Number

Numbervideo 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 %,

% incr/decre, GST• 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 problem– rounding with decimal places and significant figures– standard form, both from and into standard form and on calculator– fractions, decimals and percentages and interest

• find%, find original amount, incr/decr by %, % change, GST– integer and fractional powers applied to numbers– ratio and proportion, direct and inverse– factors, multiples, powers and roots– Rates– Currency conversions– Integers– Order of Operation

Interpreting Word problemsWhat do the following words mean?AltogetherShareLots ofMoreEachBetweenTotalLeftDifferenceSum

Addition, Sum, + or Multiplication, times, xDivision, divided, ÷Multiplication, times, xMultiplication, times, x or Addition, Sum, + Division, divided, ÷Division, divided, ÷Addition, Sum, + or Multiplication, times, xSubtraction, take away, -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)• 9.703 (4 sf)• 27.40 (3sf)• 0.036 (2 sf)• 24000 (2 sf)

Decimal Places

Count all digits after the decimal pointExamples• 3.074 (3 dp)• 0.00273 (5 dp) Ex 16.01 p 196 qn 1-2

RoundingTo 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• http://www.janbrett.com/piggybacks/rounding.htmAlways round money to the nearest cent unless otherwise stated

Examples

• 3.8194 to 2dp 3.82• 4368 to 2sf 4400• 12.895 to 2dp 12.90• 0.546 to 1 sf 0.5

Ex 16.01 p 196 3 -6

Estimation

To estimate, use 1 significant figure and do the calculations in your head.

Example230 x 4.56becomes 200 x 5 = 1000

P196 Ex 16.01 8 - 11

http://www.math.harvard.edu/~knill/mathmovies/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 FormTo 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

6.45 101 n is positive so number >1 64.51.2 102 n is positive so number > 11209 10-4 n is negative is number <1 0.00097.46 10-2 n is negative so number <10.0746

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 10n (n is positive)

• If it was less than 1, write 10-n (n is negative)

Examples

• 598 shift to 5.98 (2 places), >1 so n = 25.98 x102

• 86.5 shift to 8.65 (1 place), >1 so n=18.65 x 101

0.00678 shift to 6.78 (3 places), <1 so n = -36.78 x 10-3

0.05 shift to 5. (2 places), <1 so n = -25 x 10-2 Ex 26.01 p 376 Qn 1 -4,(every 2nd one)

Qn 6 Qn 7-8 (every 2nd one)

Standard Form and the Calculator

• To type in 5.26 x 106

• Type in 5.26 6• This will show up as

This means x10

Ex 26.02 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: +4 + +3 = +7

-4 + +2 = -

2

Examples: +4 + -2 = +2

-8 + -3 = -

11

Example:+7 - +4 = +7 + -4

= +3

Example: -3 - -5 = -3 + + 5

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 = 12 -3 x 6 = -182 x -7 = -14 -24 ÷ 3 = -8-55 ÷ -5 = 11 42 ÷ -7 = -6-3 x 2 x 4 = -6 x 4 4 x 1 x -2 x -3 = 24

P 328 Ex 23.01

Order of Operation

• B rackets• E xponents• D ivision• M ultiplication• A ddition• S ubtraction

Otherwise work from left to right

5 + 3 4 = 5 + 12 = 17

6 (3 + 2) = 6 5 = 30

6(13 – 10) + 5(20 4) = 6 3 + 5 5

= 18 + 25= 43

P 330 Ex 23.02 Qn 1 and 2, every 3rd oneQn 3-9

Powers Powers are used to show repeated multiplication of the same

number.

For powers greater than 2 use the button

Example: 25means 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 = -343 - 34 = -81

Exponent

Base

3

RootsRoots 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: = 5.192

81000

24x250

33

5 243

3 64

49

P 333 Ex 23.03 1-6 every second one7,89-11 every second one

Fractions - simplifyingYou should always simplify fractions For exampleFind 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!

204

51

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. 23 6 = 3 with 5 remainder so

Or use the calculator fraction button

3 3

623

653

653

Multiplying

• Multiply the numerators

• Multiply the denominators

• Remember to simplify the answer!

3

2415

65

43

85

Dividing• Find the reciprocal of the second fraction

(turn it upside down)• Multiply!

Simplify!4

75300

1551225

1512

525

1215

525

You can simplify at this stage if you want

43

1235

1251512

525

5

31

1

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!

32

43

128

129

1217

127

122

129

61

43

Gamma P Ex

Decimals Fractions Percentages

Into a %To convert a decimal and fraction to a percentage

multiply by 100%.Examples: 0.6 = 0.6 x 100% 0.348 = 0.348 x 100%

= 60% = 34.8%

x 100% x 100% = 32.5% = 20%

3

40134013

5151

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

10075

2516

10064

Gamma Ex 24.01 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%5040

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%43

Gamma Ex 24.02 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

= 16.8 30% 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

10024 Gamma Ex 24.01 P 346 Qn 5

Gamma Ex 24.02 P 347 Qn 14,19,20,21,23

Calculating ‘Original’ QuantitiesTo 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 = 300.2 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 = 60.15 x x = 6 x = 40 Gamma P347 Ex 24.02

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%

6015

75-60 = 15

The population of a town decreased from 600 to 540 people. What percentage decrease is this?

