data and for the division of any unit

[LO 1.4, 1.5, 3.7, 4.1] a) Comparing data by making use of ratios 1. Building costs of low cost housing developments in the Western Cape increased from R1 000 to R1 220 /m² between 2000 and 2003 as opposed to an increase from R1 330 to R2 102/m², in Gauteng. 1.1 How would you indicate that the increase is fair? 1.1.1 Express all the data as ratios. 2000 2003 GAUTENG: WESTERN CAPE:

1.1.2 Express each ratio as a decimal number or a percentage. (Make use of your calculator.) Gauteng: Western Cape: 1.1.3 Which ratio is larger? 1.1.4 Is the increase fair? 1.1.5 To what would you ascribe the difference? b) Division by making use of ratios 1. Mr Verkuil of the Langverwacht Primary School won R 150 500 in the Lotto last Saturday. He set aside R50 000 for personal use and decided to divide the rest of the money between the school's Aids project and the Helpmekaar Fund for needy farm workers. He decided to divide the money to the ration of 3 : 5. How much money will each fund obtain? This is how it can be determined: 1.1 The ratio between the Aids Project (AP) and the Helpmekaar Fund (HMF) is 3 : 5. What does this mean? The AP will receive R3 for every R5 that the HMF gets. Therefore the first step would be to divide R8. 1.3 So we can do the following: AP : HMF 3 : 5 The AP portion: ............ size 12{ { { "." "." "." "." "." "." } over { "." "." "." "." "." "." } } } {} to R = ……………………………………… The HMF portion of: ............ size 12{ { { "." "." "." "." "." "." } over { "." "." "." "." "." "." } } } {} to R …………. = …………………………… (Use your calculator and round off to the nearest cent where necessary.) Now try it yourself: 1. Compare each of the following ratios and indicate which one is bigger. Make use of percentages. 1.1 In Zimbabwe, the area of land available to black people as opposed to whites is 1 200 km² : 1,35 km². In South Africa the ratio of land available to black people to white people is 0,95 km² : 135 km². What is the difference in land ownership between Zimbabwe and S.A. expressed as a percentage? 1.2 38 : 73 and 13 : 43 2. During 2003, the Mathematics HG paper indicated a grand total of 400 marks. A mistake crept in and the paper actually counted out of 375 marks. 2.1 Express the above information as a ratio in its simplest form. 2.2 Use the information in 2.1 to convert the marks of the following learners at the Primrose Private School from a mark out of 375 to a mark out of 400: a) Sarie Neetling: 215 b) Thabo Nakane: 172 c) Maria Schmidt: 370 2.3 Calculate the average of the marks obtained for the Mathematics HG paper at this school:

<para id="para-id1165131039152">  </para> </section> <section id="id1165135674141"> <title>ACTIVITY 3

Increasing or decreasing a given ratio

[LO 3.7, 4.1] In the following activity, and in those that follow, the "recipe" that is given is of utmost importance. Recipe for success: Always set up a table. Always ask: Will the answer be more or less than what is given? 1. Increase R250 by the ratio of 2 : 3. Set up a table. Question: Where do I put down the R250? Yes, below the ratio of (2), because the amount must be increased to 3. Will the entry below ratio (3) be more or less than R250? Yes, more. Now present your information as ratios and calculate the required answer. Table: RATIO 2 3 AMOUNT 250 a

Present your information as ratios: 2 : 3 = 250 : a 2 and R250 must be in the first positions. 23 size 12{ { {2} over {3} } } {} = 250a size 12{ { {"250"} over {a} } } {} (Do crosswise multiplication ): 2 x a = 3 x 250 2a = 750 a = 375 Check your calculation: Your answer should be more than R250. 2. Now do the following. Remember the recipe for success: 2.1 Ms Radetski is experiencing financial problems and she decides to ask her domestic helper who has been working 5 days per week to only come to work 2 days per week. She decides to decrease her domestic worker's present salary of R1 250 per month to the ratio of 5 : 2. Calculate the domestic worker's adjusted salary. Table: RATIO AMOUNT

The price of a WV Polo is increased in the ratio of 3 : 5 for 2004. What will the price be in 2004 if the price in 2003 is R117 800? Table: RATIO AMOUNT

ACTIVITY 4

Comparing two different amounts

[LO 3.7, 4.1] 1. A well-known example is: 120 km/h. What is the meaning of it? 2. You drive a distance of 120 km in 2 h. What is your average speed? (km/h means km per hour or km ÷ h.) 3. When two different units are compared, in this instance km and hours (h), the answer is given as SPEED (km/h) or RATE. RATE is always indicated as ……………. / (per) …………………. 4. Try to do the following: 4.1 The Kotzes’ telephone account for July came to R 180,88 for 234 units. Calculate the cost per unit. What would the account have been if the Kotzes had used 423 units? 4.2 My car used 45,6 litres of fuel over a distance of 730 km and my sister's car used 48,4 litres over a distance of 662,4 km. Which car uses fuel more economically? 4.3 Pick ‘n Pay sells Omo washing powder in boxes of two different sizes: 1,5 kg for R25,56 and a 2 kg box for R32,44. Which one is the better buy?

