Mathematical Literacy Paper 2 Questions Question 1 1.1 Thamu is a sales representative and he needs to purchase a vehicle to use for visiting his clients. He narrowed his choice to the two vehicles shown below, a Toyota Avanza into which he could fit his surf board or a Honda Ballade, which would be more comfortable for long journeys. Both vehicles have the same engine capacity (1.5 litre) but their prices and fuel consumptions are very different. Specifications Model Price Fuel type Toyota Avanza 1.5 SX Vehicles Honda Ballade 1.5 Elegance R170 500 R193 300 Petrol Petrol Fuel tank capacity (litres) 45 45 Fuel consumption 10.5 13.2 Copyright reserved 1
(km/litre) Carbon dioxide Emission (g/km) 190 148 Ref: Car magazine, February 2012 1.1.1 Looking at the fuel consumption of the two cars, Thamu s colleague, Jane, believes that the Toyota uses less petrol but Thamu disagrees. Explain why Thamu is in fact correct. 2 1.1.2 Express the petrol consumption of the Honda in terms of the number of litres consumed for each 100km travelled. Give your answer to ONE decimal place. 2 1.1.3 If both cars are filled to capacity with petrol and driven until the fuel tank is empty, determine the difference in the distances that could be travelled by each car. 5 1.1.4 The price of unleaded petrol in Gauteng is R10.77 (4 January 2012). Determine the fuel cost per kilometre travelled for each vehicle. Give your answers to the nearest cent. 4 1.2 To help Thamu with his decision Jane plotted a graph showing the costs (purchase price and fuel costs) over given distances for the two vehicles. Copyright reserved 2
Use the graph above to answer the following questions: 1.2.1 Which line, the solid or dotted, represents the Toyota? Give a reason for your answer. 2 1.2.2 Approximate the distance that must be travelled for the breakeven point to be reached for these two vehicles. 2 1.2.3 Explain what the breakeven point means in this context. 2 1.3 The total cost for travelling a certain distance can be calculated using the following formula: Copyright reserved 3
Calculate the difference in the total costs of the two vehicles if Thamu s aim is to keep his vehicle until it has travelled 250 000km. Give your answer to the nearest R100. 6 1.4 From September 1, 2010, a new Carbon Dioxide (CO 2 ) Emissions Tax or Green Tax was introduced on all new passenger vehicles. The emission of each vehicle is measured as the number of grams of carbon dioxide emitted for every kilometre travelled, or g/km. The tax is R75 per g/km (excluding VAT) for each g/km exceeding 120g/km. This means that a Honda Ballade, which emits 148 grams of carbon dioxide per kilometre, will be taxed on 28g/km of emissions. Calculate the Green Tax (including 14% VAT) payable on the Toyota. 4 1.5 Thamu believes that it is important to be environmentally aware. Based on the price of the two cars, the fuel consumption and the carbon dioxide emissions of both, which vehicle do you think he should choose? Motivate your answer. 3 32 Question 2 2.1 When Gerry first started work at Zingela, Smith and Naaido in 2008 after leaving university, he did not understand his salary slip. Below is the first salary slip he received. ZINGELA, SMITH & NAAIDO Attorneys at Law Employee s name: Gerry Jones Identity No: 830615 5056 089 Pay date: 25 March 2008 Income Amount Deductions Amount Basic Salary R10 090.00 Medical Aid R772.00 Pension UIF (unemployment) R415.00 R100.09 Total Income R10 090.00 Tax on total income R1 126.20 Total Deductions Q2.1.4 a) Net Income Q2.1.4 b) Copyright reserved 4
2.1.1 How old was Gerry when he started working? 2 2.1.2 Explain to him the difference between Gross Income and Net Income. 2 2.1.3 He started working at the beginning of March 2008 and his contract stated that he would receive a 13 th cheque as a bonus in December. What was his gross annual income for the tax year ending February 2009? 2 2.1.4 Calculate Gerry s: a) Total deductions for March 2008, and hence 2 b) His net income for that month. 2 2.2 In 2004 Gerry inherited R50 000 from his grandmother which he invested in a fixed deposit at his local bank. He received an interest rate of 7% per annum compounded monthly on his investment. Use the formula below to calculate the value of his investment after 8 years. where A = final value P = principal value i = monthly interest rate as a decimal fraction n = term, or number of months for which the money was invested. 5 2.3 Gerry, now a successful lawyer, would like to buy a house. The house which he wishes to buy is being sold for R950 800. He was surprised to discover the costs involved in buying a house and taking out a home loan. 2.3.1 One of the costs is transfer duty. This is an amount that must be paid to the South African Revenue Services to transfer the property from the seller to the buyer. Transfer duty is currently calculated according to the following table: Purchase price R0 R600 000 0 % Transfer duty R600 001 R1 000 000 3% of value above R600 000 R1 000 001 R1 500 000 R12 000 + 5% of value above R1 000 000 More than R1 500 000 R37 000 + 8% of value above R1 500 000 Ref: www.sars.gov.za (5 February 2012) Use the above table to show how the amount of R10 524 was calculated. 3 2.3.2 What do you notice about the rates of transfer duty with respect to the purchase price of a house? Explain fully why you think this is the case? 4 Copyright reserved 5