Percentage decrease = x 100% = 10%60060

540 – 600 = -60

Percentage Change Worksheet

Increasing and Decreasing by Percentages

First change the % into a decimal (r)

Original (old)Quantity

Increased/decreased(New)Quantity

x

When you increase use (1+r)When you decrease use (1-r)

http://www.nzmaths.co.nz/Numeracy/Animations/flow_diagrams.swf

Examples:1. The price of a computer currently selling for $2500 increases

by 5%. Calculate the new selling price.

5% becomes 0.05 Increase so add 1, use 1.05Old to new so multiply

2500 x 1.05 = $26252. 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 0.1 Decrease so subtract, 1- 0.1 = 0.9New back to old so divide80 ÷ 0.9 = 88.89 (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% = 0.15 decrease so subtract, 1- 0.15 = 0.85Old to new so multiply15000 x 0.85 = $12750

4. House prices have risen 21% over the last 3 years. The market value of a house today is $170000. What was the value of the house 3 years ago?

21% = 0.21 increase so add, use 1.21new back to old so divide

170000 ÷1.21 = $140495.87 (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+ 0.15 = 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 1.15Old to new so multiply

1500 x 1.15 = $1725 2. 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 divide2700 ÷1.15 = $2347.83

Gamma p 359 Ex 24.07So amount of GST = $2700 - $2347.83 = $352.17

RatiosWhen 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 = 1:3 12:28 = 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

541183

x

http://www.nzmaths.co.nz/Numeracy/Animations/multilink_paint.swf

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 = 5 + 7 = 12

Jack’s share =

Jill’s share =

50$120125

70$120127

Remember to simplify fractions!

Gamma P364 Ex 25.01p 368 Ex 25.02

InterestSimple: I = PRTSo 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 periodSo if you invest $1000 for 5 years at 6% compounding annuallyInterest = 1000 x 0.06 = 60 after 1 year

= 1060 x 0.06 = 63.6 after 2 years = 1123.6 x 0.06 =67.42 after 3years = 1191.02x 0.06 = 71.46 after 4 years

=1262.48 x 0.06 = 75.75 after 5 yearsTotal interest = 338.22

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

hrkm

5.2200

http://www.nzmaths.co.nz/Numeracy/Animations/beanies.swf

Proportional reasoningMy 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 km2 tanks = 1680 km3 tanks = 2520kmIf we graph this we get

a straight line!The amount of petrol and the Distanced travelled are directly proportional

0.5 1 1.5 2 2.5 3 3.50

500

1000

1500

2000

2500

3000

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 tableIf 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 peopleIf 3 people take 30 mins, 6 people will take half the time2 people will take a three times as long as 6 people4 people will take half the time 2 people take5 people will take a fifth of the time 1 person takes

Students 1 2 3 4 5 6

Time taken 90 45 30 22.5 18 15

Or draw a graphIf 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 pointsWe 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

0 1 2 3 4 5 6 70

10

20

30

40

50

60

70

80

90

100

Or find an equationIf it takes three students 30 mins to spring clean

the common room, how long would it take 5?

Students x time = 90So time = 90 ÷studentsTime = 90 ÷ 5 = 18 mins

Students 1 3 6

Time taken 90 30 15

Currency conversionsThere 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?

€107.53If 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

Gardener’s Intelligences

Do Now – Standard Form

• Write in standard form3.26 645 0.00343.26 x 10 0 6.45 x 102 3.4 x 10-3

Write in ordinary form4.1 x 10 -3 6 x 106 5.321 x 102

0.0041 6000000 532.1

Make your own!

Do Now• Write in standard form 0.0023

2.3 x 10-3

• Write in ordinary form 3.243 x 104

32430• How many significant figures in 0.03450

0.035• Round to 3sf 0.004556

0.00456• If the diameter of an atom is 3.245 x 10-9, how far across will

1530 atoms be?4.96485 x 10-6

Do Now - Integers

• 3 + -4= -1• 4 x -5 = -20• 3(-2+1) = -3• -12 + 4 = 5• (3+1) x 8= 32

4

13124

Do Now – powers and roots

Find• √4• 3√27• 4.25

• 3.68

• 5√7.9

231306.9123228211.099071.511908185

http://www.math.harvard.edu/~knill/mathmovies/swf/maandpakettleaddition.html

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?

3602. 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 103 litres of fuel to travel 1 km in a

space craft. If the moon is 4.3 x 108 km away, how much fuel will you need?

• 1.29 x 1012

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 105 jelly babies and makes $3.1 x 104 how much was each jelly baby?$1.3 x 10 -1 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 in2.4560 50.00456 31.056 44020 30.00405 3

Do Now - rounding

• 54.564 to 2dp 54.56• 44500 to 2sf 45000• 342.795 to 2dp 342.80• 0.078 to 1 sf 0.08• $56.6723 to nearest cent $56.67• $56.6723 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 - ratesHelen is paid $25 per hour. If she works 5 hours,

how much is she paid?$125She pays tax of 30% on every dollar. How much

does she pay?$41.33How much does she get net (after tax)?$83.67

Multi-step Problems• Decimal and fractions Overseas• Percentages harder GST Salmon• Shopping(%) Muesli bar• Bread• Paper (std form)• Rent and hessian (fractions)• Pocket money(ratios proportion) Apples and sick• Clothing (GST)• Bus (numeric reasoning) Newletter Beach• 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 left0.4 x 4/5 = 8/25 eaten by ElizabethTotal 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?

0.6 + 1/8 = 29/401 – 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 cake20g x 30 cakes = 600g flour needed600 ÷ 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 = 50 000August price: $100 + 13% x $100 = $113August amount: 500 x (100% – 34%) = 500 x 66%= 330August: 330 x $113 = 37 290July – August amounts: 50 000 – 37 290 = 12 710

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 = 329 Deposit = 100 Leaving 229Plus interest = 229 x 1.2 = 274.80TOTAL = 374.80Option B:Cost less trade in = 352 x 1.15 – 50 = 354.80 Deposit = 150 Leaving 204.80Plus interest: 204.8 + .13 X 204.8 = 231.42TOTAL = 231.42 + 150 = 381.43Choose 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 hessianMillie 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

1071

Pocket moneyMum 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 $44.44Gina 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 $118.69 interest after tax, at what annual rate was the money invested?8.75%

ClothingDressage is a clothing shop. All prices include GST. A jacket is priced at a GST-inclusive price of $202.95. What was the price before GST was added?$176.48Sally, 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 $47.20. 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 TarltonThe 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

1700.• 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 = 68 000 Total: 68000 + 1700 x 40 = 108800Children: 3/5 of 108800 = 65280Adult: 2/5 of 108800 = 43520Entry fees:65280 10 + 43520 26 = $1 784 320.GST to claim back = $232737

Bria’s CameraBria 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 = $460Amount taken on HP = $920Interest = $220.80Total to pay = $1140.80Weekly payments = $10.97No – not within budget

Apples and Sick1. 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?

$702. Erin is sick for 1/3 of the school year. If a school

year is 195 days, how many days was Erin sick for65 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

SalmonThe following estimates were made for the salmon run on the Rangitata River in a particular year.• 3 420 adult salmon returned to the spawning grounds.• 38.6% of returning fish were female• A female salmon carries about 4 500 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

2 970 000As a result of this smolt, 5000 salmon return as adults in 3 years time.

Anglers will catch about 30% of this returning run2a. Estimate the number of salmon that will not get caught and will return

to the spawning grounds 3 500 salmon2b. 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 $10 000 to buy 10 000 smolt. Suppose that 10000 smolt were bought and that 0.5% of these smolt return to the river as adults.

3. Estimate the number of salmon from the 10000 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 1.2• 64% 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 4:5How 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 $60 000 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, $10 000 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 2011. 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%.

a You 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.b You 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:a Defence was (6%). If it had kept pace with inflation it would have gone up to 6.6%, but it has reduced to (5.8%). b Defence 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 20 000 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 – ageRecent research has shown that this formula should be modified. The new formula is:Recommended maximum heart rate = 208 – (0.7 age)

a A 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?

b The 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.

c A 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 Answersa At 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 ageStudents may also consider formulas that are less precise but easily calculated and give answers close to the optimum.

Age (years) Old formula New formula

0 220 208

10 210 201

20 200 194

30 190 187

40 180 180

50 170 173

60 160 166

70 150 159

80 140 152

90 130 145

100 120 138

Exchange StudentLiping 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).

a Liping 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?b On 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?c During 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.d Once 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 AnswersLiping 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).a 3000 SGD = 12 600 ZARb 3900 ZAR = 975 SGDc In her favour, as when she converted back she got more SGD than she would have if the exchange rate had stayed the same.d Answers 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 AgencyThe travel agency’s prices for a flight to Singapore have increased each year since 2001. 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 AnswerAccumulated increases give ’07 price of$985 1.0253 1.028 1.034 1.057 = $1191.781….. Discount needed = $1191.781.. –$985 = $206.78…at most! As a % of the November 2007 price, this is $206.78…. ÷ $1191.78… 100 = 17.35051771…. 17.4%

Recommendation:Minimum discount to satisfy the claim is 17.4%.