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Number video clip DO NOW! Skills Two or more Step problems Practice Questions
Do Now Introduction Significant Figures Rounding Standard form Decimals Fractions Integers Percentages – of an amount, find the %, incr/decr by %,of an amountfind the %incr/decr by %, % incr/decre% incr/decre, GSTGST Powers and roots Ratios Rates Proportional Reasoning Currency conversion
You will need to be able to: Use the following to solve a multi-step word problemword – rounding with decimal places and significant figures roundingdecimal placessignificant figures – standard form, both from and into standard form and on calculator standard formfromintoon calculator – fractions, decimals and percentages and interest fractionsdecimalsinterest find%, find original amount, incr/decr by %, % change, GST find%find original amountincr/decr by %, % changeGST – integer and fractional powers applied to numbers – ratio and proportion, direct and inverse ratio direct inverse – factors, multiples, powers and rootspowersroots – Rates Rates – Currency conversions Currency conversions – Integers Integers – Order of Operation Order of Operation
Interpreting Word problems What do the following words mean? Altogether Share Lots of More Each Between Total Left Difference Sum Addition, Sum, + or Multiplication, times, x Division, divided, ÷ Multiplication, times, x Multiplication, times, x or Addition, Sum, + Division, divided, ÷ Addition, Sum, + or Multiplication, times, x Subtraction, take away, - Addition +
Significant Figures To count the number of significant figures Start counting at the first non-zero digit Stop counting – When end of number if there is a decimal point – When there are only zeroes left if there is no decimal point
Examples How many significant figures in 3147 (4 sf) 306 (3 sf) (4 sf) (3sf) (2 sf) (2 sf)
Decimal Places Count all digits after the decimal point Examples (3 dp) (5 dp) Ex p 196 qn 1-2
Rounding To round a number to a given number of decimal places/significant figures: Underline the required number of digits after the decimal place/from the beginning of the number Look at the next digit – if it is less than 5, the underlined digit is the last one! – if it is 5 or more, then add one to the last digit underlined and drop all other numbers For significant figures, zeros may be needed as place holders Always round money to the nearest cent unless otherwise stated
Examples to 2dp to 2sf to 2dp to 1 sf0.5 Ex p
Estimation To estimate, use 1 significant figure and do the calculations in your head. Example 230 x 4.56 becomes 200 x 5 = 1000 P196 Ex vies/swf/galaxy_42.html
Standard Form Very large and very small numbers are written in standard form. Number in Standard Form = Number between 1-10 x 10 n Example 3.26 x 10²
Converting from Standard Form To convert a number from standard form into ordinary form, Shift the numbers so that there is only 1 digit in front of the decimal point: If the number is greater than 1, n is positive If the number is smaller than 1, n is negative 6.45 10 1 n is positive so number > 10 2 n is positive so number > n is negative is number < n is negative so number <
Converting into Standard Form To convert ordinary numbers into standard form, move the digits so there is only 1 digit in front the decimal point then count the number of spaces (n) you had to move the digits. If the number was greater than 1, write 10 n (n is positive) If it was less than 1, write 10 -n (n is negative)
Examples 598shift to 5.98 (2 places), >1 so n = x shift to 8.65 (1 place), >1 so n= x shift to 6.78 (3 places), <1 so n = x shift to 5. (2 places), <1 so n = -2 5 x Ex p 376 Qn 1 -4,(every 2 nd one) Qn 6 Qn 7-8 (every 2 nd one)
Standard Form and the Calculator To type in 5.26 x 10 6 Type in This will show up as This means x10 Ex p 380 Qn 9-19
Integers When adding – Move to the right and subtracting – Move to the left For a negative number – Move in the opposite direction! This means subtracting a negative is the same as adding! Examples: = = - 2 Examples: = = - 11 Example: = = + 3 Example: =
Multiplying and Dividing Two negatives will give a positive answer One negative will give a negative answer Or an even number of positives Or an odd number of positives Examples: - 3 x - 4 = x 6 = x - 7 = ÷ 3 = ÷ - 5 = 1142 ÷ - 7 = x 2 x 4 = - 6 x 44 x 1 x - 2 x - 3 = 24 P 328 Ex 23.01
Order of Operation Brackets Exponents Division Multiplication Addition Subtraction Otherwise work from left to right 4 = = 17 6 (3 + 2) = 6 5 = 30 6(13 – 10) + 5(20 4) = 6 5 = = 43 P 330 Ex Qn 1 and 2, every 3 rd one Qn 3-9
Powers Powers are used to show repeated multiplication of the same number. For powers greater than 2 use thebutton Example: 2 5 means 2 x 2 x 2 x 2 x 2 Bases can be a negative number and the sign depends on the value of the exponent. Remember to use brackets on the calculator for this. Examples: (-3) 4 = 81(-7) 3 = = -81 Exponent Base
Roots Roots are the reverse of powers. The square root undoes squaring.Example: = 7 The cubed root undoes cubing.Example: = 4 For other roots we use shift and this button on the calculator. Example: = 3 Remember to follow the BEDMAS rules if there is an equation under the square root sign. Example: = P 333 Ex every second one 7, every second one
Fractions - simplifying You should always simplify fractions For example Find a common factor of the numerator and denominator and divide by this – In this case 4, so you get You can use the fraction button on the calculator to do this for you!
