SCHOLAR Study Guide
National 5 Mathematics
Assessment Practice
Topic 1:
Rounding, scientific
notation, fractions, percentages
and statistics
Authored by:
Margaret Ferguson
Heriot-Watt University
Edinburgh EH14 4AS, United Kingdom.
First published 2014 by Heriot-Watt University.
This edition published in 2016 by Heriot-Watt University SCHOLAR.
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Distributed by the SCHOLAR Forum.
SCHOLAR Study Guide Assessment Practice Topic 1: National 5 Mathematics
1. National 5 Mathematics Course Code: C747 75
Acknowledgements
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created these materials, and to the many colleagues who reviewed the content.
We would like to acknowledge the assistance of the education authorities, colleges, teachers
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Grateful acknowledgement is made for permission to use the following material in the
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1
Topic 1
Rounding, scientific notation,
fractions, percentages and
statistics
Contents
1.1
Learning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Assessment practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2
TOPIC 1. ROUNDING, SCIENTIFIC NOTATION, FRACTIONS, PERCENTAGES
AND STATISTICS
By the end of this topic, you should have identified your strengths and areas for further
revision. Read through the learning points before you attempt the assessments and go
back to the Course Materials unit if you need more help.
You should be able to:
•
round to a given number of significant figures;
•
perform calculations using scientific natation;
•
calculate compound interest;
•
calculate appreciation;
•
calculate depreciation;
•
reverse percentages;
•
convert mixed numbers to improper fractions;
•
convert improper fractions to mixed numbers;
•
add, subtract, multiply and divide fractions;
•
find the quartiles of a data set;
•
identify a 5 figure summary;
•
construct a boxplot from a data set;
•
calculate the mean and standard deviation;
•
compare data sets;
•
construct a scatter graph;
•
determine the equation of the line of best fit.
© H ERIOT-WATT U NIVERSITY
TOPIC 1.
ROUNDING, SCIENTIFIC NOTATION, FRACTIONS, PERCENTAGES AND
STATISTICS
1.1
Learning Points
Significant Figures
A digit is significant if it adds value or accuracy to the number.
•
1532864 = 2000000 to 1 significant figure
•
0 · 00002598 = 0 · 000026 to 2 significant figures
•
23 · 98421 = 24 · 0 to 3 significant figures
•
6 · 32587 × 107 = 6 · 33 × 107 to 3 significant figures
Scientific Notation
To perform calculations involving scientific notation you must know how to enter them
into your calculator and how to interpret the answer.
•
8 · 96 × 109 = 8960000000
•
7 · 312 × 10-7 = 0 · 0000007312
Fractions, decimals and percentages
Compound interest
•
The method of compound interest repeatedly applies a percentage increase or
decrease.
•
4% interest p.a. for 3 years ⇒ × 1 · 043
•
8% decrease every year for 5 years ⇒ × 0 · 925
Appreciation
•
An item appreciates if it increases in value over a period of time.
•
12.5% appreciation ⇒ × 1 · 125
Depreciation
•
An item depreciates if it decreases in value over a period of time.
•
23% depreciation ⇒ × 0 · 77
Undoing percentages
•
To undo a percentage is to reverse the application of a percentage.
•
Method 1
◦
If an item has been increased 10% it is now worth 110%.
◦
Find 1% by dividing by 110.
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4
TOPIC 1. ROUNDING, SCIENTIFIC NOTATION, FRACTIONS, PERCENTAGES
AND STATISTICS
◦
•
Then find 100% by multiplying by 100.
Method 2
◦
If an item has decreased by 7% it is worth 93%.
◦
Decreased value = original value × 0·93
◦
Original value = decreased value ÷ 0 · 93
◦
Substitute and calculate.
Converting mixed numbers to improper fractions
•
Change the whole number part to an improper fraction with the same denominator
as the fraction part.
•
31 /4 = 12 /4 + 1 /4 = 13 /4
Converting improper fractions to mixed numbers
•
A fraction is just the numerator divided by the denominator
•
25 /
3
means 25 ÷ 3 = 8 remainder 1 = 81 /3
Adding & subtracting fractions
•
To add and subtract fractions you need a common denominator.
•
To subtract fractions which have mixed numbers change them to improper fractions
first.
•
Simplify your answer if you can.
Multiplying & dividing fractions
•
To multiply and divide fractions which have mixed numbers change them to
improper fractions first.
•
To multiply fractions multiply the numerators then multiply the denominators.
•
To divide fractions change the sign to a multiply and flip the second fraction over.
•
Change all solutions from improper fractions to mixed numbers.
Statistics
•
A 5 figure summary requires the lowest value, the lower quartile, the median, the
upper quartile and the highest value in an ordered list.
•
The median (Q 2 ) is the middle value in an ordered list.
•
The lower quartile (Q 1 ) is the middle value of the lower half of the list.
•
The upper quartile (Q 3 ) is the middle value of the upper half of the list.
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TOPIC 1.
ROUNDING, SCIENTIFIC NOTATION, FRACTIONS, PERCENTAGES AND
STATISTICS
•
The interquartile range IQR =Q 3 - Q1
•
The semi-interquartile range SIQR =
•
Σ means the sum of a list of numbers.
•
The mean x̄ =
•
The standard deviation s =
•
When comparing data sets you should comment on:
Σx
n
Q3 − Q1
2
where n is the number of values in the list.
Σx2 − (Σx)
n
n−1
2
◦
the averages;
◦
the spread, consistency or variability;
or s =
Σ(x − x̄)2
n−1
•
When drawing a line of best fit try to make it go through 2 definite points.
•
Use the gradient formula m =
•
Use the equation formula y − b = m (x − a) where (a, b) is a point on the line
of best fit.
•
Remember to replace x and y in the equation of the line of best fit if the x and y
axes have been defined with different letters.
© H ERIOT-WATT U NIVERSITY
y2 − y1
x2 − x1
5
6
TOPIC 1. ROUNDING, SCIENTIFIC NOTATION, FRACTIONS, PERCENTAGES
AND STATISTICS
1.2
Assessment practice
Make sure that you have read through the learning points or completed some revision
before attempting these questions. Tailor your practice by choosing the most appropriate
questions.
•
Rounding and Scientific Notation: Questions 1 to 9
•
Percentages: Questions 10 to 19
•
Fractions: Questions 20 to 29
•
Comparing: data sets using statistics Questions 30 to 40
Key point
Questions 17 to 19 and questions 30 to 40 also assess your reasoning skills.
Assessment practice: Rounding, scientific notation, fractions,
percentages and statistics
Rounding
Go online
Q1: Calculate 486 × 728
Give your answer correct to 3 significant figures.
..........................................
Q2: Calculate 0 · 15 ÷ 840
Give your answer correct to 2 significant figures.
..........................................
Q3: Calculate 142 × 698
Give your answer correct to 1 significant figure.
..........................................
Q4: Calculate π × 1252
Give your answer correct to 4 significant figures.
..........................................
Scientific notation
Q5: There are 171 · 22775 Japanese Yen to the pound.
How many Yen are there in 2·8 million pounds, giving your answer in standard form.
..........................................
Q6:
Calculate
1·9 × 10−3
,
1·18 × 106
giving your answer in full correct to 1 significant figure.
..........................................
Q7: The mission of a space probe lasted 3 years and 269 days.
Assuming no leap years, how long did it’s mission last in seconds, giving your answer in
scientific notation correct to 2 significant figures.
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TOPIC 1.
ROUNDING, SCIENTIFIC NOTATION, FRACTIONS, PERCENTAGES AND
STATISTICS
..........................................
Q8: The planet Saturn is 1 · 433 × 10 9 km from the sun and the speed of light is
2 · 99792458 × 105 km per second.
How many hours does it take for light to travel from the sun to Saturn?
Give your answer in scientific notation correct to 4 significant figures?
..........................................
Q9: A water molecule has mass 2 · 992 × 10-26 kg.
A snowflake contains approximately 1020 water molecules.
Calculate the mass of a snowflake, giving your answer in standard form correct to 3
significant figures.
..........................................
Compound interest
Q10: A bank pays compound interest at 10% per annum.
What is the total amount of interest, in pounds, received after 2 years on a deposit of
£570?
..........................................
Q11: Elaine invested £2800 in the Bank of Skye at 3·2% compound interest per annum.
How much will she have to the nearest penny if she leaves her money untouched for 5
years?
..........................................
Q12: Stuart got a bonus of £18000 from work.
He put it into the Northumberland Building society at 0·4% compound interest per month.
How much will he have to the nearest pound if he leaves his bonus alone for 3 years?
..........................................
Q13: Ian inherited £68425.
If he can get interest of 4·1% p.a. how many years would it take for him to double his
money?
..........................................
Appreciation and depreciation
Q14: A new car costs £29500.
