IAS 1.1 Numeric Reasoning

Year 11 Mathematics
IAS 1.1
Numeric reasoNiNg
NCEA
1
uLake


Explanation of the icons used in this book
Icons are used throughout this book to alert the student as to the function of each section.
Notes
At the beginning of each section an explanation of the concept being studied is included.
Examples
Straightforward examples on the concept being explained.
Solution
One for each example, to be gone through carefully as they contain teaching points.
Achievement Problems
Achievement level problems on the concept being explained.
Merit Problems
Merit level problems on the concept being explained.
Excellence Problems
Excellence level problems on the concept being explained.
Extra Notes
An alternative approach may be covered by the teacher. Space is left so that students can integrate additional material.
Extra Examples
An alternative approach may be covered by the teacher. Space is left so that students can integrate additional examples.
Bright Ideas
Where the authors have found an innovative approach for a particular concept.
Note Well
Points that should be well understood or committed to memory are signposted.
Rounding
Answers are usually rounded to the appropriate degree of accuracy. If the question involves given measurements then the answer cannot be stated more accurately than these. If the calculation involves multiplication (or division) then the least number of significant figures
is used. If the calculation involves addition (or subtraction) then the least number of decimal places is used. Usually the calculation involves both and the authors have selected the degree of accuracy they think is appropriate. The maximum accuracy expected is 4 significant figures.


Year 11 Mathematics
IAS 1.1
Numeric Reasoning
Robert Lakeland & Carl Nugent
Contents
• AchievementStandard .................................................. 2
• PrimeNumbers ....................................................... 3
• FactorsandMultiples ................................................... 4
• RoundingandEstimation ................................................ 7
• StandardForm ......................................................... 12
• OrderofOperation ..................................................... 15
• Integers(+,x,÷,–)....................................................... 17
• Fractions(+,x,÷,–)...................................................... 20
• Percentages ............................................................ 24
• Ratio................................................................... 32
• Proportion ............................................................. 37
• Rates .................................................................. 41
• Powers................................................................. 44
• CompoundingRates..................................................... 49
• PracticeInternalAssessment1 ............................................ 55
• PracticeInternalAssessment2 ............................................ 56
• PracticeInternalAssessment3 ............................................ 57
• Answers ............................................................... 58
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2 IAS 1.1 – Numeric Reasoning
NCEA 1 Internal Achievement Standard 1.1 – Numeric Reasoning
This achievement standard involves applying numeric reasoning in solving problems.
◆ This achievement standard is derived from Level 6 of The New Zealand Curriculum, Learning Media. The following achievement objectives, taken from the Number Strategies and Knowledge thread of the Mathematics and Statistics learning area, are related to this achievement standard:
❖ reason with linear proportions
❖ use prime numbers, common factors and multiples, and powers (including square roots)
❖ understand operations on fractions, decimals, percentages, and integers
❖ use rates and ratios
❖ know commonly used fraction, decimal, and percentage conversions
❖ know and apply standard form, significant figures, rounding, and decimal place value
❖ apply direct and inverse relationships with linear proportion
❖ extend powers to include integers and fractions
❖ apply everyday compounding rates.
◆ Apply numeric reasoning involves:
❖ selecting and using a range of methods in solving problems
❖ demonstrating knowledge of number concepts and terms
❖ communicating solutions which would usually require only one or two steps. Relational thinking involves one or more of:
❖ selecting and carrying out a logical sequence of steps
❖ connecting different concepts and representations
❖ demonstrating understanding of concepts
❖ forming and using a model;
and also relating findings to a context, or communicating thinking using appropriate mathematical statements.
Extended abstract thinking involves one or more of:
❖ devising a strategy to investigate or solve a problem
❖ identifying relevant concepts in context
❖ developing a chain of logical reasoning, or proof
❖ forming a generalisation;
and also using correct mathematical statements, or communicating mathematical insight.
◆ Problems are situations that provide opportunities to apply knowledge or understanding of mathematical concepts and methods. The situation will be set in a real-life or mathematical context.
◆ The phrase ‘a range of methods’ indicates that evidence of the application of at least three different methods is required.
◆ Students need to be familiar with methods related to:
❖ ratio and proportion
❖ factors, multiples, powers and roots
❖ integer and fractional powers applied to numbers
❖ fractions, decimals and percentages
❖ rates
❖ rounding with decimal places and significant figures
❖ standard form.
Achievement
Achievement with Merit
Achievement with Excellence
• Apply numeric reasoning in solving problems.
• Apply numeric reasoning, using relational thinking, in solving problems.
• Apply numeric
reasoning, using extended abstract thinking, in solving problems.
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent


IAS 1.1 – Numeric Reasoning
3
Prime Numbers
Prime Numbers
A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
A prime number has exactly two factors 1 and itself. For example 17 is a prime because it has only two factors 1 and 17.
The smallest prime number as well as the only even prime number is 2, because it is divisible by 1 and 2.
A number n greater than 1 is defined as a prime number if it is only divisible by 1 and n.
Positive numbers other than 1 that are not prime numbers are called composite numbers.
Product of Primes
It is possible to write any positive number greater than one as a product of prime numbers.
The best way to do this is to use a factor tree.
At the top of the tree you start with the number you wish to write as a product of prime numbers.
You then find two numbers that multiply to give the number. Once one of the branches of the tree has a prime number at its branch end you stop simplifying that branch.
You continue working on each branch until only a prime number remains.
If you multiply all the prime numbers at the end of each of the branches you should get the number you started with. A prime factor tree for 120 is drawn below. 80
20 x 4 4x52x2
2x2 So 80 = 2 x 2 x 5 x 2 x 2.
It does not matter what two numbers you find to multiply to give 80 (i.e. 40 x 2 or 8 x 5 or 10 x 8) you will always end up with the same prime factors at the end.
a)
b) c)
a) b)
c)
A factor is a number that divides into another number without remainder.
For example 2 is a factor of 6 because 2 divides into 6 without remainder.
Example
Copy the numbers 5, 19, 32, 37, 39, 52 and circle those that are prime.
List the next two prime numbers after 61. Draw a prime factor tree for 150.
5, 19, 32, 37, 39, 52
The next two prime numbers after 61 are 67 and 71.
Prime factor tree for 150 is as follows. 150
15 x 10 3x52x5
Primefactorsare2x3x5x5.
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4 IAS 1.1 – Numeric Reasoning
Factors and Multiples
Factors
To find the factors of a number means to find all the whole numbers that divide into that number without remainder.
The factors of 12 are 1, 2, 3, 4, 6 and 12 because 12 ÷ 1 = 12, 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3, 12 ÷ 6 = 2 and 12 ÷ 12 = 1
Highest Common Factor (HCF)
The largest common factor of two or more numbers is called the highest common factor (HCF).
The HCF of 8 and 12 is 4, because the factors of 8 are 1, 2, 4, 8 and the factors of 12 are 1, 2, 3, 4, 6, 12.
Since 4 is the largest number common to both it is the HCF.
Multiples
The multiples of a number are found by multiplying the set of natural numbers by that number.
The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, ... because
1 x 3 = 3, 2 x 3 = 6, 3 x 3 = 9
4 x 3 = 12, 5 x 3 = 15 6 x 3 = 18 etc.
Lowest Common Multiple (LCM)
The smallest multiple common to two or more numbers is called the lowest common multiple (LCM).
The LCM of 3 and 4 is 12, because the multiples of 3 are 3, 6, 9, 12, 15, ... and the multiples of 4 are 4, 8, 12, 16, ...
Since 12 is the smallest number common to both it is the LCM.
Example
Answer the following.
a) List all the factors of 18. a)
b) What is the highest common factor of 24 and b) 30.
c) List the first five multiples of 13.
d) What is the LCM of 9 and 12.
e) Terry had regular maths and science tests in class last year. Each maths test comprised 10 questions and each science test 14 questions. If Terry answered the same number of maths and science questions last year, what is
the smallest number of each type he could answer?
The set of (W)hole numbers are simply the numbers
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ...
The set of (N)atural numbers are simply the numbers
1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ...
or the whole numbers excluding 0.
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
The factors of 18 are 1, 2, 3, 6, 9, 18. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Factors are 30 are 1, 2, 3, 5, 6, 10, 15, 30. HCF of 24 and 30 is 6.
c) Multiples of 13 are 13, 26, 39, 52 and 65.
d) Multiples of 9 are 9, 18, 27, 36, 45, ...
Multiples of 12 are 12, 24, 36, 48, ...
LCM of 9 and 12 is 36.
e) Required to find the LCM of 10 and 14, that is 10, 20, 30, 40, 50, 60, 70, ... and 14, 28, 42, 56, 70. Smallest number common to both is 70.


IAS 1.1 – Numeric Reasoning
5
We can also use prime factorisation to find the HCF of two or more numbers.
For example, to find the HCF of 60 and 72 we first write both numbers as a product of primes.
72
6 x 12 2x32x6
60 = 2 x 2 x 3 x 5
The HCF of 60 and 72 are those prime numbers common to both 60 and 72, i.e. 2 x 2 x 3 = 12.
60
6 x 10 2x32x5
2x3 72 = 2 x 2 x 2 x 3 x 3
Achievement – Answer the following questions..
1. List the factors of the following numbers. a) 28 b) 42 c) 19
3. Find the HCF of the following pairs of numbers. a) 21,56 b) 45,117 c) 95,114
2. List the first four multiples of the following numbers.
a) 17 b) 21 c) 62
4. Find the LCM of the following pairs of numbers. a) 12,20 b) 6,11 c) 9,15
5. Which of the following numbers are prime and which are composite. If the number is composite give its prime factorisation.
a) 137 b) 138 c) 140
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent


6
IAS 1.1 – Numeric Reasoning
6. Two lights are flashing. The first one flashes every 8 seconds, and the second one flashes every 14 seconds. If they flash at the same time, how long will it be until they flash at the same time again?
8. The numbers 4 and 8 can be written as the sum of two prime numbers, i.e. 4 = 2 + 2, and
8 = 5 + 3. Investigate how many numbers less than 20 can be written as the sum of two prime numbers.
7. Form a 3 by 3 magic square using the prime numbers 5, 17, 29, 47, 59, 71, 89, 101, and 113. Remember all the rows, columns, and diagonals should total the same.
9. Two lengths of wire are to be cut into fixed length pieces without any wastage. One length is
448 cm long and the other 616 cm in length.
What is the greatest possible fixed length pieces they can both be cut into?
11. Find the HCF of the two numbers 48 and 108 using the method of prime factorisation.
10. Two cars are racing around a track. Car 1 takes 28 seconds to complete a lap while car 2 takes 24 seconds. If they are side by side on the grid at the start how many seconds later will they be side by side again?
12. Three different tours begin at 8.30 am each morning. The duration of each tour is
40 minutes, 48 minutes and 60 minutes. Assuming another tour, of the same duration, starts again when one finishes, how long after the start of the day will all the tours begin again at the same time?
13. At a work canteen the staff have baked
96 chicken legs, 144 chicken thighs and
224 chicken wings. The platters they are preparing have to have the same number of legs, thighs and wings. How many platters can the canteen staff make if they want the greatest number of pieces of chicken on each platter?
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent


IAS 1.1 – Numeric Reasoning 7
Rounding and Estimation
Often when we do calculations, especially when using a calculator, you need to round the answer. The two techniques are rounding to ‘n’ decimal places or to ‘n’ significant figures.
Rounding Decimal Places
• Count the required number of digits after the decimal point.
• Look at the next figure.
• If it is 5 or more, add one to the previous figure.
• If it is less than 5, leave the previous number as it is.
Significant Figures
• Count the required number of digits excluding leading zeros.
• Look at the next figure.
• If it is 5 or more, add one to the previous figure.
• If it is less than 5, leave the previous number as it is.
Example
When a number is rounded we always state how it has been rounded in brackets after the answer.
If we were rounding 2.756 to two decimal places you would write it as
2.76 (2 dp)
If we were rounding to three significant figures you would write it as
2.76 (3 sf)
When rounding, using significant figures, it is often necessary to use zeros as place holders.
For example, if we round each of the following to one significant figure we need to use zeros to place the decimal point in the correct position.
3556 = 4000 (1 sf) 417.3 = 400 (1 sf)
Round the following to the required number of decimal places as indicated.
a) 213.756 (to 1 dp) b)
a) Count 1 digit after the b) decimal point, the next
digit is a 5 therefore add 1 to the previous digit (7) giving
213.756 ≈ 213.8 (1 dp)
Example
978.9998 (to 3 dp) c)
Count 3 digits after the c) decimal point, the next digit is an 8 therefore
add 1 to the previous
digit (9) giving
978.999 8 ≈ 979.000 (3 dp)
0.078 94 (to 4 dp)
Count 4 digits after the decimal point, the next digit is a 4 therefore leave the previous digit (9) unchanged
0.078 94 ≈ 0.0789 (4 dp) as indicated.
Round the following to the required number of significant figures
a) 23.767 (to 3 sf) b) d) 27 983 (to 4 sf) e)
a) Count 3 digits, the b) next digit is a 6,
therefore add 1 to the previous digit (7) giving
23.767 ≈ 23.8 (3 sf)
d) Count 4 digits, the next e)
0.005 64 (to 1 sf) 4156 (to 1 sf)
Count 1 digit, ignore leading zeros, the next digitisa6. Add1tothe previous digit (5) giving
0.005 64 ≈ 0.006 (1 sf) Count 1 digit, the next
c) 1873 (to 2 sf) f) 0.0423 (to 2 sf)
c) Count 2 digits, the next digit is a 7, therefore add 1 to the previous digit (8)
Include 00 as place holders.
1873 ≈ 1900 (2 sf)
f) Count 2 digits, ignore
leading zeros, the next digit is a 3, therefore leave the previous digit unchanged giving
digit is a 3, therefore leave the previous
digit unchanged. Include 0 as a place holder giving
digit is a 1, therefore
leave the previous
digit unchanged. Include 000 as place holders giving
4156 ≈ 4000 (1 sf)
27983≈27980 (4sf)
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
0.0423 ≈ 0.042 (2 sf)


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IAS 1.1 – Numeric Reasoning
14. 789.875 (2 dp) 18. 0.009 75 (2 sf)
22. 26 500 (2 sf) 26. 479.8 (2sf)
15. 0.002 39 (4 dp) 19. 1998 (3 sf)
23. 43.87 (2 sf)
27. 2.7x3.12 (1dp)
16. 49.98 (1 dp) 20. 654.958 9 (4 sf)
24. 0.93 (1 sf) 28. 3.85 (1sf)
17. 67 541 (3 sf) 21. 0.047 98 (4 dp)
25. 25.06 ÷ 4.79 (3 sf) 29. 37x15x13 (3sf)
Achievement – Round the following to the precision indicated in brackets.
0.06
Rounding Sensibly
Often when we use a calculator we get an answer which has a large number of decimal places.
e.g. 15.81 ÷ 1.32 = 11.977 272 727 27....
It is unrealistic to state this as our ‘answer’ to the problem, for initially the two numbers used were only stated to an accuracy of 4 significant figures and 3 significant figures respectively.
As a guide if the question involves given measurements, then the answer cannot be stated more accurately than these.
Rounding after Multiplication
If a calculation involves multiplication (or division) of measured or rounded numbers, then the answer should be rounded to the least number of significant figures of the multiplying (or dividing) numbers.
Rounding after Addition
If a calculation involves addition (or subtraction) of measured or rounded numbers, then the answer should be rounded to the least number of decimal places of the adding (or subtracting) figures.
For example, if an area of 15.81 m2 was being divided by a length of 1.32 m, then the answer should be rounded to 3 significant figures as this is the least accurate contributing measurement.
15.81 ÷ 1.32 = 12.0 (3 sf)
In a practical context it is important to look at the figures or measurements being used. Look at the degree of accuracy of these and then round your answer accordingly.
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
When dealing with problems involving money we always round to two decimal places for the cents unless told otherwise.


IAS 1.1 – Numeric Reasoning 9
30. 3.65+2.3 34. 95.2
36.3
answer appropriately. 31. 4.1x3.5x0.6
35. 17.9 x 2.3 3.4
Units have been deliberately ignored. 32. 6.734–3.51 33.
890 23
0.0056 0.043
Achievement – Complete the calculations involving the following measurements and round your
36. 48 x 15 6.3
37.
Complete the following measurement calculations, rounding your answer appropriately and stating the units.
38. Calculate the total length of a building which is made up of the following measurements: 7.2 m, 15.14 m and 0.47 m.
40. The amount of timber left in a 12.2 m plank which has had two lengths of 3.42 m cut off it.
42. TheareaofacircleisgivenbyA=πr2. Ifa circle has radius (r) 3.7 cm, give its area.
44. James has just started his summer holidays and his mother wants him to paint the
floor of the garage. The length and width
of the rectangular garage floor are 5.6 m and 3.7 m respectively.
39. The volume of a cuboid is given by V=lxwxh. The measurements of a cuboid are l = 5.21 cm, w = 3.7 cm and h = 2.43 cm. Find its volume.
41. A 45 litre container of petrol is divided into
8 equal amounts, how much is each amount?
43. If the exchange rate is $1 NZ = $0.77641 USD, calculate how much $455.30 NZ is in USD.
d) James’ family are planning to visit Australia on holiday. James will convert his savings of NZ $200 into Australian dollars at the airport. The current exchange rate is NZ $1 = Aust $0.854 76. How much will he get in Australian dollars?
e) James’ American cousin is coming to New Zealand for Christmas. She will need to buy NZ dollars with her $2000 US dollars. If the exchange rate is $1 NZ = 0.5132 US, how much will she get in NZ dollars to the nearest dollar?
f) James needs to calculate the volume of water in the swimming pool so he can add some chlorine. His brother calculated the volume of water as 52.789 59 m3. The pool is a cuboid shape with length 10.54 m, width 3.71 m and depth 1.35 m. Is his brother’s answer correct? Comment.
g) If the pool in f) above was filled at the rate of 50 litres per minute, how long would the pool take to fill, to the nearest hour?
a) Find the area of the garage.
b) Paint covers 13 m2 per litre and can be purchased in multiples of 1 litre. How many one litre tins are required for 2 coats of paint?
c) James has to replace a plaster cast ornament he has broken. It is in
the shape of a cone. The measurements of the cone are: diameter 5.08 cm and height8.1cm.
Find its volume.
8.1 cm
5.08 cm
V= 1πr2h 3
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent


10 IAS 1.1 – Numeric Reasoning
Estimation
When we do any calculation, it is often a good idea to estimate the answer in order to ensure that the calculated answer is similar and therefore likely to be correct.
When we estimate our answer we usually round each number to one or at most two significant figures. We then do the calculation and use this as the estimate.
45. 16.86 x 5.9 49. 137x19
14.7
46. 7.345 x 11.45 50. 16.5x27.3
9.5
52.
Example
Estimate the answers to the following. a) 29.1x6.5 b)
380x 27 c) 19
1.7x3.4x23.8
Round to 1 sig. figure.
1.7x3.4x23.8≈2x3x20 = 120
48. 58.7 x 0.05 x 3.2
4657 23.7
a) Round to 1 sig. figure. b) so 29.1x6.5≈30x7 so
= 210
Achievement – Estimate the following.
53. Simon earns $14.90 per hour. If he works for 9.5 hours a day, estimate his daily wage.
55. Estimate the cost of purchasing a box of 54 pens if each pen costs $1.85.
57. A person earns $980 per month, estimate their annual salary.
59. A tube of acrylic paint costs $14.95. An artist requires 21 tubes. Estimate the cost.
54. Nick’s monthly repayments on his car are $509. If he still has 19 months to go before his car is paid off, estimate the amount still owing.
56. Estimate the cost of a single CD if a pack of 11 cost $29.95.
58. A carton measures 28 cm by 9 cm by 4.6 cm. Estimate the volume of the carton.
60. Jan walks the same route 5 days a week for every week of the year. The distance she walks is 4.6 km. Estimate how far she walks in a year.
Round to 1 sig. figure. c) 380x 27≈ 400x 30 so
19
20
= 600
47. 4415.2 8.21
51. 23x47x8 16.2 x 3.8
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IAS 1.1 – Numeric Reasoning 11
61. It takes Susan 18 minutes to cycle to school and the same time home. She attends school
198 days each year, estimate how long she spends travelling to and from school.
63. Maree pays $73.87 per month for Sky television and $127.50 per month to rent a plasma television. Estimate how much she spends per year on TV watching.
64. Liz is planning to purchase a car. She has seen a suitable vehicle costing $19 995 and intends to buy the vehicle on hire purchase and pay the car off over 18 months. The finance package she has agreed on is:
20% deposit with the balance to be paid over 18 months. Interest at a rate of 9.5% will be charged on the outstanding amount less the deposit, when she initially purchases the vehicle.
Other costs she has to take into account are registration which is $187.50 per year and insurance of $412.50 per year.
She thinks she will use at most 20 litres of fuel per week which is currently $1.77 per litre.
As well the car will need to be serviced every 6 months and her garage tells her to budget $235 per service.
Liz needs to estimate what her per monthly costs will be, over next the 18 months, if she goes ahead and purchases the vehicle.
Show all your estimated calculations and clearly explain what you are doing at each step.
62. Howard needs to fill his boat with petrol. If the cost per litre of 95 octane fuel is $1.81 and the capacity of the boat’s fuel and reserve tank is 93 litres, estimate the cost to fill the tanks.
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12
IAS 1.1 – Numeric Reasoning
Standard Form
Standard Form
We use standard form as a concise way of writing very large and very small numbers. A number in standard form is written as a number between 1 and 10, multiplied by a power of 10.
Consider 3 127 686.2
You first move the decimal point so the number has a value between 1 and 10, i.e. 3.127 686 2
You now multiply by 1 000 000 or 106 to make the number equal 3 127 686.2
In standard form 3 127 686.2 = 3.127 686 2 x 106
With very small numbers we work similarly.
Consider 0.0045
Move the decimal point, so the number has a value between 1 and 10, i.e. 4.5.
You now need to divide this number by 1000 or multiply by 0.001 = 10–3 to make 4.5 equal 0.0045.
In standard form 0.0045 = 4.5 x 10–3
The decimal point needs to move 3 places left from
4.5 so the power is –3.
0 . 0 0 4 5 = 4.5 x 10–3
This number must always be between 1 and 10.
Example
Write the following in standard form. a) 1 320 000 b)
To enter a number in standard form on a graphics calculator we use the EXP button (Casio 9750GII) or the EE button (TI-84 Plus),
e.g. for 3.127 686 2 x 106 we enter
3.1276862 EXP 6 Casio 9750GII EE
3.1276862 2nd , 6 TI-84 Plus
To write any number in standard form first move the decimal point, so that the number has a value between 1 and 10
then multiply by an appropriate power of 10, so the number has the same value (found by counting the number of decimal places the decimal point has
moved). To convert a number into standard form on your calculator use the scientific
mode. Each calculator is a little
different.
On the TI-84 Plus press MODE then choose SCI and then the number of decimal places you want to display.
On the Casio 9750GII press SHIFT MENU then scroll down and select Display and choose SCI and then the number of decimal places you want to display.
A number displayed as 8.89E+05 means 8.89 x 105.
A number displayed as 8.89E–05 means 8.89 x 10–5.
0.000 56 a) 1.32x106 b) 5.6x10–4
c) 2.1
c) 2.1x100
c) 3.91 x 105 c) 391 000
as100 =1
Example
Write the following as an ordinary number.
a) 2.4x10–3 b) 8.647x102
a) 0.0024 b) 864.7
Merit – Write the following in standard form, calculating the answer first if required. 65. 41 500 66. 591 67. 12.75 68. 0.045
69. 0.592 70. 7 71. 12 700 000 72. 0.000 0095 6
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IAS 1.1 – Numeric Reasoning
13
75. 0.5436 ÷ 12 000
79. 4.5x101 ÷9.0x10–2
85. 4.85x10–4 86. 7.89x107 87. 3.25x10–1
89. 9.876x102 90. 1.423x10–1 91. 1.64x102 x 1.1x10–1
76. 4.26 ÷ 0.003 55
80. 1.8x100 – 2.6x10–2
84. 4.0x10–2 – 3.5x10–3
88. 1.98 x 10 0
92. 5.28x101 ÷ 4.0x103
73. 9320
77. 3.7 x 2.4
0.2
81. 4.73x106 +2.95x105
Write the following as ordinary numbers, calculating the answer first if required.
74. 395 x 4
78. 2.3x102 x 3.1x10–3
82. 5.7x102 x67000
83. 0.0025÷5.0x10–1
Merit – Calculate the answer to each of the following problems, rounding your answer appropriately.
93. The distance from the earth to the sun averages 1.496 x 1011 m. Light travels at about 3.00 x 108 m/s. How many seconds does it take light to travel from the sun to the earth?
95. There are 31 557 600 seconds in a year and light travels at about 3.00 x 108 m/s. How far does a beam of light travel in a year (a light year) in kilometres (1000 m = 1 km)?
97. Einstein’s theory of relativity relates the mass lost in a nuclear reaction to the energy given off. The formula is E = mc2 where E is energy in joules, m is the mass in kg and c is the speed of light (3.00 x 108 m/s). Calculate the energy released when 2.1 kg of matter is converted to energy.
94. The energy from the sun comes from the conversion of 3.9 x 106 tonnes of hydrogen
into helium every second. The sun is estimated to have 3.8 x 1024 tonnes of hydrogen. There
are 31 557 600 seconds in a year. How many years will the hydrogen last for?
96. The total water in the world consists of about 1.57 x 1020 kg of hydrogen and 1.24 x 1021 kg of oxygen (12.4 x 1020 kg). What is the total mass of water on earth? Remember to round appropriately.
98. There are 8766 hours in a year. If the world’s first interstellar rocket travelled at 85 500 km/h, how long will this rocket take to travel to the nearest star (Proxima Centauri) which is
3.97 x 1013 km from earth?
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent


14
IAS 1.1 – Numeric Reasoning
99. A blade of grass grows on average at a rate of 2.0 x 10–8 metres per
second. How much
will a blade of grass
grow in one week in millimetres?
100. A country’s national debt is $1.5 x 1012. If the population of the country
is 285 million, how much is
owed per individual?
101. Use standard form to calculate the following information about 75 year old Martin. Round all your answers to 3 significant figures.
One estimate of the number of cells in the average human body is seventy-three point eight million
million.
a) Write this as an ordinary number.
b) Write this number in standard form.
c) The population of the Earth is approximately 6 150 000 000 people.
Calculate the total number of human cells for the entire population of Earth.
d) The average human weighs 65.5 kg. What is the average mass of one cell in grams?
(1000 g = 1 kg).
e) The mass of a single hydrogen atom is
1.66 x 10–24 g. Calculate how many times heavier a single human cell is than a hydrogen atom.
A person’s heart beats, on average, 85 beats per minute for every minute of their life.
f) How many minutes has Martin lived for? Ignore leap years and give your answer in standard form.
g) How many times has Martin’s heart beaten?
h) Each beat of the heart pumps about 67 mL.
How many litres has the heart pumped in Martin’s life?
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent


IAS 1.1 – Numeric Reasoning
15
Order of Operation Order of Operations
When we are required to solve a problem that involves more than one operation, i.e. +, x, ÷, –, it is necessary to do the problem in a pre-defined order. We can use the mnemonic BEDMAS to remember the order.
B Brackets. If you are not using a calculator and there are more than one set of brackets in the expression, do the innermost brackets first.
E Exponents
D/M Division and Multiplication are done
from left to right.
A/S Addition and Subtraction are done from left to right.
We can also use a calculator to do such problems. It is important to press the ‘=‘ button only at the end of the calculation. The calculator has the pre-defined order built in as part of its logic.
Example
Calculate the following. a) 3 + 2 x 8 ÷ 4
a) Question =3+2x8÷4 1stMultiply =3+16÷4
2ndDivide =3+4
As Multiplication and Division are done from left to right.
3rd Add = 7
3+2x8÷
Casio 9750GII
Example
Calculate the following. 5+22 x3+(12–9)2 –3
Quest. =5+22 x3+(12–9)2 –3 1stBracket =5+22 x3+32 –3
2nd Exp = 5 + 4 x 3 + 9 – 3
3rd Mult. = 5 + 12 + 9 – 3
4th Add/Subt. = 23
Most scientific calculators will do the problem correctly if you key it in as it is written down. Even an expression such as 3(4 + 6) is ‘understood’ by the calculator to be 3 x (4 + 6).
The only exception is –32 which a calculator will evaluate as –9. It has to be entered as (–3)2 = 9.
(4 + 2)
(5 – 3)
4 EXE
b) Bracketslefttorightfirst
1stBrackets =9+4x5+18÷3
2ndMultiply =9+20+18÷3 3rd Divide = 9 + 20 + 6
4th Add = 35
9+(6–2 )x5+(1 3+5)÷3
ENTER TI-84 Plus
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
b)
When you’re checking your answers
in a test or examination always do the calculation a second time to check your answer and avoid the possibility of ‘key in’ error.
9 + (6 – 2) x 5 + (13 + 5) ÷ 3
Problems such as 4 + 2 have implied
brackets. 5 – 3
You must treat the problem as
.
x 3 +(12–9
) x2 – 3 EXE Casio 9750GII
5 + 2 x2


16
IAS 1.1 – Numeric Reasoning
Achievement – Evaluate the following using your calculator.
102. 5+8x 3–4 106. 9–8÷2+12
110. (13–6)2 +(19–12)2 114. ((12–5)+2)x3+23
118. ⎛⎛3–2⎞+1⎞x4 ⎜⎝⎜⎝4 5⎟⎠ 2⎟⎠ 5
122. 3 x 16+ 2 x 48 4 3
126. 4+5–3x 2 24–8x2÷4
103. 32 +16÷2+12÷3 107. 42 +24 ÷2+12x 3
104. 42÷6+12–(8+7) 105. (9+3)x(6–2)+42
108. 33 x 23 +24 –82 112. ((9+5)–8)2 x 23
109. (15–2)x(16–5)+52 113. ((((15–7)+8)x 3)÷2)
117. 3x2+⎛1–2⎞ 5 3 ⎜⎝2 5⎟⎠
121. 2–1x2+3 5 4 3 8
111. (42 +22)x (3+12x 2)
115. (((3+6)x4)–2)x5+7 116. 2+1x3
3 2 4
119. ⎛ 7 –1⎞÷⎛2+1⎞ ⎜⎝10 4⎟⎠ ⎜⎝3 2⎟⎠
120. 3+2÷1x3 4 3 5 8
124. 48–(2+4)2 52 +5
128. 2(6+4) 5(8–2)
123.
127.
12+3 x 5 5–2x3
3+22 x 8 42 +3x 23 +5
125.
129.
(7–5) x 12+32 3+42 –5
(4–7) x 3+8 2(7–3)2
130. Find and circle the first mistake in the working below and rewrite the working with the error corrected.
Quest. =71– 36 x 8+7x (4–22)–3 = 71 – 6 x 8 + 7 x (4 – 4) – 3
=71–6x 15x 0–3 =71–90x 0–3 =74
131. Find and circle the first mistake in the working below and rewrite the working with the error corrected.
Quest.=32 + 9 –42 +5x (5–3)2 –2 = 9 + 3 – 16 + 5 x 4 – 2
= 9 + 3 – 16 + 5 x 2 = 9 + 3 – 16 + 10 = 12 – 16 + 10
= –4 + 10
=6
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent


IAS 1.1 – Numeric Reasoning
17
Integers (+, x, ÷, –) Integers (+, x, ÷, –)
Adding and Subtracting
To add and subtract integers we use the first number as our starting point. If the operation following this number is a minus we move to the left on the number line and if it is a positive we move to the right on the number line. The number of steps we move left or right is dependent on the number following the operation.
Consider the integer problem –4 + 6
• the –4 is our starting point.
• the + indicates that we move to the right.
• the 6 tells us the number of steps to move.
We can use a calculator to calculate all integer problems. Using the (–) button on the calculator enables us to
enter a negative number.
Using the calculator to evaluate –3 x 5 we would
Starting point
Move to the right 6 steps Answer (2)
–5–4–3–2–10 1 2 3 4 5
If the problem has two signs in between the numbers, then we need to replace the signs by a single sign using the rules for the ‘Gap’ below, then proceed as before.
Rules for the ‘Gap’
+ +actsas+ + –actsas– – +actsas– – –actsas+
To add or subtract integers we can use a number line to help us.
Consider –3 + –2 = –5
You begin by rewriting the problem as –3 – 2 since by our rules the inner signs + – can be changed to a –.
So for –3 – 2
• the–3isourstartingpoint.
• the – indicates that we move to the left.
• the2tellsusthenumberofstepstomove.
Move to the left 2 steps Answer (–5) Starting point
–5–4–3–2–10 1 2 3 4 5
There is an easy way to remember these rules. If the two signs are the same the answer is positive. If the two signs are different the answer is negative.
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
enter:
(–) 3 x 5 EXE For –20 ÷ –4 we would enter:
Casio 9750GII
(–)
20 ÷
(–) 4
ENTER
TI-84 Plus
Multiplying and Dividing
To multiply or divide integers we use the rules
detailed below
positive the answer is negative (= –15). Consider –20 ÷ –4.
Since we are dividing a negative by a negative the answer is positive (= 5).
Consider
Rules for x and ÷
÷ x + –
+ – + – – +
–3 x 5.
Since we are multiplying a negative by a