ACTIVITY 5

Differentiating between a direct proportion and an indirect proportion

[LO 1.5, 3.7, 4.1] The recipe for success is also important in this exercise. Set up a table. The question is: More-more or less-less? The answer is obtained from your table. (A): Direct proportion: More-more or less-less as the answer to the question. [DIVIDE] (B): Indirect Proportion: More-less or less -more as answer to the question. [MULTIPLY] (A): E.g.: 6 chocolate bars cost R55,45. How much will 13 bars cost? Table: CHOCOLATE BARS 6 13 COSTS 30 a

Your question: Will 13 chocolate bars cost more or less than R30,00? Your answer : More. Therefore: 6 ---- to R30 -> MORE 13 ---- to R a -> MORE This therefore is direct proportion. “DIVISION” Solution: 630 size 12{ { {6} over {"30"} } } {} = 13a size 12{ { {"13"} over {a} } } {} (crosswise multiplication) 6a = 13 x 30 6a = 390 a = 65 Therefore: 13 chocolate bars cost R65. (B): 6 men complete a task in 12h. How long will it take 8 men to do the same task? Table: MEN 6 8 TIME (H) 12 a

Your question: Will 8 men need more or less time to complete the task? Your answer: Less. Therefore: 6 ---- to 12 h -> MORE 8 ---- to a h -> LESS This is an indirect proportion. “MULTIPLY” Solution: 6 x 12 = 8 x a 72 = 8a 9 = a Now do the following. Indicate whether there is a direct or indirect proportion. The steps are given with no. 1, but you will have to do the rest yourself. 1. 2 dozen eggs cost R25,50. What do 7 eggs cost? Table:

Your question: Your answer: Therefore: ---- to -> (more/less) ---- to -> (more/less) This therefore is “ ” Solution: 2. A 3,5 m-long stick casts a shadow that measures 5,20 m on the ground What is the height of a flagpole that casts a 29,20 m-long shadow? 3. François of 7th Avenue walks at a speed of 5 km/h and cycles at 15 km/h. If he cycles, he reaches the Coffee Den in 15 minutes. How long does he take when he walks? 4. The woodwork educator can cut 12 mm-long strips of wood of length 190 mm from a single length of wood. How many 250 mm-strips could he cut from the same length of wood? 5. A Boeing 747 of the SAA flies from the Cape Town International Airport to London in 17 hours, at an average speed of 1 200 km/h. What will the average speed be if the time is reduced to 13 hours?

Assessment LO 3 Space and Form (geometry)The learner is able to describe and represent features of and relationships between two-dimensional forms and three-dimensional objects in a variety of orientations and positions. We know this when the learner: 3.2 describes and classifies geometric figures and three-dimensional objects in terms of properties in contexts inclusive of those that can be used to promote awareness of social, cultural and environmental issues, including:3.2.1 sides, angles and diagonals and their relationships, focusing on triangles and quadrilaterals (e.g. types of triangles and quadrilaterals); 3.3 uses vocabulary to describe parallel lines that are cut by a transverse, perpendicular or intersection line, as well as triangles, with reference to angular relationships (e.g. vertically opposite, corresponding);3.4 uses a pair of compasses, a ruler and a protractor for accurately constructing geometric figures so that specific properties may be investigated and nets may be designed;3.5 designs and uses nets to make models of geometric three- dimensional objects that have been studied in the preceding grades and up till now;3.7 uses proportion to describe the effect of expansion and reduction on the properties of geometric figures;3.8 draws and interprets sketches of geometric three-dimensional objects from several perspectives, focusing on the retention of properties. LO 4 MeasuringThe learner is able to use appropriate measuring units, instruments and formulas in a variety of contexts. We know this when the learner: 4.1 solves more complicated problems involving time, inclusive of the ratio between time, distance and speed;4.2 solves problems involving the following:4.2.1 length;4.2.2 circumference and area of polygons and circles;4.2.3 volume and exterior area of rectangular prisms and cylinders; 4.3 solves problems using a variety of strategies, including:4.3.1 estimation;4.3.2 calculation to at least two decimal points;4.3.3 use and converting between appropriate S.I. units; 4.5 calculates the following with the use of appropriate formulas:4.5.1 circumference of polygons and circles;4.5.2 area of triangles, right angles and polygons by means of splitting up to triangles and right angles;4.5.3 volume of prisms with triangular and rectangular bases and cylinders; 4.7 estimates, compares, measures and draws triangles accurately to within one degree.

Memorandum ACTIVITY 1 :3:4:5 72 size 12{ { {7} over {2} } } {} : 92 size 12{ { {9} over {2} } } {} = 7:9 :70:75 = 14:15 :7 500:500 = 15:1 :3 000:300 = 10:1 6. 3:7 = x:2 520 = 37 size 12{ { {3} over {7} } } {} = x2520 size 12{ { {x} over {"2520"} } } {} 7x = 3 × 2 520 x = 32×5207 size 12{ { {"32" times "520"} over {7} } } {} = 1 080 :1:35 315  35 = 9 :23 × 35 = 805 :32 – (700  35) = 12 :12 × R4 982,55 = R59 790,60 ACTIVITY 2 a) 1.1.1 2000 2003 Gauteng: 1 330 2 102 Western Cape: 1 000 1 220 Gauteng: 13302102 size 12{ { {"1330"} over {"2102"} } } {} = 0,63 / 63,3% Western Cape: 1000:1220 = 10001220 size 12{ { {"1000"} over {"1220"} } } {} = 0,82 / 81,97% = 82% Western Cape Own conclusion Own conclusion b) 1.3 VP part of 38 size 12{ { {3} over {8} } } {} of R100 500,00 = R37 687,50 HMF part of 58 size 12{ { {5} over {8} } } {} of R100 500 = R62 812,50 Now you can try: 1.1 Zimbabwe: 12001.35 size 12{ { {"1200"} over {1 "." "35"} } } {} = 888,90 South Africa: 0.95135 size 12{ { {0 "." "95"} over {"135"} } } {} = 128,30 1.2 3873 size 12{ { {"38"} over {"73"} } } {} × 1001 size 12{ { {"100"%} over {1} } } {} = 52,1% / 1343 size 12{ { {"13"} over {"43"} } } {} × 1001 size 12{ { {"100"%} over {1} } } {} = 30,2% 375:400 = 375400 size 12{ { {"375"} over {"400"} } } {} = 1516 size 12{ { {"15"} over {"16"} } } {} = 15:16 a) 215  15 × 16 = 229 172  15 × 16 = 183 370  15 × 16 = 395 ACTIVITY 3 2.1 Ratio 5 (less than) 2 Amount 1 250 (less than) x 5:2 = 1 250:x 52 size 12{ { {5} over {2} } } {} = 1250x size 12{ { {"1250"} over {x} } } {} 5x = 2500 x = R500.00 ACTIVITY 4 Drives 120 km in 1 hour 1202 size 12{ { {"120"} over {2} } } {} = 60 km/h 4.1 a) 180,88  234 = R0,77/unit 423 × 0,77 = R325,71 45,6 ℓ = 730 km = 16 km/ℓ Sister: 48,4ℓ = 662,4 km = 13,69 km/ℓ A: 25,56  1,5 = R17,04/ kg B: 32,44  2 = R16,22/ kg = Best buy ACTIVITY 5 1. Table: dozens (number) 2(24) (less) 7 Price 25,50 (less) x Therefore: 24 to 7 = less 25,50 to 7,44 = less It is therefore an indirect Solution: 24:7 = 25.50:x 247 size 12{ { {"24"} over {7} } } {} = 25.50x size 12{ { {"25" "." "50"} over {x} } } {} 24x = 178,50 x = R7,44 2. Length 3,5 m (more) x Shadow 5.20 m (more) 29,20 m 3,5: x = 5,2:29,2 3.5x size 12{ { {3 "." 5} over {x} } } {} = 5.229.2 size 12{ { {5 "." 2} over {"29" "." 2} } } {} 5,2x = 102,2 x = 19,7 m 3. Walk 5 km/h (more) 15 km/h Cycle x (less) 1560 size 12{ { {"15"} over {"60"} } } {} = 312 size 12{ { {3} over {"12"} } } {} = 14 size 12{ { {1} over {4} } } {} h 5x = 15 × 14 size 12{ { {1} over {4} } } {} x = 0,75 h = 34 size 12{ { {3} over {4} } } {} h 4. Pieces 1 2 (less) x mm 190 (more) 250 12 × 190 = 250x 9.12 = x 9 pieces 5. Time 17 (less) 13 Speed 1200 (more) x 17 × 1200 = 13x 1569 km/h = x