Mixed and Improper Fractions It is often better to use Improper fractions, (where the numerator is larger than the denominator) rather than a mixed number. rather than Divide the numerator by the denominator e.g = 3 with 5 remainder so Or use the calculator fraction button
Multiplying Multiply the numerators Multiply the denominators Remember to simplify the answer!
Dividing Find the reciprocal of the second fraction (turn it upside down) Multiply! Simplify! You can simplify at this stage if you want
Adding and Subtracting Must have same denominator, if not find an equivalent fraction that does! + = + = But of course, you can just use the fraction button on the calculator! Gamma P Ex
Decimals FractionsPercentages Into a % To convert a decimal and fraction to a percentage multiply by 100%. Examples: 0.6 = 0.6 x 100%0.348 = x 100% = 60% = 34.8% x 100% x 100% = 32.5%= 20%
From a % To convert a percentage to a decimal or fraction, divide by 100 ( and simplify it a fraction if required). 75% = 64%= = 0.75 Gamma Ex P 346
One Amount as a % of Another Amount Write the information into a fraction and then change to a percentage. Examples: 1.40 out of 50 pupils love mathematics. What percentage of pupils love mathematics. Percentage= x 100% = 80%
In the last mathematics test a student scored 45 marks out of a possible 60 marks. What percentage did they get in the mathematics test? Percentage = x 100% = 75% Gamma Ex p 347 qn 2,3,4,5,7,…..
Calculating Percentages ‘of’ Quantities To calculate a percentage of a given quantity, multiply the quantity by the percentage (as a fraction or a decimal). Examples: 24% of 70 = x 70 = % of the Year 10 pupils at OGHS (160 pupils) are left handed. How many Year 10 girls are left handed? Left handed girls = 30% of 160 = 0.3 x 160 = 48 Gamma Ex P 346 Qn 5 Gamma Ex P 347 Qn 14,19,20,21,23
Calculating ‘Original’ Quantities To calculate the original quantity we reverse the process of working out percentages of quantities. We express the percentage as a decimal and write an algebraic equation to solve. Examples: 30 is 20% of some amount. What is this amount? 20% of x= x x= 30 x= 150
15% of the students in a class are left handed. If there are 6 students who are left handed, how many students are in the class? 15% of x = x x= 6 x= 40 Gamma P347 Ex Qns 8, 9, 10, 11, 22,24,25
Calculating Percentage Increases/Decreases To calculate a percentage increase or decrease: Percentage =difference in values x 100% original amount Examples: The number of senior girls boarding at the hostel increases from 60 to 75 girls. What percentage increase is this? Percentage increase = x 100% = 25% = 15
The population of a town decreased from 600 to 540 people. What percentage decrease is this? Percentage decrease = x 100% = 10% 540 – 600 = -60 Percentage Change Worksheet
Increasing and Decreasing by Percentages First change the % into a decimal (r) Original (old) Quantity Increased /dec reased (New) Quantity x When you increase use (1+r) When you decrease use (1-r) ons/flow_diagrams.swf
Examples: 1.The price of a computer currently selling for $2500 increases by 5%. Calculate the new selling price. 5% becomes 0.05Increase so add 1, use 1.05 Old to new so multiply 2500 x 1.05 = $ Coca Cola reduced the caffeine content of their coke drink by 10%. They now contain 80g/L of caffeine. How much did they contain before the reduction? 10 % becomes 0.1Decrease so subtract, = 0.9 New back to old so divide 80 ÷ 0.9 = (2dp) 5 ÷100 = 0.05
3.A car depreciates 15% over a year. It was worth $15000 at the start of the year. What was it worth at the end of the year? 15% = 0.15decrease so subtract, = 0.85 Old to new so multiply x 0.85 = $ House prices have risen 21% over the last 3 years. The market value of a house today is $ What was the value of the house 3 years ago? 21% = 0.21increase so add, use 1.21 new back to old so divide ÷1.21 = $ (2dp) Gamma p353 Ex 24.04