2.3.3 Gerry is able to pay a deposit of 20% of the purchase price of the house. Using the information below given to him by the estate agent, calculate the a) The deposit, and hence b) The total costs that he will need to pay before he moves into his new house. Cost details Cost Transfer duty R10 524 Legal fees R15 450 Bank and bond costs R15 792 Deposit 20% of purchase price Q2.3.3 a) Total (deposit and other costs) Q2.3.3 b) Ref: www.ooba.co.za (5 February 2012) 4 2.4 Gerry s bank agreed to lend him R760 640 at 10.5% per annum over 25 years to pay for his house. The bank gave him the factor table below to show him how his monthly bond repayments will be calculated. The table gives various factors which are dependent on the interest rate and the period of the bond. Factor table Period of loan Interest rate 20 years 25 years 8.5% 8.68 8.05 9.0% 9.00 8.39 9.5% 9.32 8.74 10.0% 9.65 9.09 10.5% 9.98 9.44 11.0% 10.32 9.80 11.5% 10.66 10.16 12.0% 11.01 10.53 Ref: www.absa.co.za To calculate Gerry s monthly repayment the bank uses the following formula: Monthly repayment = (loan amount 1000) factor Example For a loan of R435 000 at 11.5% interest over 25 years, the bond repayment is calculated as follows: Monthly repayment = (R435 000 1 000) 10.16 = R4 419.60 2.4.1 Use this table to determine Gerry s monthly loan repayments. 3 2.4.2 How would Gerry s monthly loan repayments be affected if he chose to take his loan over 20 years rather than 25 years? Give ONE advantage of making this choice. 2 2.5 Gerry was given the graph below showing the outstanding amount on his 25-year loan with time if he pays the required monthly instalment. Copyright reserved 6
Ref:www.fin24.com/tools/Calculators.aspx?Calc=Home Use the graph to answer the following questions: 2.5.1 Estimate how much he will have paid off his loan after 12.5 years. 3 2.5.2 Approximately how long will it take him to repay half of his home loan? 3 2.5.3 Explain why it takes longer to pay off the first half of the loan than the second half of the loan. Remember Gerry will be paying the same amount every month for 25 years and each payment is made up of capital, interest and bank charges. 2 39 Question 3 Mr and Mrs Marais have a small farm in the Karoo where water is often scarce. They have a borehole, which provides water for the farm. The water is pumped into their reservoir by a windmill. Both the reservoir and the windmill need to be repaired. 3.1 The reservoir is circular, has a flat bottom and made of concrete. The internal diameter is 4.5 m and the depth is 185cm. Copyright reserved 7
3.1.1 Mr Marais needs to calculate the capacity of this reservoir. Determine the volume of the reservoir using the formula below given to him by a neighbouring farmer: Volume (m 3 ) = diameter (m) diameter (m) depth (m) 4 5 3 3.1.2 How does the volume he obtained in question 3.1.1 compare with the volume if it is calculated using the formula for a right circular prism given below? Volume of a cylinder = π radius 2 height, π = 3.14 3 3.1.3 Which formula do you think is more useful to Mr Marais? Give TWO reasons for your choice. 3 3.2 The inside of the reservoir needs to be painted with a waterproof sealant to stop water from leaking out. The chemical required to do this is sold and priced according to the table below: Tin size Price (inc VAT) 1 litre R74.95 2 litre R129.99 5 litre R345.90 1 litre of sealer will seal 6.5 m 2 3.2.1 Use the following formula to determine the surface area, to the nearest square metre, of the inside of the reservoir: Surface Area = πr(2h + r), where r = radius, h = height and π = 3,14 3 3.2.2 Determine the most economical combination of tins of sealant required to paint the inside of the reservoir. Show all your working. 6 3.3 The windmill on the farm should have 16 metal vanes like the one shown alongside. One vane had fallen to the ground. It has the shape and dimensions shown below: Copyright reserved 8