18
IAS 1.1 – Numeric Reasoning
Example
Calculate the following. a) –6 + 12
a) start at –6 move right 12
–6+12= 6
b) –13 – –24
b) change inner signs to + (– – acts as +)
–13––24= –13+24 = 11
c) c)
27 ÷ –3
positive ÷ negative is negative
27÷ –3= –9
135. 85––98
132.
136. 140.
144.
148.
152. 156.
160. 164.
–27+56
13 x –12 –12+–34––56
(– 4)3
– 6 x – 27 18
–25 – –18 + –20
–35+– 67 3
133. –94–35 137. –15 x –16
141. –18x12x–3
134. –28+–37 138. –288 ÷ 18
139. 143.
147.
151.
155.
159.
–216 ÷ – 8
23–56 –47+36
16 x – 5 2x–2x–4
– 47 – – 47
–323 ÷ 19 + –37
–98 – –89 – –99
123 +5 712
Achievement – Calculate the following.
142. 240÷–3÷–10
145. (–14+27)x–3 146.
– 155 5
149. 5+2x (–6+3) 153. – 38 x –12 ÷ – 4
157. (–15 + –8)2
161. (–273 – –155) x –43
150. (4 + – 7) x (11 – – 7) 154. 64 – –78 – 93 + –15
158. 2403 ÷ (–3)3 162. (–2)3 – 34 + 52
650÷–13+–9x –4 –2970 ÷ 15 + –97
163. 166. (–25+78)–(–12––5) 167.
5
165. 47x –32+–57––19
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
(– 3)
2 (9)


IAS 1.1 – Numeric Reasoning 19
Achievement/Merit – Answer the following questions.
168. A customer has a balance of $6700 Dr (–$6700) on their personal loan. They make a loan repayment of $1950. What is the new balance of their loan?
170. In 2013 a company makes a net loss of
–$18.2 million and in 2014 a net profit of $14.3 million. How much more did the company make in 2014 compared to 2013?
172. A submarine hovers at a depth of 280 m below sea level. If it descends a further 145 metres and then ascends 390 metres what is it new depth?
174. The air temperature in the atmosphere decreases at a rate of 9˚ C for every
300 metres. What elevation would a plane have to fly at to achieve a temperature of
–58.5 ˚ C assuming the temperature at sea level is 0 ˚C.
176. The table below shows some dates in history. Answer the following questions.
169. A customer has a balance of $345 000 Dr
(–$345 000) on their mortgage. They make fortnightly repayments of $842. What would be the balance of their mortgage after one year, excluding any interest charges?
171. A plane is at an altitude of 6540 m. In preparation for landing it loses altitude at a rate of 350 m per minute. What is its altitude after 15 minutes?
173. A Roman emperor was born in 58 BC and died in 23 AD. How many years did he live for?
175. The chiller at a storage facility is set to –15 ˚C. If the temperature outside is 27 ˚C what is the difference in temperature between the chiller and outside temperature?
a)
b)
c)
How many years elapsed between the wheel being used in Mesopotamia and iron used for weapons?
How many years before the black plague in Europe was Julius Caesar assassinated?
How many years elapsed between
the earliest and latest events in the table?
Event
Approx. year
Wheel used in Mesopotamia
3500 BC
Iron used for weapons
1130 BC
Pythagoras starts his school
532 BC
Julius Caesar assassinated
44 BC
Doomsday Book written
1086
Black plague in Europe
1348
Pilgrims arrive in Plymouth
1620
Germany declares war on France
1914
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent


20 IAS 1.1 – Numeric Reasoning
Fractions (+, x, ÷, –)
All fraction calculations can be done on a calculator but we have explained the steps involved, as the process is used in manipulating algebraic fractions. We leave it to the student to decide whether to set out the problems in full or to use their calculator.
Fractions (+, x, ÷, –) Adding and Subtracting
To add or subtract fractions together they must have the same bottom line or denominator.
Consider 34 + 52
We begin by rewriting the fractions with the same
The TI-84 Plus does not have a fraction button like the Casio 9750GII. Instead to
enter the problem 3 + 2 we press: 4 5
denominator (20 in this case, since 4 x 5 = 20)
so =15+8 3÷4+2÷5
20 20 = 23
Frac
MATH 1 ENTER
The answer is 23 . 20
When using a calculator always do the calculation twice to check your answer and avoid the possibility of ‘key in’ error.
The TI-84 Plus does not have a fraction button like the Casio 9750GII. Instead to
20
= 1 3 20
Alternatively we can use a calculator to add or subtract fractions.
The ab/c button on the calculator is the fraction button. To calculate the problem above we would
enter
3 ab/c 4 + 2 ab/c 5 EXE
Multiplying
To multiply fractions together we multiply the two numerators and the two denominators together.
34 x 52 3x2
4x5 6
20 3
Consider
so =
=
=
enter the problem 3 x 2 we press: 10 45
Alternatively we can use our calculator to multiply fractions.
3÷4x2÷5 To evaluate using the calculator we enter Frac
3 ab/c 4 x 2 ab/c 5 EXE
MATH 1 ENTER
The answer is 3 . 10
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent


IAS 1.1 – Numeric Reasoning 21
The reciprocal of ab is ba . When dividing fractions only the second fraction is
inverted.
The TI-84 Plus does not have a fraction button like the Casio 9750GII. Instead to
enter the problem 34 ÷ 52 we press:
Frac
The answer is 15 . 8
Fractions (+, x, ÷, –) cont... Dividing
To divide fractions we multiply by the reciprocal.
34 ÷ 52 = 34x52
Consider so
= 178
Alternatively we can use our calculator to divide
fractions.
To evaluate using the calculator we enter
Mixed Numbers
A mixed numeral or number is one that includes an integer and a fraction, i.e. 1 78 , – 2 34 etc.
The calculator automatically displays fractions as mixed numerals. To key a mixed number into a calculator we repeatedly use the fraction key.
Improper Fractions
An improper (or top heavy) fraction is one where the numerator is greater than the denominator (ignoring the signs)
i.e. 15, –11 84
A calculator may automatically simplify an improper fraction to a mixed number.
The calculator can display a mixed numeral as an improper fraction by pressing:
For example if we have 1 78 on the display then pressing:
3
EXE
a b/c
=
15 8
4
÷
2
a b/c
5
1
a b/c
7
a b/c
8
EXE
1
5
a b/c
8
EXE
SHIFT
FD
SHIFT
FD
toggles it back to an improper fraction.
(
3
÷
4
)
÷
(
2
÷
5
)
MATH
1
ENTER
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent


22
IAS 1.1 – Numeric Reasoning
177. 49–13 181. 1 12 + 2 34
185. 1+3+2 275
178.
182.
186.
73 + 58 179. 4 18 – 3 35 183.
2x4x1 187. 578
56 x 34 213x132
15–1–2 1959
⎛3⎞2 ⎜⎝ 4 ⎟⎠
180. 74÷15 184. 3 56 ÷ 1 14
188. 4÷3÷1 572
45 192. 3
7
196. 4 of 812 7
Example
Calculate the following. a) 1 73 + 32
3 2 10 2 a) 17+3 = 7 +3
= 30 + 14 21 21
= 44 21
= 2 2 21
Calculate the following. a) 34 x 72
a) 3x2 = 3x2 474x7
= 6 28
b) b)
2 49 – 15
4 1 22 1
Operation
Mixedtoimproper Common denominator Simplify Mixednumeral
ConverttoMult.
Achievement – Calculate the following giving your answer as a simplified fraction.
Example
=3
14 5
151 311
189. 32–16–18 190. 48÷22÷13 191.
193. 4 194. 2 of 4 plus 1 195. ⎛33⎞3 9 592⎜⎝5⎟⎠
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
b)
b)
45 ÷ 92 4÷2
29–5 = 9 –5 = 110 – 9
= 4x9 595x2
45 45 = 101
= 211 45
= 36 10
45
Mult. top and bottom =33 Simplify


IAS 1.1 – Numeric Reasoning 23
Achievement / Merit – Solve the following number problems, involving fractions.
197.
199.
201.
203.
205.
207.
209.
Sam swims 80 lengths of the pool at training. 198. What fraction of his training has he completed
after swimming 35 laps?
35 of a pie is left over from tea. If Mike’s 200. Mum serves 14 of the remaining pie to him for supper, how much of the original pie has he
eaten for supper?
32 of a farmer’s herd comprises Jersey cows 202. and 4 of the remaining herd are Ayrshires.
Diane spends 75 minutes at the gym each day. When she has completed 4 of her time, how
many minutes has she left?
Jake spends 52 of his weekly wage on rent and 14 on food and entertainment. What fraction of
his wage is left?
A company has 420 employees, 112 of
5
which are administrative. What fraction 5 of the workforce in the company is NOT
What fraction of the herd is Ayrshires?
45 of a box of apples are shared evenly among 3 people. What fraction does each person get?
administrative?
204. How many pieces of tape 4 32 cm in length, can be cut from a roll 70 cm in length?
A worker quality checked 13 of a consignment 206.
of products before lunch and 72 after lunch.
This leaves him 720 more to do. How many were there in the original consignment?
What has to be added to the product of 208. 174 and 234 toget 632?
A traveller completes 72 of his journey one day 210. and 35 the next day. If he has travelled 806 km,
how much further does he have to go?
The product of two fractions is 24. If one of the fractions is 3 35 , what is the other?
A girl spends 52 of her savings and has $75 left. How much did she have initially?
How much greater than 2 12 is the difference between392and745?
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent


24 IAS 1.1 – Numeric Reasoning
211. Keith is a used car salesman and at the d) beginning of a month has 120 vehicles in stock.
a) b)
c)
5
Keith sells 24 of his stock in the month. How
many vehicles does he have left at the end of the month?
1 of the vehicles are commercial vehicles, such 3 5
as vans and utes, and 8 of these cost less than $10 000. How many of Keith’s commercial
vehicles cost less than $10 000?
23 of the vehicles are cars and of these 34 have an
engine capacity less than 2000 cc. How many cars does Keith have with an engine capacity greater than 2000 cc?
f) Keith sells 5 of his stock in a month and 2 of 24 5
Percentages
Percentages to Fractions
To convert a percentage to a fraction we write the percentage as a fraction out of 100 and then simplify the fraction if possible.
Consider 35%
7 20
Decimal to a Percentage
To convert a decimal to a percentage we multiply by 100%.
Consider 0.05
The word percentage is made up of the prefix ‘per’ meaning out of and ‘centage’ from the same root as ‘century’ meaning
100. Percentage therefore means ‘out of 100’.
By representing figures as a percentage we can easily make a comparison between two or more sets of figures.
To simplify a fraction on the
Casio 9750GII enter it into your calculator as a fraction and then press EXE. The calculator will automatically reduce
it down to its simplest form. If the simplified fraction is a mixed numeral press SHIFT F D
to convert it to an improper fraction. On the
TI-84 Plus to simplify a fraction like 35 we enter
3 5 ÷ 1 0 0 MATH
Frac
1
As a fraction Simplify =
Mult. by 100%
= 0.05 x 100% = 5%
ENTER
To convert a decimal to a percentage move the decimal point two places
to the right. On a Casio 9750GII to convert a fraction to a percentage on the
= 35 100
Fraction to a Percentage
To convert a fraction to a percentage we multiply
by 100%. Consider
Mult. by 100% Simplify
calculator enter
3 ab/c 5 x 1 0 0 EXE
On the TI-84 Plus to convert a fraction to a
ENTER
Keith sells one of his cars for $27 000. The purchaser puts down a deposit of 15 and pays
the balance off over 24 months at 0% interest. How much does the purchaser pay per month?
e) 45 of the customers who bought vehicles one
month paid cash for them. If 20 customers paid cash, how many vehicles in total did Keith sell during the month?
the remaining stock the following month. What fraction of the original stock of vehicles remain after the two months and how many cars does Keith have left?
3 5
= 35 x 100%
300%
5 3÷5x100
=
= 60%
percentage enter
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
100


IAS 1.1 – Numeric Reasoning
25
Percentage of an Amount
To find a percentage of an amount, write the expression as a mathematical statement and solve.
Consider
Write as statement
Simplify
a)
100
= $4.50
Convert the following to or
Example
Example
What percentage is 15 out of 40?
from percentages. 16% as a fraction.
b)
b)
3
8 as a percentage.
Multiply by 100% 3 = 3 x100%
a)
b) Find 16% of $25.
16% = 16 100
a)
Simplify = 4
8 8
15
a) 15÷40 = 40 x100% b) 16%x25=0.16x25
Find 15% of $30 = 15 x 30
100 = 450
=37.5% Achievement – Write the following percentages as simplified fractions.
=$4.00
215. 75% 219. 12.5%
212. 38% 213. 155% 216. 6% 217. 0.5% Write the following as percentages.
214. 17.5% 218. 6.8%
220. 0.72 224. 1.8
221. 0.435 225. 0.025
222. 45 223. 226. 38 227.
230. 18.5% of $1750 231. 234. 1.2% of 395 235.
15 12
1 20
25
= 37.5%
Calculate the following percentages.
228. 15% of $120 229. 36% of 375 kg
232. 12.5% of $180 233. 0.5% of 11 000
9% of 85 litres
115% of 240
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent


26
IAS 1.1 – Numeric Reasoning
Answer the following.
236. Shani pays 20% of her weekly wage of $132 in tax. How much tax does she pay?
238. 56% of the students in Nathan’s school are girls. If the school roll is 1350, how many students are girls?
240. Melinda scored 95 out of 135 in her maths test. What is this as a percentage?
242. David plans to buy a new MP3 player costing $250. If he receives a 15% discount how much will he save?
244. Callum scored 87 runs from 115 times at bat during a softball season. Calculate his percentage strike rate for the season.
246. A home handyman wishes to line part of his lounge with wood panelling. The panelling
is $4.44 per metre including GST (15%) and the rimu beading to go on top of the panelling after it is fitted is $3.75 per metre including GST.
a) If the handyman requires 189 metres of panelling and 20 metres of beading for the job calculate the total cost of the timber required.
The handyman can choose one of two payment options – cash or fixed term.
b) If the handyman chooses cash he receives a 7.5% discount on both the beading and panelling. How much will he save?
237. Jim plans to reduce his 86 kg weight by 15%. How much weight is he planning to lose?
239. Byron is eligible for a 5% discount on a new computer listed for $2100. How much discount will he receive?
241. Karl received a $190 cash rebate on a purchase of $3800. What percentage is this?
243. Samantha plans to buy a new blouse costing $136. The store is offering a 12.5% discount on all stock. How much will Samantha save if she buys the blouse during the sale?
245. 52% of the students in Gwyneth’s school are girls. If the school roll is 1250, how many of the students are boys?
c) If the handyman chooses fixed term he has
12 months to pay off the total amount. However he will pay 9.5% interest on the total amount owing. How much interest will the handyman pay if he chooses this option?
d) How much to the nearest dollar will the handyman pay per month if he chooses the fixed term option?
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent


IAS 1.1 – Numeric Reasoning
27
Percentage Change
Increase or Decrease by a Percentage
To increase or decrease by a percentage we could calculate the percentage of the amount. If we are increasing, we add it to the original amount. If we are decreasing, we subtract it from the original amount.
To calculate the result of a percentage increase or decrease, on a scientific calculator with a percentage key, we could use the expression
Start x (100 ± Change)% = Result
For example, to increase $20 by 12% we calculate
20 x (100 + 12)% = Result On the calculator we key:
20x(100 +12)%
which gives the answer $22.40.
For example, to decrease $20 by 12% we calculate 20 x (100 – 12)% = Result
On the calculator we key: 20x(100
–12)% which gives the answer $17.60.
When dealing with problems that involve money we always round to two decimal places for the cents. We do NOT round the cents to the nearest
5¢. Similarly percentages are usually rounded to one decimal place.
To go backwards and find the original amount, using a scientific calculator with a percentage key, we first substitute in the formula
Start x (100 ± Change)% = Result
For example, to find the pre-GST price of an article that is now $87 (where GST is 15%) we substitute
Consider:
% of amount Adding =
=
Increase $20 by 12%
=
$2.40
$20 + $2.40 $22.40
Alternatively we could use the formula in the ‘Bright Idea’. The advantage of this is that you are applying the same formula for both a percentage increase and a percentage decrease.
Percentage Increase or Decrease
(Profit or Loss)
To find the percentage increase or decrease (profit or loss) we use the formula
% increase (decrease) = increase (decrease) x 100% original amount
A painting is purchased for $1200 and sold one year later for $1000. What is the percentage loss?
% decrease
= 200 x 100 1200
= 16.7%(1dp)
Finding the Original Amount
An amount sometimes includes a tax or percentage mark-up. Often we wish to be able to find the pre- tax or pre-mark-up price of an article or service. To do this it is not sufficient to calculate the percentage of the amount and subtract it. We must divide the amount by 100% + tax (mark-up)%.
= $75.65
Start x (100 + 15.0)% Start x 115%
Consider:
Original price = 1.15
= 87
= 87
=87 115%
= $75.65
87÷115%
A retailer is selling an article for $87 which includes GST of 15%. What is the pre-GST price of the article?
Start On the calculator we key:
87
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28
IAS 1.1 – Numeric Reasoning
a)
An article is bought for $58 and sold for $70. What is the percentage increase?
a) Start x (100 ± Change)% = Result 58 x (100 ± Change)% = 70
(100 ± Change)% = 1.2069
(100 ± Change) = 120.69 as a %
Example
Calculate the following.
b) Increase $255 by 16.5%
b) Start x (100 ± Change)% = Result 255 x (100 + 16.5)% = Result 255 x 116.5% = Result
Result = $297.08 (2 dp)
b) Find the pre-GST amount when an article selling for $250 includes GST of 15%.
a) a)
Decrease $95 by 15%.
Start x (100 ± Change)% = Result
95 x (100 – 15)% = Result 95 x 85% = Result Result = $80.75
Example
Change = 20.7% (1 dp)
Achievement – Answer the following percentage questions.
247. Increase $210 by 23%
249. Decrease $47.50 by 12.5%
251. Alec has to pay a 15% surcharge on a meal costing $96.50. How much will he pay altogether?
248. Decrease 194 by 35%
253. Due to a flu epidemic 12% of the pupils in a
school are absent one day. If the school roll
is normally 575 pupils, how many pupils for? are present?
255. A school’s roll has increased by 6.5% over the last 10 years. If ten years ago it had a roll of 1450, what is its roll now?
256. The value of a car has depreciated in value by 55%. If it was initially purchased for $42 500 what is it worth now?
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
Start x (100 ± Change)% Start x (100 + 15.0)% Start x 115.0%
Start
= Result
= 250
= 250
= 250 115.0%
= $217.39
250. Decrease by 30% a shirt that costs $67.50
252. A plasma screen usually retails for $3999. If a purchaser pays cash they are eligible for a 12.5% discount. How much does the plasma screen cost with the cash discount?
254. A house sells for 34% above its government valuation of $485 000. How much does it sell


IAS 1.1 – Numeric Reasoning 29
Merit – Answer the following percentage questions.
257.
259.
261.
263.
265.
267.
269.
An article is bought for $350 and sold for $460. 258. What is the percentage profit?
A person buys some shares valued at $4200. 260. Some time later they are sold for $4650. What
is the percentage increase in the shares’ value?
A house is purchased for $540 000 and sold 262. 12 months later for $745 000. What is the percentage profit?
Find the original amount when a $210 article 264. includes a mark-up of 15%.
Find the original cost price of a book which 266. has been increased in price by 50% and is now
being sold for $67.50.
An article of clothing selling for $185 includes 268. a sellers commission of 10%. What is the price exclusive of commission?
An article costing $595 includes GST of 15% 270. and a sellers commission of 8.5%. What is the
price of the article exclusive of all taxes and commission?
An article is bought for $1275 and sold for $980. What is the percentage loss?
A painting is purchased for $120 000 and later sold for $95 000. What is the percentage loss?
The price of a barrel of oil increased from $58 to $85. What is the percentage increase?
A mountain bike costs $495 and includes GST of 15%. What is the pre-GST price?
A quoted price for some work is $2400 and includes GST of 15%. What is the pre-GST price?
A bottle of perfume sells in NZ for $158 and includes duty of 18%. What would be the pre-duty price?
The cash price of a car is $29 995. The car can be purchased over 12 months for $2800 per month. What percentage above the cash price would a person buying the car over 12 months be paying?
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30 IAS 1.1 – Numeric Reasoning
271. A student has a job at a supermarket earning d) $13.50 an hour on weekdays and $14.00 an hour
at the weekends.
a) Calculate the gross (total) amount the student would earn in a week if he worked from 4 pm to 7 pm Monday to Friday and 1 pm to 5 pm on Saturday and Sunday.
b) From his weekly gross pay, tax is deducted at the rate of 19%. What is the amount (net take home pay) after tax has been deducted?
c) After working for a period of 6 months his gross pay increases to $350.00 a week. What percentage increase in gross pay has he received?
After receiving his pay rise in c) he finds that his tax rate is to increase to 24%. Is he better off now, or prior to his recent pay rise? Explain your answer and calculations clearly.
272. A couple decide to sell their business by listing it with a real estate firm. They approach two companies to find their commission rates. Company A charges $400 + 4.5% of the purchase price up to $100 000 and thereafter 2.5%, plus GST of 15%. Company B charges $250 + 4% of the purchase price up to $80 000 and thereafter 3.5%, plus GST of 15%.
The couple expect their business to sell for approximately $210 000.
a) Show by completing the table that the commission that the couple would have to pay is $8797.50 (including GST) if they sold the business through Company A.
c)
What savings would the couple make by listing their business with company A?
Commission – Company A
Flat fee of $400
4.5% of $100 000
2.5% of $110 000
Total (excluding GST)
Total (including GST of 15%)
$210 000
$ $ $ $ $
d) Express the commission paid to Company B (incl. GST) as a percentage of the selling price of the house. Your answer should be accurate to one decimal place.
b) Calculate the commission the couple would have to pay (including GST) if they sold the business through Company B. Use the table provided.
Commission – Company B
Flat fee
Total (excluding GST)
Total (including GST of 15%)
$210 000
$ $ $ $ $
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
e) The student’s expenses for the week include hire purchase $24.50, clothing $20 and entertainment $15. The rest is savings. What percentage of the student’s take home pay in d) above is saved after his pay rise?