Its value depreciates by 13% in the first year and by 7% in the second.
What is the value of the car after 2 years (in £s)?
..........................................
Q15: Mr & Mrs Gray bought a house 3 years ago for £231,500.
It appreciated by 6·5% in the first year and 8·2% in the two subsequent years.
How much is the house worth now?
..........................................
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TOPIC 1. ROUNDING, SCIENTIFIC NOTATION, FRACTIONS, PERCENTAGES
AND STATISTICS
Q16: A sports club purchased equipment for £5995.
It depreciates by 20% each year and is replaced when it’s value falls below £1000.
How many years will it be before the equipment needs to be replaced?
..........................................
Reversing percentages
Q17: There are 1081 pupils in Lynemouth High School.
This is 15% more than the school role last year.
What was the school role last year?
..........................................
Q18: Linda bought a new car a year ago and found that having depreciated by 22% it
is now worth £13104.
How much was the car when she bought it?
..........................................
Q19: The worldwide membership of the Pony Club in 2013 was 105570 members which
was an increase of 3·5% on 2012.
How many members were there in 2012?
..........................................
Fractions
Only answers given as fractions will be accepted and must be given as a mixed number
where specified.
Answer these questions without a calculator.
Q20: What is
45
7
as a mixed number?
..........................................
Q21: What is 8
11
12
as an improper fraction?
..........................................
Q22: What is
5
6
+
5
7
as a mixed number?
..........................................
Q23: What is 2 15 − 1 34 ?
..........................................
Q24: What is
11
12
× 3
1
3
as a mixed number?
..........................................
Q25: What is 1
1
2
÷ 2 47 ?
..........................................
Fractions in context
Answer these questions without a calculator.
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TOPIC 1.
ROUNDING, SCIENTIFIC NOTATION, FRACTIONS, PERCENTAGES AND
STATISTICS
Q26: A rectangle measures 5 12 metres by 3 23 metres.
What is the area of this rectangle as a mixed number?
..........................................
Q27: A triangle has sides of length 1 13 metres, 2
What length is the perimeter of this triangle?
3
4
metres and 3
1
2
metres.
..........................................
Q28: The footpath sign shows the distance to Amble and Warkworth from here.
What is the distance between Amble and Warkworth as a mixed number?
..........................................
Q29: A piece of ribbon 12 23 cm long is cut into 6 equal pieces.
What length would each piece be?
..........................................
Comparing means and standard deviations
The two
are:
formulaefor standard deviation
2
( x)2
x −
(x − x̄)2
n
and s =
s =
n−1
n−1
The maximum temperatures ( ◦ C) over a five day period were taken at Direlton and
Southwold.
Direlton
Southwold
18
21
19
24
21
25
17
28
Q30: What is the mean temperature at Direlton?
..........................................
Q31: What is the standard deviation at Direlton?
..........................................
Q32: What is the mean temperature at Southwold?
..........................................
Q33: What is the standard deviation at Southwold?
..........................................
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27
9
10
TOPIC 1. ROUNDING, SCIENTIFIC NOTATION, FRACTIONS, PERCENTAGES
AND STATISTICS
Q34: Which place had the higher average temperature?
..........................................
Q35: Which place had the most consistent temperatures?
..........................................
Comparing datasets
The results of an exam are as follows:
Class
5A
40
29
15
31
45
32
48
28
30
22
19
Class
5B
45
18
39
25
12
38
47
37
10
41
37
Q36: Using the data for Class 5A answer the following questions.
a) What is the minimum value of the data set?
b) What is the lower quartile (Q1) value of the data set?
c) What is the median (Q2) value of the data set?
d) What is the upper quartile (Q3) value of the data set?
e) What is the maximum value of the data set?
f) What is the semi-interquartile range of the data set?
g) Create a boxplot to illustrate the data.
..........................................
Q37: Using the data for Class 5B answer the following questions.
a) What is the minimum value of the data set?
b) What is the lower quartile (Q1) value of the data set?
c) What is the median (Q2) value of the data set?
d) What is the upper quartile (Q3) value of the data set?
e) What is the maximum value of the data set?
f) What is the semi-interquartile range of the data set?
g) Create a boxplot to illustrate the data.
..........................................
Q38: Which class had the higher average exam results?
..........................................
Q39: Which class’s results were most variable?
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1.
ROUNDING, SCIENTIFIC NOTATION, FRACTIONS, PERCENTAGES AND
STATISTICS
Line of Best Fit
Q40: A group of people of different ages test their reaction times. The scatter graph
shows the results.
Mark two points on the graph to indicate your best fit line. Do this in pencil so that you
are able to change the location of the points if necessary.
Work out the equation of your line and use it to find the estimated value of a person of
10 years of age.
..........................................
..........................................
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12
ANSWERS: TOPIC 1
Answers to questions and activities
1 Rounding, scientific notation, fractions, percentages and statistics
Assessment practice: Rounding, scientific notation, fractions, percentages and
statistics (page 6)
Q1:
354000
Q2:
0·00018
Q3:
100000
Q4:
49100
Q5:
4 · 79 × 108
Q6:
0 · 000000002
Q7:
Steps:
•
How many days in 3 years and 269 days? 1364
•
How many hours is this? 32736
•
How many minutes is this? 1964160
•
How many seconds is this? 117849600
•
Round this answer to 2 significant figures and express it in scientific notation.
Answer: 1 · 2 × 108
Q8:
Steps:
•
Using your knowledge of speed, Distance & Time, how long did it take for light to
travel in seconds? 4779·973484
•
What is this in minutes? 79·66622474
•
What is this in hours? 1·32777. . .
•
Round this answer to 4 significant figures and express it in scientific notation.
Answer: 1 · 328 × 100
2 · 99 × 10-6
Q9:
Q10:
Steps:
•
How much money will be in the bank after 2 years? £689·70
•
Remember money has 2 decimal places.
Answer: £119·70
Q11: 3277·60
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ANSWERS: TOPIC 1
Q12:
Steps:
•
How many months are there in 3 years? 36
•
What do you multiply by to find the amount after 1 month? 1·004
•
Use these answers to find the amount, remembering to round to the nearest £.
Answer: £20782
Q13:
Steps:
•
What is double £68425? 136850
•
Use trial and error until you reach this amount or just over it.
Answer: 18
Q14:
Steps:
•
To calculate the value of the car after 2 years, you must first find the value of the
car after 1 year.
•
The car’s value depreciates by 13% in the first year. What is the value of the car
after 1 year (in £s)? 295000 × ·87
Answer: £23868·45
Q15: £288639·08
Q16:
Hint:
•
Use trial and error until the equipment is worth £1000 or just below.
Answer: 9
Q17:
Steps:
•
What percentage is the school role now? 115
•
How many pupils are 1%? 9·4
•
Use this answer to find 100%
Answer: 940
Q18:
Steps:
•
What percentage is the car worth now? 78
•
How much is 1%? 168
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ANSWERS: TOPIC 1
•
Use this answer to find 100%
Answer: £16800
Q19:
Steps:
•
What percentage was the Pony Club membership in 2013? 103·5
•
How many members are 1%? 1020
•
Use this answer to find 100%
Answer: 102000
Q20: 6
Q21:
3
7
107
12
Q22:
Hint:
•
To add these fractions you need a common denominator.
Answer: 1
23
42
Q23:
Hint:
•
To subtract these fractions you need improper fractions with a common
denominator.
Answer:
9
20
Q24:
Hint:
•
To multiply these fractions you need improper fractions.
Answer: 3
1
18
Q25:
Hint:
•
To divide these fractions you need improper fractions then multiply by the reciprocal
of the second fraction.
Answer:
7
12
Q26:
Hint:
•
Area of a rectangle = length × breadth
Answer:20
1
6
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ANSWERS: TOPIC 1
Q27:
Hint:
•
The perimeter is the total length of the outside edge of the triangle.
Answer: 7
7
12
Q28:
Hint:
•
The difference between the two distances is the distance between the two places.
Answer: 1
11
12
Q29:
Hint:
•
For 6 equal pieces divide by 6.
Answer: 2 19
Q30: 19
Q31: 1·58
Q32: 25
Q33: 2·74
Q34: Southwold (because it has the highest mean temperature)
Q35: Direlton (because it has the lowest standard deviation)
Q36:
a) 15
b) 22
c) 30
d) 40
e) 48
f) 9
g)
Q37:
a) 10
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16
ANSWERS: TOPIC 1
b) 18
c) 37
d) 41
e) 47
f) 11·5
g)
Q38: Class 5B (The class with highest median.)
Q39: Class 5B (The class with highest SIQR.)
Q40: This is a possible answer.
The equation of the line is y = 8 · 82x + 2 · 57
From the equation, the reaction time of a person of 10 years age is: 90·7
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