A AB = 90cm B 10 cm C 20 cm 10cm Area of a triangle = ½ base perpendicular height Area of a rectangle = length breadth Pythagoras theorem: a 2 + b 2 = c 2 (in a right angle triangle) 3.3.1 Use Pythagoras theorem to show that the perpendicular height (AC) of this vane is 89.44cm. 3 3.3.2 Hence calculate the area of the windmill vane in square centimetres. 4 3.3.3 The metal required for these vanes is purchased in sheets that are 100 cm wide and 1.5m long. Draw an accurate diagram of one such sheet using the scale 1:10 (i.e. 1cm on paper representing 10cm in real life). On this diagram indicate how many vanes could be cut from each of these sheets. 4 29 Question 4 Population statistics are collected in all countries. This information is used by government departments to plan for the future and to predict trends. In South Africa a census was undertaken in 2001 and another in 2011. The latest census information is not available yet but the Department of Statistics has provided a mid-year estimate of the population of South Africa in 2011. 4.1 Below is a table giving the total population of South Africa in 2001 and that estimated in mid-2011, divided into male and female: 2001 2011 (est) Male 21 434 040 24 515 036 Female 23 385 737 26 071 721 Total 44 819 778 50 586 757 ref: www.statssa.gov.za 4.1.1 Compare the percentage increase in the male population with the increase in the female population from 2001 to 2011 using the equation given below. 4 Copyright reserved 9
4.1.2 Give TWO possible reasons for the difference in the population increase of males and females over this period? 2 4.1.3 The total population of South Africa increased by 12,59% from a census in 1996 to 2001. Calculate the total population of South Africa, to the nearest thousand, on census day in 1996. 3 4.2 The number of males and females living in South Africa is broken down into an age distribution of five-year intervals and this information is represented in charts known as Population Pyramids, or age-gender pyramids. Use the population pyramids below for 2001 and 2011 to answer the questions that follow: Copyright reserved 10
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4.2.1 Both Pyramids have a wide base and a narrow top. Explain what this tells us about the age distribution of the South African population. 2 4.2.2 i) Estimate, in millions, the number of children in the 0 4 age group in 2001. 3 ii) In which age group would these children be represented in 2011? 1 iii) Estimate the number of children, in millions, in this group. 2 iv) Give TWO possible reasons why you think there is such a large difference in numbers between this group of children in 2001 and 2011. 2 4.2.3 It is indicated that there are more females than males who were older the 80 years in both gender pyramids. Estimate the number of males and females in this age category in 2011 and determine the ratio of males to females in the form 1:n. 4 4.3 The following statistics about employment and the labour force in South Africa are given in the Quarterly Labour Force Survey: Quarter 3, 2011 by the Department of Statistics. 1000s Total population mid-2011 (estimated) 50 587 Working age population ( people aged 15 64) 32 435 Labour force (number of people from the working age Population who are economically active) 17 663 Number of people employed 13 125 Number of people unemployed 4 538 [Source: www.statssa.gov.za] 4.3.1 How many people in South Africa were younger than 15 years or older than 64 years in 2011? 3 4.3.2 From the table above it can be calculated that 45.54% of the Working Age Population is NOT economically active. a) Show how this was calculated 4 b) Give TWO possible reasons why someone between the ages of 15 and 64 years may not be economically active. 2 4.3.3 The unemployment rate is calculated as follows: Unemployment rate = Number of people who are unemployed x 100 Number of people in the labour force Calculate the unemployment rate for mid-2011. 3 4.3.4 The South African population in 2011 was 2.86 times that of the labour force. If the state of the economy improved dramatically, would you expect this factor to increase or decrease? Explain your choice. 3 38 Question 5 The map below shows the national roads joining major towns and cities in South Africa. The distance table provides the shortest distance between destinations. For example, to travel from Pretoria/Tshwane to Nelspruit one goes via the N4, the shortest distance will be 342km. Copyright reserved 12
[Ref: Fundamental Mathematical Literacy, Level 4, L. Bowie, et al] Scott decided to arrange a party in East London after his matric exams and invited friends from all around the country to join him. 5.1 Marc will be travelling down from Johannesburg and is wondering whether to travel to East London via Durban or via Bloemfontein. 5.1.1 According to the table above, how far is Johannesburg from East London? 1 Copyright reserved 13
5.1.2 Determine the distance from Johannesburg to East London via Durban. 2 5.2 Scott needs to give Tarryn instructions on how to reach East London from Springbok, north of Cape Town. Describe one possible route that Tarryn could take; tell her the direction in which she will be travelling, the towns or cities she will pass and the road numbers. 3 5.3 Rhett is late leaving Port Elizabeth and needs to be in East London for the party which begins at 7:30 pm. He has to travel 300 km and his car cannot go faster than 110km/hr. 5.3.1 Using the formula Time = distance speed, calculate the earliest possible time of his arrival in East London if he leaves Port Elizabeth at 5:00 pm. 4 5.3.2 Is he likely to arrive at that time? Explain. 2 12 Copyright reserved 14