IAS 1.1 – Numeric Reasoning 31
Merit / Excellence – Answer the following percentage questions.
273. $5000 is invested at the rate of 12% per annum for 3 years. At the end of each year the interest is added to the amount already invested. What is the total value of the investment after this period?
275. The cash price of a new model car is $28 000. If it is bought on hire purchase the deposit is 18% and the monthly repayments are $750 a month for 36 months.
What percentage difference is there between paying cash for the car and buying it on hire purchase?
274. A motor vehicle is purchased for $37 000. It depreciates at the rate of 14% per annum on the amount it is worth at the start of each year. What will be the value of the vehicle in
three years time?
276. A store sells teddy bears at $35.10
including 15% GST (retail price). They pay
$15 (wholesale price) for the bears then put their mark-up on and then add GST to get the $35.10.
a) What is the pre-GST price of a teddy bear? b) What is the mark-up the store puts on the
c) What % is this mark-up of the wholesale price?
wholesale price of a teddy bear?
277. A mother offers her daughter two investment options towards the cost of her university studies.
Option 1 - $5000 invested at 9% per annum for the next five years. The money is compounded which means that each year they get interest on the original amount and the interest already earned. For example, in the first year they earn $450 interest, so in the second year they will earn interest on $5450.
Option 2 - A cash sum of $275 in the first year, $550 in the second year, $1100 in the third year etc. for five years.
a) Show clearly the value of each option after the five years and decide which is better and by how much (use the table provided).
b) What is the percentage difference between Option 1 and Option 2?
c) What approximate interest rate would Option 1 have to be, for it to yield a similar value to Option 2?
a)
Year 1
Year 2
Year 3
Year 4
Year 5
Option 1
$5450
Option 2
$275
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32
IAS 1.1 – Numeric Reasoning
Ratio Ratio
A ratio is a way of dividing or splitting quantities. Although ratios can be represented as a fraction we often use a colon (:) to separate the different quantities in a ratio.
A ratio of 2 : 3 means that for every 2 units of the first quantity we use 3 units of the second quantity. Therefore if we have a ratio of 2 : 3 it is exactly the same ratio as 4 : 6 or 10 : 15.
Ratios can be simplified if we can find a common factor that divides into all the quantities.
The ratio 10 : 15 can be simplified by dividing through by the common factor of 5 giving the ratio 2 : 3.
The units for ratios should always be the same. If we are mixing liquid in the ratio 1 : 5 it does not matter if it is 1 mL to 5 mL or 1 L to 5 L.
We can also divide a quantity into a given ratio. Consider: Split $24 into the ratio 3 : 5
We begin by adding the 3 and 5 to get 8 parts. Therefore the $24 is to be split into 38 and 58 . So $24x38 =$9
and $24x58 =$15
One person receives $9 and the other person $15.
Example
Simplify the ratios.
Ratios are meant to simplify problems, so we don’t usually have any decimals in a ratio and the ratio is expressed in its simplest form.
a) a)
Casio 9750GII
1 8 ab/c 3 0 EXE
Always check that the final quantities total the original amount.
In this case $9 + $15 = $24.
b) 100mL : 3L Give your answer with the same units.
b) We convert to the same units (mL) i.e. 100 : 3000.
We then divide by the highest common factor of 100 and 3000 which is 100 to get the simplified ratio 1 : 30.
18 : 30
We identify the highest common factor of 6 and divide both 18 and 30
by 6.
18:30 =3:5
Alternatively we can simplify using our calculator by entering the ratio as a fraction, i.e.
TI-84 Plus
1 8 ÷ 3 0 MATH 1
ENTER
Therefore 18 : 30 simplifies to 3 : 5
Frac
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IAS 1.1 – Numeric Reasoning 33
278. 2:4 282. 64:4
286. 15m:25m 290. 20:12:4:2
279. 0.5:8 283. 200:500
287. 100mL:5L 291. 50g:1.5kg
280. 17: 51 284. 6: 8:20
288. 0.5m:3m 292. 10mm:1km
296. 18:7=6:y
300. Divide $128 in the ratio 35 : 55
304. Divide $17.60 in the ratio 3 : 5
281. 18 : 27 285. 12:27:9 289. 8:12:20
293. 450 kg : 1.5T 297. 3:9=4:z
301. Divide 5 L in the ratio 19 : 1
305. Divide $143 in the ratio 6 : 5 : 2
Achievement – Simplify each ratio as much as possible. The final ratio should consist of whole numbers with the same units. Units should not be given as part of the answers.
Find the unknown variable.
294. 2:7=10:w 295. 4:0.5=x:18
Share each quantity in the given ratio.
298. Share $32 in the ratio 3 : 5
302. Split 5.2 m in the ratio 3 : 6 : 4
299. Split 40 L in the ratio15:5
303. Split $35.80 in theratio5:3:8:4
306. Two students work after school to clear a building site. Alysia works a total of 15 hours and Barbara works 10 hours. They get paid $240 in total. How much should each student get?
308. Two people enter into a partnership. One contributes $20 000 and the other $30 000. How should a profit of $8500 be shared between them?
307. Three friends Clare, Dennis and Elliot contribute $16 , $12 and $20 respectively towards the cost of an antique. They then sell the antique for a profit of $80. How much profit should each friend get?
309. One person puts in $2160 and another $900 as part of an investment. How should a profit of $680 from the investment be shared between them based on their contributions?
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34
IAS 1.1 – Numeric Reasoning
310. The price of a cinema ticket for a child compared to an adult can be represented by the ratio 4 : 5. If an adult pays $18.75 for a ticket what would a child pay?
312. A farmer has a 2 : 7 ratio of cows to sheep on his property. If he was to sell 42 sheep, how many cows would he have to sell to keep the same ratio?
314. It is the first day of Deeana’s holidays and she has a number of things to do prior to leaving on her Australian holiday. Use ratios to solve Deeana’s problems.
a) Deeana needs to “shock dose” the swimming pool with chlorine. The container says that for a 9000 L pool she should add 200 g. What dose should she add for their 60 000 L pool?
b) Her neighbour has to add 1.5 kg of chlorine to their pool. How big a pool must she have?
c) Deeana’s friend Crystal has asked her to purchase some Australian clothes. The ratio of Australian money to New Zealand money is
A $0.825 : NZ $1. How much Australian money will Crystal’s NZ $300 equate to?
d) A fashion outfit cost A $185. How much is this in New Zealand dollars?
e) To get to Crystal’s place Deeana uses her motorcycle. Her bike requires 50 mL of oil to be added for every 2 litres of petrol. How much oil should she add to 6.5 L of petrol?
311. A recipe has a ratio of water to milk of 3 : 2. If the recipe requires a total of 2.6 litres of liquid, how many litres of water is required?
313. A fertiliser comprises potash and super in the ratio 7 : 12. How much potash does 152 kg of fertiliser contain?
f) The map of Auckland is at a scale of 10 mm for every 1 km of real distance. Deeana estimates that it is 7.5 cm to Crystal’s place on the map. How far must she travel on her motorcycle?
g) Australia won 16 gold Olympic medals in 2000. The population of New Zealand was 3.8 million and that of Australia was 19.5 million. How many gold medals should NZ have won.
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IAS 1.1 – Numeric Reasoning
35
Example
A clothing store sells three different sizes of board shorts, S(mall), M(edium) and L(arge). They always purchase them in theratioof2:3:4. Ifthestore purchases 18 M(edium) pairs of board shorts how many did they purchase altogether?
Let x = S size shorts and y = L size shorts.
S=2=x M 3 18
so x = 12
i.e. 12 S(mall) pairs of shorts
and M = 3 = 18 L4y
so 3y = 72 y = 24
i.e. 24 L(arge) pairs of shorts.
Total sold = 12 + 18 + 24 = 54 pairs of shorts
Example
The total number of Year 9 students who sign up for volleyball is 57 and the ratio of girls to boys is 4 : 15. How many boys would have to choose another sport and leave volleyball for the ratio of girls to boys to be 4:11.
Number of boys who initially sign up for volleyball is 15 x 57 = 45 boys, so there must be 12 girls.
The required ratio of girls to boys is 4 : 11,
so 11 x 45 = 33, which is the number of boys
Merit/Excellence – Answer the following questions.
315. The ratio of three different coffees sold in a café are 2 : 4 : 5 (latté, cappuccino and flat white). If the café sells 48 cappuccino’s in one day how many coffees did they sell in total?
317. An alloy is composed of three metals, copper,tinandironintheratio17:2:3.Ifthe alloy contains 19 units of tin, how many units of the other metals are required to make the alloy?
316. The total number of people at a night class course is 54 and the ratio of men to women is 15 : 12. How many men would have to leave the course if the required men to women ratio had to be 5 : 6?
318. The weight of dry ingredients in a recipe is 675 grams and the ratio of flour to sugar is 8 : 7. How much sugar would have to be added for the ratio to be 9 : 10?
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
19
15
required.
Hence 45 – 33 = 12 boys would need to choose another sport.