Goods and Services Tax (GST) GST is a tax on spending (15% or 0.15) Increased Quantity - With GST Original Quantity - No GST x GST is always an increase so always use ( = 1.15)
Examples: 1.A DVD player is advertised at $1500 excluding GST. What price will you pay for the DVD player once the GST is included? With GST means an increase so use 1.15 Old to new so multiply 1500 x 1.15 = $ A new computer package costs $2700 including GST. How much of the price is GST? With GST means an increase so use 1.15 With GST (new) to without GST (old) so divide 2700 ÷1.15 = $ Gamma p 359 Ex So amount of GST = $ $ = $352.17
Ratios When two quantities measured in the same units are compared they give a ratio. Example: There are 32 lambs to 21 ewes. Write as a ratio. 32:21 Ratios can be simplified, just like fractions, by finding a factor that goes into both ratios. Examples: 5:15 = 1:312:28 = 3:7 Equivalent ratios can find missing parts of a ratio. Example: The ratio of teachers at a primary school is 1:18. If there are 3 teachers how many students are there? 3: x students 1: 18 students You can use the fraction button on your calculator – but must be improper fraction acy/Animations/multilink_paint.s wf
To share a given ratio, work out fractions of the quantity. Example: Jack and Jill have $120 to split into the ratio of 5:7. How much does each person get? Total number of shares = = 12 Jack’s share = Jill’s share = Remember to simplify fractions! Gamma P364 Ex p 368 Ex 25.02
Interest Simple:I = PRT So if you invest $1000 for 5 years at 6% I = 1000 x 0.06 x 5 = $300 Compound: the interest is added to the capital at the end of each period So if you invest $1000 for 5 years at 6% compounding annually Interest = 1000 x 0.06 = 60 after 1 year = 1060 x 0.06 = 63.6 after 2 years = x 0.06 =67.42 after 3years = x 0.06 = after 4 years = x 0.06 = after 5 years Total interest = Amount= Principal(1+r)
Rates Rates compare quantities that are measured in different units. Example: The distance from Invercargill to Dunedin is 200 km. If it takes 2½ hours to cover this distance, what is my average speed in km/hr? Speed = = 80km/hr /Numeracy/Animations/be anies.swf
Proportional reasoning My car can travel 840km on a full tank of petrol. How far could it travel on 3 tanks? More petrol means the car will go further 1 tank = 840 km 2 tanks = 1680 km 3 tanks = 2520km If we graph this we get a straight line! The amount of petrol and the Distanced travelled are directly proportional
Inverse Proportional Reasoning If it takes three students 30 mins to spring clean the common room, how long would it take 5? Find out how long it takes 1 first! If it takes 3 students 30 mins, it would take 1 student 3 times as long, so 90 mins It would take 5 times a fifth of the time so 90 divided by 5 = 18 minutes Gamma p 371 Ex 25.03
How else could you do it? Create a table If it takes three students 30 mins to spring clean the common room, how long would it take 5? I person will take 3 times as long as 3 people If 3 people take 30 mins, 6 people will take half the time 2 people will take a three times as long as 6 people 4 people will take half the time 2 people take 5 people will take a fifth of the time 1 person takes Students Time taken
Or draw a graph If it takes three students 30 mins to spring clean the common room, how long would it take 5? 1 student will take 3 times as long so 90 mins, and 6 students half the time, so 15 mins. We can graph these 3 points We get a curve because the time is inversely proportional to the time (more people means less time) You can read off for 5 people - 18 minutes
Or find an equation If it takes three students 30 mins to spring clean the common room, how long would it take 5? Students x time = 90 So time = 90 ÷students Time = 90 ÷ 5 = 18 mins Students136 Time taken903015
Currency conversions There are generally 2 rates for currency. One when you are buying a different currency and one when you are selling. If you can buy 1 euro for $1.86, how many euros can you get for $200? € If you have come back from your holiday and can sell € 1 for $1.84, how much will you get for €234? $430.56
What do I need to know? Significant Figures Rounding Standard Form Revise – integers, powers, roots and BEDMAS Fractions Percentages -of a quantity -original quantity -increase and decrease -GST Rates Ratios Converting currency
Fry’s Nine use cards (A – 9) How many ways can you do it? What can’t go in the hundreds columns?
Dicing with Decimals Throw 6 sided dice. Can put digit on either column in row. Aim to get close to 9.9 Can also do 3 rows with 6 rolls to get 9.9 (use both columns) instead of 6 rolls and 6 rows
Multi digit game board Use 10 sided dice, aim to put numbers in, 1 row at a time in order to get largest value.
Create a concept map or Buzan Map for Number
Why is accuracy important
Do Now – Standard Form Write in standard form x x x Write in ordinary form 4.1 x x x
Make your own!
Do Now Write in standard form x Write in ordinary form3.243 x How many significant figures in Round to 3sf If the diameter of an atom is x 10 -9, how far across will 1530 atoms be? x 10 -6
Do Now - Integers = -1 4 x -5 =-20 3(-2+1) = =5 (3+1) x 8=32 4
Do Now – powers and roots Find √4 3 √ √
Do Now - % of an amount 20% of my Year 13 Stats class failed their assessment. If I have 30 students in the class, how many failed? 6 students
Do Now – write as % 1.40% of the students of Gale High School walk to school. If the school has 900 students, how many of them walk? Sharon scored 30 out of 40 in a test. What was her mark as a percentage? 75% 3.Stephen, Troy and Ricky scored 42, 45 and 32 out of 60 in an English test. Write each of these scores as a percentage to the nearest whole number. 70%, 75%, 53.3% (1dp)
Do Now - Ratios Tara and Casey are sharing out their M and Ms. As Tara paid less, they are sharing them in the ratio 3:5. If there are 40 M and Ms, how many do they each get? Tara gets 15, Casey gets 25 It takes 3 x 10 3 litres of fuel to travel 1 km in a space craft. If the moon is 4.3 x 10 8 km away, how much fuel will you need? 1.29 x 10 12
Do Now - GST If my new watch cost $67, how much of the cost was GST? Excluding GST price is $58.26, so GST = $8.74 If the shop is advertising a discount of 5% today for their ipods, how much would it have cost yesterday to the nearest dollar, if it costs $325 today? Reduction so.95, new to old so divide$342
Do Now – revision Sarah is baking cakes. She needs 3 cups of flour to make 10 cakes. How many cups of flour does it take to make 25 cakes? 7.5 cups Georgina is trading Jelly babies on Trade Me. If she has 2.3 x 10 5 jelly babies and makes $ 3.1 x 10 4 how much was each jelly baby? $1.3 x or 13c
Do Now – incr/decr by % A shop has 15% discount. If the jeans originally cost $130, what is the new discounted price? $110.5 A book is advertised as $16.90, a 20% reduction has been given. What was the ORIGINAL price? $21.13
Do Now - %incr/decr Last week 16 students did their homework, this week 14 did their homework. What is the percentage decrease in the number of students who do their homework? 12.5%
Do Now – find amount 70% of the Year 11 Maths class did their homework. – If there are 20 people in the class, how many did their homework? – 14 students – If the class had 7 people who did their homework how big was the class – 10 students
Do Now – Sig Figures How many significant figures in
Do Now - rounding to 2dp to 2sf to 2dp to 1 sf0.08 $ to nearest cent$56.67 $ to nearest dollar$57
Do Now – proportional reasoning 1.Ms Boomer can run a 8 km race in 36 minutes. If she runs at the same rate, how long should it take her to run a 5 km race? 2.If it takes 5 people 6 hours to wash the windows at the Sun Office building, how long will it take 4 people to do the job? 22.5 minutes 7.5 hours
Do Now - Ratios 1.Roi grows orchids for export. The ratio of flower stems to reject stems is 4:7. She exports 332 stems, How many stems are rejected?211 orchids 2.Tessa bought 3 computer disks for $2.40. Kate bought 4 computer disks for $3.75. Whose was the best buy? Tessa’s is better value
Do now - rates Helen is paid $25 per hour. If she works 5 hours, how much is she paid? $125 She pays tax of 30% on every dollar. How much does she pay? $41.33 How much does she get net (after tax)? $83.67
Multi-step Problems Decimal and fractions Overseas Decimal and fractionsOverseas Percentagesharder GST Salmon Percentagesharder GSTSalmon Shopping(%) Muesli bar Shopping(%)Muesli bar Bread Paper (std form) Rent and hessian (fractions) Pocket money(ratios proportion) Apples and sick Pocket money(ratios proportion)Apples and sick Clothing (GST) Bus (numeric reasoning) Newletter Beach Bus (numeric reasoning)NewletterBeach Farm (ratio)
Bria made herself a birthday cake. Lynda ate a fifth of the cake and Elizabeth ate 0.4 of the remainder. How much of the cake was eaten? 1 – 1/5 = 4/5 cake left 0.4 x 4/5 = 8/25 eaten by Elizabeth Total eaten: 1/5 + 8/25=13/25
Bria’s cake was chocolate. She used 0.6 of a large block of chocolate to make the cake and grated one eighth of the block to decorate the top. She ate the chocolate that was left over. How much did she eat? /8 = 29/40 1 – 29/40 = 11/40 left
A recipe for 12 small cakes uses 240 g of flour. A bag contains 1.5 kg of flour. What percentage of the flour is used to make 30 small cakes ? 240 ÷ 12 = 20g per cake 20g x 30 cakes = 600g flour needed 600 ÷ 1500 x 100 = 40%
A shop buys rugs from a factory. In July, each rug costs $ 100 and the shop buys 500. In August, the cost of each rug had increased by 13 % and the shop bought 34 % less. Calculate the difference between the amount the shop spend in July and August. July price: $100 x 500 = August price: $ % x $100 = $113 August amount: 500 x (100% – 34%) = 500 x 66%= 330 August: 330 x $113 = July – August amounts: – =
Which is the best option? Anna is investigating the options available to purchase a ipod Option A: Cost $ 399 including GST. Trade in of $70 on old ipod. Deposit of $100. Interest of 20 % on the balance. Option B: Cost $ 352 excluding GST. Trade in of $50 on old ipod. Deposit of $150. Interest of 13 % on the balance. Which option should Anna accept? Option A: Cost less trade in = 329Deposit = 100Leaving 229 Plus interest = 229 x 1.2 = TOTAL = Option B: Cost less trade in = 352 x 1.15 – 50 = Deposit = 150Leaving Plus interest: X = TOTAL = = Choose Option A
Shopping Mum bought a skirt which was reduced to ⅔ of the original price. Dad bought a shirt in a 35% off sale. Who got the better percentage reduction and by how much? Dad by 1.67% (2dp) Julie bought an iPod and an iPod case for a total price of $261. If the case cost $19, what percentage of the total price was the iPod? 92.7%
Rent and hessian Millie spent 2/5 of her wages on rent and ½ of her remaining wages on other household expenses. What fraction of her wages is left after wages and household expenses? 3/10 7/10 of a metre of hessian costs $3.50. What is the cost of m of hessian? $8.50
Pocket money Mum gives out pocket money to her three children in the ratio 3:5:7. The older a child is, the greater the amount of pocket money received. If Mum gives out $90 per month, how much pocket money does the oldest child receive per month? $42 The time taken to fill a swimming pool with water is inversely proportional to the rate at which the water is delivered to the pool. It takes 9.6 hours to fill the pool when the water is delivered at 12.5 cubic metres per hour. How long would it take to fill the pool if water is delivered at 15 cubic metres per hour? 8 hours
Bus Jane takes the bus to school every day. Hannah takes the bus every third day and Cleo takes the bus every fourth day. One day all the three friends are on the bus together. How many bus trips before this will happen again? 12
Bread The number of loaves of white bread baked in a supermarket is proportional to the number of loaves of brown bread it bakes. On Monday the supermarket baked 600 loaves of white and 360 loaves of brown bread. On Tuesday the supermarket baked 500 loaves of white bread. How many loaves of brown bread did it bake on Tuesday? 300 loaves
Interest Sammy and Beth go into business together, but they need to borrow money. Sammy borrows $8000 from her parents at 5% simple interest per annum for years. Beth borrows $8000 from the back for 3 years at 4.6% compound interest. Who pays the most interest Sammy or Beth? Beth by $44.44 Gina invests $5000 for four months, at a certain annual rate of interest. The interest is compounded monthly(ie each month the interest is added onto her investment). At the end of her investment she pay 19.5% tax on the total interest earned from the investment. If Gina makes $ interest after tax, at what annual rate was the money invested? 8.75%
Clothing Dressage is a clothing shop. All prices include GST. A jacket is priced at a GST- inclusive price of $ What was the price before GST was added? $ Sally, the owner of Dressage, decides to have a sale. She will reduce all clothing in the sale by 35%. What is the price of a $130 shirt? Sally gives a bag the sale price of $ What was the price before it was marked down? $84.50 and $72.60
Practice Questions Kelly Tarlton Bria’s Camera Blu-ray Heartbeat Rock Concert Mathland Exchange Student Travel Agency
Kelly Tarlton The amount received from entry fees at Kelly Tarlton’s Antartic Encounter and Underwater World over the past year depends on the number of visitors and the cost of the entry fees. The cost of an adult entry fee is $26. The cost of a child’s entry fee is $10. The visitors to the underwater world were made up of adults and children in the ratio of 2 : 3. On average the number of visitors per week in the 12 weeks of school holidays were twice as many as the other weeks of the year. The average number of visitors per week to the underwater world during term time is Kelly Tarlton’s Antartic Encounter and Underwater World can claim back the GST included in all the entry fees. How much can they claim back? 3400 12 = Total: x 40 = Children: 3/5 of = Adult: 2/5 of = Entry fees:65280 26 = $ GST to claim back = $232737
Bria’s Camera Bria is visiting Auckland and wants to take photos of her trip She decides to buy a digital camera priced at $1380. The camera shop offers her hire purchase with the following terms: one third deposit 12% interest per annum for two years equal weekly payments. Bria has budgeted $10 per week for the repayments. Will she be able to meet her budgeted amount if she accepts the hire purchase? $ Deposit = $460 Amount taken on HP = $920 Interest = $ Total to pay = $ Weekly payments = $10.97 No – not within budget
Apples and Sick 1.Bob has 100 apples. 1/5 get stolen. O.5 of the remaining apples are rotten. John split the non- rotten apples into the ratio 1:3 with Cathy. If John sold his for $1.00 and Cathy sold her’s for $2.00, how much did they make together? $70 2.Erin is sick for 1/3 of the school year. If a school year is 195 days, how many days was Erin sick for 65 days
Overseas.... Emma was in Japan and decided to buy rice cakes. She bought 18 rice cakes. 1/3 of them has soy bean powder on them and 1/6 of them has sweet bean paste and the remainder had seaweed. What fraction had seaweed? ½ Rachel was in Kenya and decided to go look at the wildlife. On the way she got attacked by a pride of lions. 1/3 of her was eaten by lions, and 1/7 by hyenas. Suddenly a stampede of zebras came and squashed one half of what was left of Rachel. Later the vultures came and ate the rest. What was left for the vultures? 11/42
Salmon The following estimates were made for the salmon run on the Rangitata River in a particular year adult salmon returned to the spawning grounds. 38.6% of returning fish were female A female salmon carries about eggs. About 50% of eggs will hatch successfully and produce smolt (baby salmon) 1.Estimate the number of smolt that were produced by the salmon run As a result of this smolt, 5000 salmon return as adults in 3 years time. Anglers will catch about 30% of this returning run 2a. Estimate the number of salmon that will not get caught and will return to the spawning grounds salmon 2b. What is this number as a % of the original number of smolt? 0.1%
Anglers frequently suggest that hatchery reared smolt be released into the river in order to increase the salmon run. It would cost $ to buy smolt. Suppose that smolt were bought and that 0.5% of these smolt return to the river as adults. 3. Estimate the number of salmon from the extra smolt the anglers would catch 0.1% What is the effective cost to produce each fish caught?$666.67
Muesli Bars Lucia was investigating the packaging on Oaties Muesli Bars. She measured the package and found it was 18.0cm long, 3.6cm wide and 14.5cm high. She calculated the volume to be 940cm3 (2sf). The packet had 8 muesli bars in it. Each was 10cm long, 3.5wide but the thickness varied from 1.2cm to 1.6cm. Find the volume of 8 muesli bars as a % of the packet’s volume. 48% if use 1.4cm 43% if use % if use 1.6
Blu-Ray Jenna bought a blu-ray player. It was now $50 off and only $399 or $5.39 weekly (Weekly payments were based on a10% deposit and included a $30 booking and insurance fee) Jenna paid a $40 deposit, if it takes her 2 years to pay it off, how much did she pay for the blu-ray player excluding the $30 booking and insurance fee? While Jenna was in the store she entered a competition “win your height in DVDs” If Jenna is 1.76m tall and the average DVD is $30 and about 1cm thick, estimate to the nearest $100 how much Jenna could win.
Ellie MacDonald’s Farm Ellie has a small farm. On her farm she has sheep and cows but altogether she has less than 20 animals. She has only 24 bales of hay each week to feed her animals Each sheep needs a bale of hay per week Each cow eats 2 bales of hay per week She can make $45 profit per sheep and $150 per cow. The ratio of cows :sheep is 4:5 How much can Ellie make? 4 cows, 5 sheep so $825
Paper Sheets of paper have a gsm rating, where gsm stands for grams per square metre. This means that one square metre of 45gsm will weigh 45 grams. The school newsletter was made from 2 types of paper. The cover used 85gsm paper and unfolded to measure 0.3m by 0.6m. The inside 8 pages were all 70gsm and each sheet measured 0.3m by 0.42m. There were 850 newsletters printed. Calculate the weight of all the newsletters together.
Beach A building company owns land by a beach. It plans to develop the land into 16 sections, and sell then at $ each. The planned development is a rectangle. 375m long and 45m wide. If some of the sections are used as parks, the remaining sections need to be sold at a higher price, $ more for every park. What number of parks would give the developer the greatest return and what is the value of the return?
Mathland In Mathland, the people are worried about being attacked by the nearby Mathfobics. The national defence budget for Mathland was $30 million for The total budget for that year was $500 million. In 2012, the defence budget was $35 million while the total budget was $605 million. Inflation during the period covered was 10%. aYou are invited to give a lecture for a pacifist society in Mathland. You intend to explain that the defence budget decreased over this period. Explain how you would do this. bYou are invited to lecture to a Mathland military academy. You intend to explain that the defence budget increased over this period. Explain how you would do this.
Mathland answer Answers will vary – examples of arguments follow: aDefence was (6%). If it had kept pace with inflation it would have gone up to 6.6%, but it has reduced to (5.8%). bDefence has increase by (16.7%), which is significantly more than inflation.
Rock Concert For a rock concert, a rectangular field of size 100 m by 50 m was reserved for the audience. The concert was completely sold out and the field was full with standing room only. What is the best estimate of the total number of people attending the concert?
Rock Concert Answer Answers will vary as there are a number of ways to estimate the floor area needed by one person. For example, if you estimate that four people can stand in 1 sq m, then people could stand in the field.
Heartbeat For health reasons people should limit their efforts, for instance during sports, in order not to exceed a certain heartbeat frequency. For years, the relationship between a person’s recommended maximum heart rate and the person’s age was described by the following formula: Recommended maximum heart rate = 220 – age Recent research has shown that this formula should be modified. The new formula is: Recommended maximum heart rate = 208 – (0.7 age) aA newspaper article stated: ‘A result of using the new formula instead of the old one is that the recommended maximum number of heart beats per minute for young people decreases slightly and for old people it increases slightly.’ From which age onwards does the recommended maximum heart rate increase as a result of the introduction of the new formula? bThe new formula for maximum heart rate is also used to determine when physical training is most effective. Research has shown that physical training is most effective when the heartbeat is at 80% of the recommended maximum. cA sports centre is using this to refine its exercise programs. Your task is to provide them with a method for calculating the heart rate for most effective physical training for any age person.
Heartbeat Answers aAt age 40 years – a range of methods could be used to justify this, eg a table of values: b. Various methods – eg optimum heart rate = 0.8 (208 – (0.7 age)) Optimum heart rate = 166 – 0.56 age Students may also consider formulas that are less precise but easily calculated and give answers close to the optimum. Age (years)Old formulaNew formula
Exchange Student Liping from Singapore was preparing to go to South Africa as an exchange student for 3 months. She needed to change some Singapore dollars (SGD) into South African rand (ZAR). aLiping found out that the exchange rate between Singapore dollars and South African rand was 1 SGD = 4.2 ZAR. Liping changed 3000 Singapore dollars into South African rand at this exchange rate. How much money in South African rand did she get? bOn returning to Singapore after 3 months, Liping had 3900 ZAR left. She changed this back to Singapore dollars, noting that the exchange rate was now 1 SGD = 4.0 ZAR. How much money in Singapore dollars did Liping get? cDuring these 3 months, the exchange rate had changed from 4.2 to 4.0 ZAR per SGD. Was it in Liping’s favour that the exchange rate now was 4.0 ZAR instead of 4.2 ZAR when she changed her South African rand back to Singapore dollars? Give an explanation to support your answer. dOnce she gets back home, Liping wonders if it would be possible to change money from Singapore dollars to South African rand, spend enough money to live in South Africa for 3 months, and exchange the South African rand that she had left for the same amount of Singapore dollars that she started with. Investigate to see under what conditions, if any, this would be possible. Explain your answer clearly.
Exchange student Answers Liping from Singapore was preparing to go to South Africa as an exchange student for 3 months. She needed to change some Singapore dollars (SGD) into South African rand (ZAR). a3000 SGD = ZAR b3900 ZAR = 975 SGD cIn her favour, as when she converted back she got more SGD than she would have if the exchange rate had stayed the same. dAnswers will vary. For example, if the exchange rate started at 4.2 and she started with 3000 SGD, and she spent the same amount, then the exchange rate would need to drop to 1.3 for her to end up with 3000 SGD when she converts back.
Travel Agency The travel agency’s prices for a flight to Singapore have increased each year since In November 2001, the price was $985. During each of the next three years there was an increase of 2.5% on the previous year’s price. By November 2005, the price had increased by 2.8% on the previous year’s price. By November 2006, the price had increased again – this time by a further 3.4%. By November 2007, the price had increased yet again – this time by a further 5.7%. In December 2007, the travel agency is planning to advertise Singapore flights at “less than November 2001 prices”. There are serious penalties for false advertising claims, so the travel agency want to be sure their advertisement is not false. Calculate the minimum percentage discount they will have to offer on their November 2007 price for their claim to be true..
Travel agency Answer Accumulated increases give ’07 price of $985 = $ ….. Discount needed = $ –$985 = $206.78…at most! As a % of the November 2007 price, this is $206.78…. ÷ $ … 100 = …. 17.4% Recommendation: Minimum discount to satisfy the claim is 17.4%.