36
IAS 1.1 – Numeric Reasoning
319.
A company’s ratio of sales of three different 320. types of calculators are 6 : 3 : 2 (scientific,
graphic and algebraic). If the company sells 15 graphic calculators in one month how many scientific and algebraic calculators did they
sell?
Thecostratioinbuildinganewhouseis 322. 21 : 5 : 4 (construction : decor : landscaping).
If the Smith’s spend $112 500 on landscaping,
and decor what is their construction costs and
the total cost of the new house?
A company has a ratio of male to
female employees of 15 : 11. Currently the company has 208 employees, but is offering voluntary redundancy to any male staff so that the ratio reduces to 12 : 11. How many male staff can accept voluntary redundancy?
Iftheratio4:x=x:9,findthevalueofx.
Merit/Excellence – Answer the following questions.
321.
323. Iftheratio25:4x=x:4,findthevalueofx.
324. The cost ratio in printing a number of books is4:2:1(printing:paper:cover). Ifa publisher spends $45 000 on printing plus covers, how much would the paper component be and what is the total printing cost?
325. Iftheratio9:2x–4=5:x,findthevalueofx.
326. A company has a sales ratio of Workbooks
to Homework Books of 11 : 8. Last year it sold 53 200 books in total. By how much would its Homework Book sales have to drop so its sale ratio became 77 : 50?
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IAS 1.1 – Numeric Reasoning
37
Proportion Proportion
A proportion is a part considered in relation to a whole or a statement of equality between two or
more ratios.
i.e. aa=cc
bb dd
Directly Proportional Problems
Directly proportional problems are problems where a change in one quantity causes a proportional change in another quantity.
Two quantities y and x are in direct proportion if by whatever y changes, x changes by the same proportion or multiplier.
We write y ∝ x, which is read as y is directly proportional to x, this means y = kx, where k is a constant.
E.g. The cost of pens is directly proportional to the number of pens you buy. If two pens cost $1.50, how many pens can you buy for $10.50?
First we find k, 1.50 = 2k
k = 0.75 (each pen costs 75 cents)
To calculate how many pens we can buy for $10.50 we divide 10.50 by the cost of a single pen 0.75 which equals 14 pens.
Inversely Proportional Problems
Inversely proportional problems are problems that are similar to directly proportional problems except that when x increases y will decrease and vice versa.
Two quantities y and x are inversely proportional if their product always remain constant, i.e. xy = k or
y = kx where k is a constant.
E.g. If it takes 4 men 6 hours to dig a drain, how
long will it take 7 men to do the same job?
First we find k, which is 4 x 6 = 24 (total number of man hours).
To find how long it will take 7 men to dig the drain we divide 24 (total number of man hours)
by 7 = 3 73 hours.
A good test of an inverse proportional
problem is to ask yourself,
“If one quantity doubles, will the other half”, i.e. if x increases by a multiplier, y will decrease by the same divisor.
Directly proportional problems can also be set up as ratios, but make sure that the two ratios are written in the correct order.
In the problem on the left,
if x = the number of pens then
Solving this equation gives and then
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
Two quantities x and y are said to be inversely proportional if their product xy always remains constant.
In the problem on the left,
if x = the number of hours then 4 men x 6 hours = 7 men x x
24 = 7x
x = 3 73 hours
The ratio of the number of men = the inverse ratio of the number of hours.
i.e.
The more men the less time to complete the job, hence this is an inversely proportional problem.
When x increases, y decreases.
4 : 7 = x : 6
4 x 7 = 6 7x = 24
x = 3 73 hours
1..5
2 = x
1.5x = 21 x = 14
10..5


38
IAS 1.1 – Numeric Reasoning
Example
If Paul ran 200 metres in 34 seconds, how long would it take him to run three metres?
This is a directly proportional problem as double the distance will double the time. The further Paul runs the longer it will take or conversely the shorter the distance Paul runs the less time it will take.
Setting up as ratios, if x = the number of seconds
200 = 3 34 x
200x = 102
x = 0.51 seconds
Example
If it takes 5 men six hours to repair a transformer how long will it take 7 men to do the same job if they work at the same rate?
This is an inversely proportional problem. The more men on the job the less time it should take to complete the task.
327.
329.
331.
To make 5 apple pies requires 3 kg of apples. How many kg of apples would be needed to make 8 apple pies? (Directly proportional)
A farmer has enough feed for 50 head of stock for 8 days. If the farmer sells 10 head of stock, how long will the feed last for now? (Inversely proportional)
A tramping group has enough food for
four people for 12 days. If two more people decide to join the tramp, how long will the food now last? (Inversely proportional)
328. If a 108 kg man on earth weighs 18 kg on the moon what would a 7 kg boy on the moon weigh on earth? (Directly proportional)
330. A family of four has budgeted enough money to last them for a month (30 days). If two visitors turn up for the month how long will their budgeted amount now last? (Inversely proportional)
332. Last year Jason travelled 12 500 km in his car and used 1500 litres of fuel. This year he predicts he will travel a total of 15 000 km. How much more fuel will he use this year? (Directly proportional)
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
So
5 men x 6 hours = 7 men x T 30 = 7T
T = 4.3 hours (1 dp)
Achievement/Merit – Answer the following directly proportional and inversely proportional questions.


IAS 1.1 – Numeric Reasoning 39
Achievement/Merit – Answer the following directly proportional and inversely proportional questions.
333. The current A in an electric circuit is inversely 334. proportional to the resistance R in the circuit.
When the resistance is 7.5 ohms, the current
is 5 amperes. Find the resistance if the current
is 9 amperes.
The speed Simon bikes at is inversely proportional to the time it takes him to get to school. It takes Simon 20 minutes to get to school at an average speed of 12 km/h. If
he wants to reach school in 15 minutes what should his average speed be?
335. Lee is planning a trip to Thailand. The 336.
exchange rate is 22.5 Thai baht for $1 NZD.
How many baht would Lee get for $1000
NZD? to dollars. How much would she get in NZD?
Molly took a trip to Mexico. She exchanged $125 NZD and got 1125 pesos. Later in the trip she wanted to exchange 4500 pesos
337. Tony had a photo measuring 48 cm in 338. width and 125 cm in height. He decides to
reduce its size in proportion. If the reduced
width is 40 cm what will be the new height?
339. Water usage is directly proportional to the 340. number of people living in a house. A family
of four have an average water usage of
1800 litres per day. If two additional people
visit, how much would their daily water usage be now?
341. The ingredients needed to make 24 scones are:
600 g of flour 250 g of butter
250 ml of milk 100 g of dried fruit
a) How much dried fruit is needed for a batch of 8 scones?
b) How much flour would be required for a batch of 40 scones?
c) How much milk is required for a batch of 36 scones?
d) Howmuchbutterisrequiredforabatchof 18 scones?
Ming gets charged an additional $37.35 for her baggage being overweight by 4.5 kg. How much does her friend Lin pay if she is overweight by 6.2 kg?
The number of bales of hay a farmer uses in
a season is directly proportional to his herd size. If he requires 1000 bales of hay for a herd of 160, how many bales in total will he need if the size of his herd increases by 40 head?
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40 IAS 1.1 – Numeric Reasoning
Merit/Excellence – Answer the following questions involving proportions by first identifying whether they are directly or inversely proportional.
342. Tom is working on an essay. He can type at the rate of 35 words per minute and in 2.6hourscankeyintheessay. IfJanecantype at 23 words per minute how long would it take them working together (at 58 words per minute) to enter the essay?
344. A company has enough money in its bank account to pay the wages of 25 workers for 40 days. If the company took on another five workers, how long could it afford to pay the entire work force?
346. A building has a shadow that is 45 metres long, while a vertical flag pole that is 6 metres long makes a shadow of 8 metres. How high is the building?
348. The tax on a property valued at $650 000
is $29 250. What would be the value of a property with a tax assessment of $35 000?
343. A building crew of 15 tradesmen can construct a house in 90 days. If the owner of the house wantedtohavethehousebuiltin50days,how many extra tradesmen would he have to hire?
345. If three-sevenths of a tank can be filled in five minutes, how long would it take to fill the entire tank?
347. Water is leaking out of a tap at a constant rate. If after 3 minutes 10 litres has leaked, how long will it take for 25 litres to leak out?
349. A picture that currently measures 35 cm high by 50 cm wide is to be enlarged so that the width is 90 cm. What will be the new height of the frame?
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IAS 1.1 – Numeric Reasoning
41
Rates Rates
A rate is a ratio which identifies how long it takes to do something, such as walking. To walk 5 kilometres in one hour is to walk at the rate of 5 km/h.
The fraction expressing a rate has units of
distance in the numerator and units of time in
the denominator. With a rate the two terms are in different units. For the example above the distance is measured in kilometres and the time in hours.
The difference between a ratio and a rate is that a ratio is a comparison of two numbers with the same unit, so units are not required.
Example
A tap with a worn washer is leaking at the rate of 4.5 L/h.
a) How long will it take for the tap to leak
36 litres of water?
b) How much water will be wasted in one day?
a) 36 ÷ 4.5 (rate) = 8 hours
b) 24 (hours in a day) x 4.5 = 108 litres
Example
Api is boxing up books at a printery. He has 270 boxes to pack and believes he can do it in 5 hours.
a) What rate per hour does he need to pack at to complete his task?
b) After 3 hours he has packed 151 boxes, but then takes a 15 minute break. What rate does he have to pack at for the remaining time to complete the task?
a) Rate = 54 boxes/hour
b) 119 boxes remaining to pack, with 1 hour 45
minutes left (which is 1.75 hours) so Rate = 119 ÷ 1.75
= 68 boxes/hour
Example
If it costs four people a total of $5456 to stay at a hotel for 11 days, how much wold it cost five people to stay at the same hotel for ten days?
$5456 ÷ 4 ÷ 11 = rate per person per night = $124 so for 5 people to stay for 10 days would cost
5 x 10 x $124 = $6200
Example
Eighteen workmen dig a drain 80 metres long in
five days. How long will it take 24 workmen to dig a drain 64 metres long?
More workmen means less time.
18 workmen dig 80 metres in 5 days which equates to 0.888... metres per day per workman (i.e. 80 ÷ 5 ÷ 18).
So 24 workmen would dig 21.3333 metres per day (i.e. 24 x 0.8888...). So it would take 3 days to dig 64 metres because 64 ÷ 21.3333 = 3 days
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42 IAS 1.1 – Numeric Reasoning
Achievement/Merit – Answer the following questions.
350. A robot welds 150 seams over a 20 minute period.
a) Whatistheweldingrateoftherobotper minute?
b) How many seams could the robot weld in 45 minutes?
c) How long would it take the robot to make 1000 welds?
352. A car is travelling at an average of 95 km/h. a) Howfarwouldthecartravelinfouranda
halfhours?
b) How far does the car travel each second in metres?
c) How long would it take the car to travel 100 metres?
354. Neil has to sort 150 kg of carrots over a six hour period.
a) What rate per hour does Neil need to sort at to complete his task?
b) After one and a half hours he has sorted
45 kg of carrots. What rate does he have to sort at for the remaining time to complete the task?
356. If a bale of hay lasts three horses four days,
351. Anne earns $767 for a week’s work comprising 52 hours.
a) WhatisAnne’shourlyrate?
b) How much would Anne earn if she worked a total of 214 hours over a one month period?
c) How many hours would Anne have to work to earn $5015?
353. The water flow from a household tap is 0.25L/sec.
a) Howlonginminuteswouldittaketofilla bath with 280 L of water?
b) How long in hours would it take to fill the family pool if it holds 35 000 litres?
c) What is the water flow rate in litres per hour?
355. A creeper on average grows at the rate of 8.5 cm/day.
a) What is its rate of growth per hour?
b) Howlongwouldittaketogrow2metres?
357. If 12 workers can plant 1320 seedlings in five hours, how long would it take
a) 8 workers? b) 15 workers? c) x workers?
how long would it last a) one horse? b) c) x horses?
10 horses?
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent


IAS 1.1 – Numeric Reasoning 43
Merit/Excellence – Answer the following questions.
358. Fifteen labourers can lay 85 metres of cable in 10 days. How long would it take
a) 12 labourers to lay 85 metres of cable? b) 8 labourers to lay 100 metres of cable?
360. A magazine comprising 120 pages can be produced by a staff of 15 in 30 days. How long would it take for
a) 9 people to produce the same magazine?
b) 12 people to produce a 208 page magazine?
c) The publisher has decided to produce the 120 page magazine every two weeks (10 working days), how many employees do they need?
359. If it takes four people three days to assemble ten TVs, how long would it take
a) 6 people to assemble 10 TVs?
b) 5 people to assemble an order of 15 TVs?
c) The company has an order for 50 TVs
in 4 days time, how many people do they need to employ to assemble them?
361. A gang of 60 workers can lay 900 metres of railway track in 20 days.
a) What is their rate of track laid/worker/day?
b) How long would it take a gang of
50 workers to lay one kilometre of track?
c) The railway company wish to complete a section of track, 1.8 km in length in
24 days, how many workers do they require?
363. One decorator can decorate a room in twelve hours and another decorator can decorate it in eight hours. How long would it take the two decorators working together to decorate the room?
362. An object travels at a speed of 15 m/s.
a) How long will it take the object to travel
a distance of 3 km?
b) What is the speed of the object in km/h?
c) If the object travels at a speed of 15 m/s for 30 minutes and at a speed of 20 m/s for the next 30 minutes, what is its average speed in km/h?
364. Two electricians are wiring a house. The first electrician could complete the job in 6 hours on his own, while the apprentice would take 8 hours. They work together for the first two hours, but then the first electrician works
on his own to complete the job, while the apprentice goes off on another job. How long will it take to wire the house?
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent


44
IAS 1.1 – Numeric Reasoning
Powers
Powers and Square Roots
Powers
A Power is another name for an index or exponent.
We can raise numbers and variables to a power. Consider 34 (read as 3 to the power of 4)
= 3x3x3x3
= 81
Ingeneral an = ax ax ax ax ...x a(ntimes)
We can use a calculator to find a number raised to a power. The power button on the calculator is ^
A number raised to a negative power is the same as the reciprocal (turned upside down) raised to the same power.
2–a = 21 2–3 = 21 a3
Square Roots
The symbol used to represent the positive square root is the radical sign .
The square root of a number is the number that multiplied by itself gives us the value under the radical sign.
Consider 9 (the positive square root of 9) =3 (since 3x3=9)
A number to a fractional root (i.e. a half) is the same as the square root of that number.
.
The following rules apply to powers.
Rule Example
21/a = a 2 Example
a) Evaluate 303
c) Evaluate 2500
21/2 = 2
b) Evaluate ⎛ 3 ⎞ 3 ⎜⎝ 4⎟⎠
cont... 27
b) Using the calculator we get 64 Casio 9750GII
d) Estimate 12 a) Using the calculator we get 27 000
Frac
MATH 1 ENTER
c) Using the calculator we get 50
Casio 9750GII
3 0 ^ 3 EXE
TI-84 Plus
3 0 ^ 3 ENTER
Casio 9750GII SHIFT
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
am x an am÷an
(am)n
21/a
a–m
a0
=am+n 23x25
=28 =23
= 215 = 2
=1 23
= 1
=am–n = amn
= a 2
=1 2–3
= 1
am
( 3 ab/c 4 ) EXE
^ 3 (3÷4)^3
TI-84 Plus
2 5 0 0 EXE 2ND 2500ENTER
d) The first square number less than 12 is 9 and the first square number greater than 12 is 16 so 12 is between 3 and 4.
TI-84 Plus
25÷22 (23)5
20
21/2


IAS 1.1 – Numeric Reasoning 45
⎛1⎞3 ⎜⎝ 2 ⎟⎠
⎛4⎞3 ⎜⎝ 7 ⎟⎠
17 0
⎛1⎞2 ⎜⎝ 34 ⎟⎠
400
⎛ 1⎞2 366. ⎜⎝ 5⎟⎠
370. 54
⎛ 2⎞2 367. ⎜⎝ 3⎟⎠
371. 37 375. 4–2
⎛ 2 ⎞ –1 379. ⎜⎝ 3⎟⎠
⎛ 2⎞3 368. ⎜⎝ 5⎟⎠
372. 28 376. 3–3
Achievement – Evaluate the following using your calculator. Give your answer as a fraction where possible.
365.
369.
373.
377.
381.
385.
389.
Estimate (between two consecutive whole numbers) the answer to the following.
393. 10 394. 30 395. 42 397. 150 398. 300 399. 950
Evaluate the following rounding your answer appropriately.
374.
378.
382.
5–1
⎛ 1 ⎞ –2 ⎜⎝ 2⎟⎠
784
383. 7225 387. 4
380.
384.
⎛ 3 ⎞ –3 ⎜⎝ 4⎟⎠
9216
1
4 16 9 49
1
25
386.
196 390. 21 391. 27 392. 214
388.
400 4 9 25
396. 112 400. 1000
403. 380 407. 950
409. 1580
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
401. 37 405. 150
402. 198 406. 300
404. 620 408. 1000 412. 12 890
410. 2512
411. 9780


46 IAS 1.1 – Numeric Reasoning
Integer Fractional and
Negative Powers
We can extend our knowledge of powers by including the use of fractional and negative exponents.
In the previous section we touched on negative powers and identified that a number raised to a negative power is the same as the reciprocal (turned upside down) raised to the same power.
i.e. a–m = 1 so 2–3=21 am 3
Fractional exponents like 1/n take the nth root of a number.
i.e. a1/n=na so 641/3=364=4
More complicated fractional exponents where a term is raised to the power of m/n, can be broken up into two parts – first the whole number part m, and then the fractional part 1/n.
Therefore a fractional exponent like m/n, means take the nth root and then raise to the mth power
or
raise to the mth power and then take the nth root.
i.e am/n=nam so 642/3=3642 =16
If you are calculating a term involving a fractional exponent it is easier to take the nth root first and then raise your answer to the power of m, rather than the other way around, even though both will result in the same answer. By finding the nth root first, you deal with smaller numbers.
For example 3 642 can be calculated as 642 = 4096 and then finding the cube root of this, which is 16,
or
finding the cube root of 64 which is 4, and then squaring this to get 16. Obviously this calculation is simpler than the first.
It is also possible that we will have to deal with negative fractional exponents.
i.e. a–m/n= 1 = 1 am/n n am
so64–2/3= 1 = 1 =1 642/3 3 642 16
The following rules apply to powers.
Rule
a–m = 1 am
Example
2–3 = 21 3
a1/n =na
641/3 = 3 64
am/n =nam
a–m/n =na1m
642/3 = 3 642
64–2/3 = 3 1 2 64
Investigate your calculator now.
Make sure you are capable of calculating negative powers and fractional indices.
On the TI-84 Plus to find 5 243 you enter:
On the Casio 9750GII to find 5 243 you enter:
On the TI-84 Plus to find 64–1/2 you enter:
To convert your answer to a fraction enter:
Frac
On the Casio 9750GII to find 64–1/2 you enter:
To convert your answer to a fraction enter:
FD
ENTER
5
x
2
4
3
EXE
6
4
(–)
1
2
^
)
(
÷
ENTER
MATH
1
ENTER
6
4
^
(–)
1
2
)
(
a b/c
EXE
a0 = 1, if a ≠ 0, so 20 = 1 , 30 = 1, 40 = 1 etc.
5
MATH
5
2
3
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
4


IAS 1.1 – Numeric Reasoning
47
Example
a) Evaluate 811/2 c) Evaluate163/4
a) 811/2 = 81 =9 b) 5–2 = 1 = 1
b) Evaluate 5–2
d) Evaluate64–3/2
On the Casio 9750GII to find 811/2 you enter:
8 1 ^ ( 1 ab/c 2 ) EXE
On the TI-84 Plus to find 5–2 you enter: 5 ^ (–) 2 ENTER
To convert your answer to a fraction enter:
Frac
MATH 1 ENTER
On the Casio 9750GII to find 163/4 you enter: 1 6 ^ ( 3 ab/c
4 ) EXE
On the TI-84 Plus to find 64–3/2 you enter: 6 4 ^ ( (–) 3
÷ 2 ) ENTER
To convert your answer to a fraction enter:
Frac
1 ENTER
415. 10–2 418. 274/3
421. ⎛1⎞–2 ⎜⎝ 2 ⎟⎠
424. (–8)2/3 ⎛1⎞–1/3
427. ⎜⎝8⎟⎠
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
52 25 c)163/4=4163=8
d) 64–3/2 =
1 = 1 643 512
413. 4–3 416. 363/2
Achievement – Evaluate (leave fractional answers as fractions). 414. 243 1/5
417. 25–3/2 420. 811/4
423. ⎛4⎞–1/2 ⎜⎝ 9 ⎟⎠
419. 64–1/3 422. ⎛3⎞–1
425. ⎜⎝ 4 ⎟⎠
⎜⎝ 4 ⎟⎠ ⎛1⎞–1/2
⎛1⎞1/3 426. ⎜⎝8⎟⎠
MATH


48
IAS 1.1 – Numeric Reasoning
Achievement – Evaluate (leave fractional answers as fractions). 429. 512 2/3
436. (–27)2/3 ⎛16⎞3/2 ⎛1⎞–4/3
Achievement – Evaluate the following using your calculator
428. 8–2/3 431. 163/4 434. 216–1/3
430. 100 –1/2 433. 125–2/3
432. 8–4/3
435. 2563/4
438. ⎛ 1 ⎞–1/3 ⎜⎝ 2 7 ⎟⎠
441. ⎛3⎞–2 ⎜⎝ 4 ⎟⎠
437. ⎜⎝ 9 ⎟⎠ ⎛ 2⎞–3
439. ⎜⎝ 8 ⎟⎠
⎛ 5 ⎞ –1
440. ⎜⎝ 3 ⎟⎠
442. ⎜⎝ 8 ⎟⎠
443. (0.16)–3/2 446. (0.008)2/3
449. (–0.008)–1/3 452. (–0.125)–1/3
444. (0.04)5/2
445. (0.01) –1/2 448. (2.25)3/2
451. (–0.064)2/3 454. (0.0001)5/4
457. z 3/4 460. z–2/3
463. 5z4/3 466. 10z–2/3
447. (0.125)–4/3
450. (–0.001)–2/3
453. (0.0081)3/4
Merit – Write as positive exponents under a radical sign. 456. y 2/3
461. 4x1/3 464. 2x–1/2
455. x1/3 458. x–1/2
459. y–3/2
462. 3y1/2
465. 8y–3/4
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent