australian curriculum mathematics year 7 · at year 6 level make connections between equivalent...
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AUSTRALIAN CURRICULUM
MATHEMATICS YEAR 7
Percentages
MATHEMATICS YEAR 7
Percentages
Student’s name: ________________________________
Teacher’s name: ________________________________
First published 2012
ISBN 9780730744382 SCIS 1564078
© Department of Education WA 2012 (Revised 2020)
Requests and enquiries concerning copyright should be addressed to:
Manager Intellectual Property and Copyright Department of Education 151 Royal Street EAST PERTH WA 6004
Email: [email protected]
This resource contains extracts from The Australian Curriculum Version 3.0 © Australian Curriculum, Assessment and Reporting Authority 2012. ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that:
the content descriptions are solely for a particular year and subject
all the content descriptions for that year and subject have been used
the author’s material aligns with the Australian Curriculum content descriptions for the relevant year andsubject.
You can find the unaltered and most up to date version of this material at www.australiancurriculum.edu.au. This material is reproduced with the permission of ACARA.
creativecommons.org/licenses/by-nc-sa/3.0/au/
Graphics used in this resource are sourced from http://openclipart.org under the creative commons license http://creativecommons.org/publicdomain/zero/1.0
This product will be registered through the National Copyright Unit for use in all Australian schools without remuneration.
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Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 1
Contents
Signposts ....................................................................................................................................2
Introduction...............................................................................................................................3
Curriculum details....................................................................................................................4
1. What is a percentage?.........................................................................................................7
2. Fraction and decimal equivalents....................................................................................13
3. Finding a percentage of a quantity..................................................................................19
4. Finding a percentage of a quantity again .......................................................................25
5. Writing one quantity as a percentage of another...........................................................31
6. Problem solving.................................................................................................................37
7. Solving harder problems..................................................................................................43
8. Summary............................................................................................................................47
9. Review tasks ......................................................................................................................49
Solutions...................................................................................................................................57
Percentages Year 7 Mathematics
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Signposts Each symbol is a sign to help you.
Here is what each one means:
The recommended time you should take to complete this section.
An explanation of key terms, concepts or processes.
A written response. Write your answer or response in your journal.
Correct this task using the answers at the end of the resource.
Calculators may not be used here.
Make notes describing how you attempted to solve the problem. Keep these notes to refer to when completing the Self-evaluation task. Your teacher may wish you to forward these notes.
Year 7 Mathematics Percentages
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Introduction This resource should take you approximately two weeks to complete. It comprises seven learning sections, a summary section and a review task section.
The learning sections have the following headings:
Key wordsThese are the main words that you need to understand and use fluently to explain yourthinking.
Warm-upWarm-up tasks should take you no longer than 10 minutes to complete. These are skillsfrom previous work you are expected to recall from memory, or mental calculations thatyou are expected to perform quickly and accurately. If you have any difficulties inanswering these questions, please discuss them with your teacher.
ReviewSome sections have reviews immediately after the warm-up. The skills in these reviewsare from previous work and are essential for that section. You will use these to developnew skills in mathematics. Please speak to your teacher immediately if you are havingany trouble in completing these activities.
Focus problemFocus problems are designed to introduce new concepts. They provide examples of thetypes of problems you will be able to solve by learning the new concepts in this resource.Do not spend too long on these but do check and read the solutions thoroughly.
Skills developmentThese help you consolidate new work and concepts. Most sections include skillsdevelopment activities which provide opportunities for you to become skilled at usingnew procedures, apply your learning to solve problems and justify your ideas. Pleasemark your work after completing each part.
Correcting your work
Please mark and correct your work as you go. Worked solutions are provided to show how you should set out your work. If you are having any difficulty in understanding them, or are getting the majority of the questions wrong, please speak to your teacher immediately.
Journal
Please keep an exercise book to record your notes and to summarise your learning. At the end of each section, write definitions for the key words that were introduced for that section.
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Curriculum details
Content Descriptions This resource provides learning and teaching to deliver the Australian Curriculum: Mathematics for the following Year 7 Content Descriptions.
Connect fractions, decimals and percentages and carry out simple conversions (ACMNA157)
Find percentages of quantities and express one quantity as a percentage of another, with and without digital technologies. (ACMNA158)
Content Descriptions 1 2 3 4 5 6 7 R
ACMNA157
ACMNA158
Indicates the content description is explicitly covered in that section of the resource.
Previous relevant Content Descriptions
The following Content Descriptions should be considered as prior learning for students using this resource.
At Year 6 level
Make connections between equivalent fractions, decimals and percentages (ACMNA131)
Investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and without digital technologies (ACMNA132)
Proficiency strand statements at Year 7 level At this year level:
Understanding includes describing patterns in uses of indices with whole numbers, recognising equivalences between fractions, decimals, percentages and ratios, plotting points on the Cartesian plane, identifying angles formed by a transversal crossing a pair of lines, and connecting the laws and properties of numbers to algebraic terms and expressions
Fluency includes calculating accurately with integers, representing fractions and decimals in various ways, investigating best buys, finding measures of central tendency and calculating areas of shapes and volumes of prisms
Problem Solving includes formulating and solving authentic problems using numbers and measurements, working with transformations and identifying symmetry, calculating angles and interpreting sets of data collected through chance experiments
Year 7 Mathematics Percentages
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Reasoning includes applying the number laws to calculations, applying known geometric facts to draw conclusions about shapes, applying an understanding of ratio and interpreting data displays
General capabilities
General capabilities 1 2 3 4 5 6 7 R
Literacy
Numeracy
Information and communication technology (ICT) capability
Critical and creative thinking
Personal and social capability
Ethical behaviour
Intercultural understanding
Indicates general capabilities are explicitly covered in that section of the resource.
Cross-curriculum priorities
Cross-curriculum priorities 1 2 3 4 5 6 7 R
Aboriginal and Torres Strait Islander histories and cultures
Asia and Australia’s engagement with Asia
Sustainability
Indicates cross-curriculum priorities are explicitly covered in that section of the resource.
This resource contains extracts from The Australian Curriculum Version 3.0 © Australian Curriculum, Assessment and Reporting Authority 2012. ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that:
the content descriptions are solely for a particular year and subject
all the content descriptions for that year and subject have been used
the author’s material aligns with the Australian Curriculum content descriptions for the relevant year and subject.
You can find the unaltered and most up to date version of this material at www.australiancurriculum.edu.au. This material is reproduced with the permission of ACARA.
creativecommons.org/licenses/by-nc-sa/3.0/au/
Percentages Year 7 Mathematics
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Year 7 Mathematics Percentages
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1. What is a percentage?When you complete this section you should be able to:
understand what a percentage is of a whole.
Keywords
percentage
Warm-up 1
1. Is 4 a factor of 20? ___________
2. 8 + 4 = ___________
3. What is the missing number?
a = ___________
4. Circle the greater fraction.2
3or
6
6
5. What is a half of 18? ___________
6. 6 – 4.3 = ___________
7.
5 . 4
5
8. Write 0.333… as a fraction. ___________
9. Complete the next number: 5, 10, 15, 20 ___________
10. The truck is at (1, 4).
If the truck moves 2 units right,
where will it then be?
x1 2 3 4 5
y
12345
30-3a
Percentages Year 7 Mathematics
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Review 1
Example
Write each of these numbers as fractions with denominators of 100.
(a) 0.23 (b) 0.7
(c)9
50(d)
56
200
Solution
(a)23
100(b) 0.7 = 0.70 =
70
100
(c)9 9 2 18
50 50 2 100
(d)
56 56 2 28
200 200 2 100
1. Write each of these numbers as fractions with denominators of 100.
(a) 0.09 ______________ (b) 0.88 ______________ (c) 0.2 _____________
(d) 12
50_______________(e)
12
200______________
2. Write the answers as fractions with a denominator of 100.
(a)5 87
100 100 ___________________________________________________
(b)13 8 29
100 100 100 ___________________________________________________
(c)51 15
100 100 ___________________________________________________
(d) 0.25 + 0.72 ___________________________________________________
(e) 0.2 + 0.37 ___________________________________________________
(f)3 4
10 50 ___________________________________________________
In (c), 50 × 2 = 100 so the numerator and
denominator had to both be multiplied by 2.
Year 7 Mathematics Percentages
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Focus problem 1
Dan and Melinda were watching some games of basketball.
They knew there were one hundred players altogether.
Of the hundred players, they counted 40 female players, 64 blond haired players, 15 players over 185 cm tall and 23 wearing a green uniform.
Melinda said it would be easy to write these as percentages since a percentage is the same as a fraction with the denominator of 100.
Dan started to change them to percentages. He preferred to use the shorthand symbol for a percentage, %.
He knew that of the players, 40 out of the 100 were females so 40% were females. Melinda then noted that if that were the case then the percentage of males players would be 60%.
Help Dan and Melinda by completing the following table of percentages for players with the different characteristics.
Female players 40%
Male players
Blond haired players
Players without blond hair
Players over 185 cm tall
Players not over 185 cm tall
Players wearing green uniforms
Players wearing other colour uniforms apart from green
All the players
Check your work before continuing.
In the 15th century when writing a percent the shortened pc was used after the number. By the 17th century with some flourishes in the writing it had evolved to the symbol o
oand this eventually changed to the symbol we use today, %.
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Skills development 1
One whole can be written as the fraction 100
100 or as the percentage 100%.
Example
If 42% of a class of students is boys, what percentage are girls?
Solution
The percentage of girls is 100% – 42% = 58%.
1. A grid has 100 squares. 18 of the squares are coloured red, 26 are coloured green and 41are coloured yellow. The remainder are uncoloured.
Write the percentage of each of the following.
(a) green squares ________________________________________
(b) uncoloured squares ________________________________________
(c) squares that are not red ________________________________________
(d) squares that are not yellow or red ________________________________________
(e) purple squares ________________________________________
2.
(a) On the grid above shade:
(i) 10% of the squares using sloped lines (////)
(ii) 25% of the squares using vertical lines (|||||)
(iii) 4% of the squares using a solid colour. ( )
(b) What percentage of squares are now:
(i) shaded ________________________________________
(ii) unshaded? ________________________________________
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3. Answer the questions below using the following grid.
(a) How many squares are in the grid? _ ____________________________
(b) What percentage of the grid is one square? ____________________________
(c) What percentage of the grid is:
(i) unshaded ____________________________
(ii) shaded with sloped lines (\\\\) ____________________________
(iii) shaded with a solid colour ____________________________
(iv) shaded with cross hatches ( ) ____________________________
4. How many squares need to be shaded if you were asked to shade 20% of a grid that haseach of the following number of squares?
(a) 100 squares
__________________________________________________________________
(b) 50 squares
__________________________________________________________________
(c) 25 squares
__________________________________________________________________
(d) 200 squares
_________________________________________________________________
(e) 1000 squares
__________________________________________________________________
Check your work before continuing.
Percentages Year 7 Mathematics
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Year 7 Mathematics Percentages
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2. Fraction and decimal equivalentsWhen you complete this section you should be able to:
convert between fractions, decimals and percentages.
Warm-up 2
1. Is 4 a common factor of 24 and 32? ___________
2. 13 – 5 = _________
3. The temperature was 1 degree but it dropped 4 degrees.
What is the new temperature? _________
4. Insert <, > or = to make the following sentence true.
1 2
2 4
5. 1
210 = _________
6. Round 6.9 to a whole number. _________
7. 8.4 4 = _________
8. Write 1
21 as a decimal. _________
9. Complete the next number. 0.7, 1.4, 2.1, 2.8, ________
10.A six-sided die is rolled.
Express, as a fraction, the probability that it lands on an odd number. _________
0
5
10
20
°C
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Review 2
Example
(a) Write the following as fractions with a denominator of 100.
(i)13
50(ii) 0.93 (iii)
425
500(iv) 0.6
(b) Use your calculator to find the equivalent decimals to the following fractions.
(i)27
50(ii)
17
20(iii)
5
8
Solution
(a) (i) 13 2 26
50 2 100
(ii)
93
100(iii)
425 5 85
500 5 100
(iv)
600.60
100
(b) (i) 27 ÷ 50 = 0.54 (ii) 0.85 (iii) 0.625
1. (a) Write the following as fractions with a denominator of 100.
(i)47
50 ___________________________________________________________
(ii) 0.19 ___________________________________________________________
(iii) 0.3 ___________________________________________________________
(iv)176
400 ___________________________________________________________
(v)7
25 ___________________________________________________________
(b) Use your calculator to find the equivalent decimals to the following.
(i)13
20 ___________________________________________________________
(ii)102
300 ___________________________________________________________
(iii)9
16 ___________________________________________________________
(iv)35
56 ___________________________________________________________
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Focus problem 2
Leith, Hugh and Sam were talking about their love of music. They decided to compare what proportion of their pocket money they spent on music last month.
Leith said she spent 1
4, Hugh said 0.27 and Sam said he spent
22% of his pocket money on music.
It would be easier to compare the proportions if they were all in the same form.
Percentages have been used for a long time to easily compare values so the friends decided to use percentages for their comparisons.
Since percentages are equivalent to fractions out of 100 the idea would be to first convert the decimal and the fraction to fractions with a denominator of 100. Once they have the denominator of 100 the percentage value could be written easily.
(a) (i) Complete the following calculations.
1 %
4 100 0.27 %
100
(ii) Now write the three percentages in order from the smallest to the largest.
_______________________________________________________________________
(iii) Who had the largest percentage value? _______________.
Jemma came in later and she said her proportion was 11
40 because she spent $11 on music out
of her $40 pocket money. This was harder to change quickly to a fraction out of 100 and hence a percentage.
Sam suggested she change it to a decimal using her calculator.
To change 11
40 to a decimal using a calculator do the division 11 ÷ 40.
(b) (i) Write down the decimal equivalent of 11
40 from the calculator.
_______________
(ii) How do you change the decimal to a percentage?
______________________________________________________________________
______________________________________________________________________
Percentages Year 7 Mathematics
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Check the following working.
1111 40 0.275
400.275 100 27.5%
11So 27.5%
40or
11(11 40 100)% 27.5%
40
(c) To find the percentage of pocket money spent on music in each of the followingcases, write down the calculation you would perform on your calculator then writethe percentage answer.
Complete the calculation on your calculator to find the percentage answer.
Pocket money Amount spent on music Calculation Percentage
$10 $2.70
$25 $7.50
$32 $28.00
Check your work before continuing.
Year 7 Mathematics Percentages
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Skills development 2
Converting fractions, decimals and percentage
Fraction or decimal to percentage:
Percentage to fraction or decimal:
Example
1 Change each of the following to percentages.
(a) 0.78 (b) 0.323 (c)4
5(d)
5
8
2. Change these percentages to simple fractions.
(a) 15% (b) 2.5%
3. Change these percentages to decimals.
(a) 85% (b) 2% (c) 1
333 %
Solution
1. (a) (0.78 × 100)% = 78% (b) (0.323 × 100)% = 32.3%
(c)4
(4 5 100)% 80%5 (d)
5(5 8 100)% 62.5%
8
2. (a) 15 3
100 2015% (b)
2.5 25 1
100 1000 402.5%
3. (a) 85% = 85 ÷100 = 0.85 (b) 2% = 2 ÷100 = 0.02
(c) 1
333 % 33.3... 100 0.333...
fraction
decimal
percentagenumerator ÷ denominator × 100
fraction
decimal
percentage
write the percent with a denominator
of 100, then simplify if possible
÷ 100
Percentages Year 7 Mathematics
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1. Change each of the following to percentages. Show your working.
(a) 0.61 ____________________________________________________________
(b) 0.04 ____________________________________________________________
(c) 1.13 ____________________________________________________________
(d)2
5____________________________________________________________
(e)5
16____________________________________________________________
(f)2
75____________________________________________________________
2. Change these percentages to simple fractions.
(a) 45% ____________________________________________________________
(b) 0.5% ____________________________________________________________
(c) 125% ____________________________________________________________
3. Change these percentages to decimals.
(a) 23% ____________________________________________________________
(b) 1.5% ____________________________________________________________
(c) 2
366 % ____________________________________________________________
4. In each of the following cases change all the numbers to the same form: fraction, decimalor percentage. Then arrange the original numbers in order from the smallest to thelargest.
(a) 47% 0.4512
25 _______________________________________________________________________
_______________________________________________________________________
(b) 19
20 92%
15
160.975 94.8%
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Check your work before continuing.
Year 7 Mathematics Percentages
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3. Finding a percentage of a quantityWhen you complete this section you should be able to:
find a given percentage of a quantity.
Warm-up 3
1. Circle the composite number. 5, 7, 9, 11
2. 9 7 = _________
3. What is the missing number?
a = __________
4. Locate 5
10 on the number line.
5. What is a quarter of 20? ________
6. Estimate the difference by first rounding to whole numbers. 8.9 – 4.1 _______
7. 1.2 4 = _________
8. Write 20
100 as a percentage. _________
9. Write the next number.1 3 5
8 8 81 , 1 , 1 , _________
10.
Which shape is at (4, 3)?
_______________
0 1
-3-4 -1 0a
x1 2 3 4 5
y
1
2
3
4
5
Percentages Year 7 Mathematics
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Review 3
Finding a fraction or a decimal of a quantity is revised in these examples.
Example
Find the following.
(a) 1
5 of 35 km (b)
3
5 of 35 km (c) 0.3 of 35 km
Solution
(a) 1
5 of 35 km = (35 × 1 ÷ 5) km = 7 km
(b) 3
5 of 35 km = (35 × 3 ÷ 5) km = 21 km
(c) 0.3 of 35 km = (35 × 0.3) km = 10.5 km
1. Find the following fractions or decimals of 72 litres of water.
(a)1
8________________________________________________________________
(b)3
4________________________________________________________________
(c) 0.6 ________________________________________________________________
(d) 0.09 ________________________________________________________________
2. Find the following fractions or decimals of the given quantity.
(a) 0.15 of $70
______________________________________________________________________
(b) 2
3 of 51 kg
______________________________________________________________________
(c) 0.06 of 150 minutes
______________________________________________________________________
(d) 11
25of 94.5 km
______________________________________________________________________
Notice that the word 'of' means
to multiply.
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Focus problem 3
A nursery was growing a number of different trees. It was known that for fruit trees 5% of the seeds that were planted wouldn’t sprout.
1. Calculate the following for fruit trees.
(a) How many seeds out of 100 wouldn't sprout? _________
(b) How many seeds out of 200 wouldn't sprout? _________
(c) How many seeds out of 300 wouldn't sprout? _________
(d) How many seeds out of 260 wouldn't sprout? _________
(e) How many seeds out of 480 wouldn't sprout? _________
When growing wattles all the seeds sprouted but 2% of the
young trees died before maturity.
2. Calculate the following for wattle trees.
(a) How many young trees out of 100 died before maturity? _________
(b) How many young trees out of 200 died before maturity? _________
(c) How many young trees out of 50 died before maturity? _________
(d) How many young trees out of 550 died before maturity? _________
3. Explain how you worked out your answers for:
(a) the fruit trees
___________________________________________________________________
___________________________________________________________________
(b) the wattles.
___________________________________________________________________
___________________________________________________________________
Check your work before continuing.
Wattles
Wattles are the most widely spread of any of the native plants in Australia. There are over 700 species throughout the country. At any time of the year you can find a wattle flowering somewhere. They are especially suited to our harsh climate.
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Skills development 3
To find a percentage of a quantity do one of the following.
• Multiply the quantity by the percentage number and then divide by 100. This is thesame as multiplying by the fraction equivalent with a denominator of 100.
• Multiply the quantity by the percentage written as a decimal.
• Multiply the quantity by the fractional equivalent of the percentage.
Example
Find the following.
(a) 15% of $340
(b) 20% of 45 litres
Solution
(a) 15% of $340 = $(15 × 340 ÷ 100) = $51
or
15% of $340 = $(0.15 × 340) = $51
(b) By using the known fact that 20% is equivalent to the fraction 1
5 the calculation
will be finding 1
5of 45 litres.
1
5 of 45 litres = 9 litres
1. Find 20% of $35 by:
(a) writing the percentage as a decimal first
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
(b) writing the percentage as a fraction with a denominator of 100.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
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2. Calculate each of the following percentages. Show the calculations you used.
(a) 15% of 27 litres
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
(b) 25% of 472 people at a concert
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
(c) 9% of 15 km
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
(d) 1.2% of $2300
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
3. A property had an area of 540 hectares. The house and sheds occupied 1.1% of theproperty and 28% was used for cattle grazing. A further 2% was fenced off to protectsome rare bush orchids and 17% was uncleared scrub. Find how many hectares wereused for each of the purposes.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Check your work before continuing.
Percentages Year 7 Mathematics
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4. Finding a percentage of a quantityagain
When you complete this section you should be able to:
solve harder problems involving finding a percentage of a quantity.
Warm-up 4
1. What is the next number? 2, 3, 5, 8, 12, __________
2. 42 7 = _________
3. The temperature was minus 2 degrees but it went up 4 degrees.
What is the new temperature? __________
4. Express the value of w as a fraction.
__________
5. 1
330 = _________
6. 5201 1000 = _________
7. 4 8.24 _________
8. Write 50% as fraction. __________
9. Find the next number. 75, 64, 53, ________
10. Determine the size of the missing angle.
__________
10
w
?40°
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Review 4
Example
1. Find 2
3 of each of the following amounts.
(a) 69 kg (b) 18 hours (c) $22
2. Find 12% of each of the following amounts.
(a) 120 km (b) $5.00 (c) 95 litres of fuel
Solution
1. (a) (69 ×2
3) = (69 × 2 ÷ 3) kg = 46 kg
(b) (18 × 2 ÷ 3) hours = 12 hours
(c) $(22 × 2 ÷ 3) = $14.67 (to the nearest cent)
2. (a) (120 × 0.12) km = 14.4 km
(b) $(5 × 0.12) = $0.60
(c) (95 × 0.12) litres = 11.4 litres
1. Find 3
5 of each of the following amounts.
(a) 75 kg ________________________________________________________
(b) 105 minutes ________________________________________________________
(c) $13 ________________________________________________________
2. Find 7% of each of the following amounts.
(a) 500 hectares ________________________________________________________
(b) 56 kg ________________________________________________________
(c) $4.20 ________________________________________________________
Notice that in the money calculation the answer has been rounded to the nearest
cent.
Year 7 Mathematics Percentages
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Focus problem 4
Lance wants to buy a new bike but the shop says he needs to make a
deposit of1
333 % of the asking price of $264 before they order it from
the manufacturer.
How can Lance find 1
333 % of the price?
It is hard to write that percentage as a fraction with a denominator of 100. The decimal value is a repeating decimal so you can't enter that easily on the calculator without doing some rounding.
1. How would you suggest he calculates the deposit?
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
2. Now think back to earlier work. Where did you see 1
333 % before?
_____________________________________________________________________
3. What is the simple fraction equivalent to 1
333 % ? __________
That is a good fraction equivalent to remember along with some others so that you can work quickly when finding percentages of quantities.
4. Complete this table of common fraction and percentage equivalents.
Percentage Fraction
100% 1
10% 1
10
5%
25%
1
8
1
3
Percentages Year 7 Mathematics
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5. Once you know 1
8 you may be able to recognise other relationships with fractions with a
denominator of 8. This should make some of your calculations easier to complete.
Make use of the percentage equivalent of 1
8to find the percentage equivalent to each of
the following fractions.
(a)3
8 _________________
(b)5
8 _________________
(c)7
8 _________________
6. You should now recognise that 1 1
3 333 % .
(a) What will be the percentage equivalent to 2
3? __________________
(b) What will be the percentage equivalent to 4
3? ___________________
7. How much was the deposit in dollars, that Lance had to make for the bike he wanted?
______________________________________________________________________
______________________________________________________________________
Check your work before continuing.
Year 7 Mathematics Percentages
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Skills development 4
If the percentage required is over 100% it means there has been a percentage increase from the original value.
Example
Calculate:
(a) 1
333 % of 45 kg
(b) the value of an antique originally valued at $40 if it increased to 120% of the originalvalue.
Solution
(a) 1
333 % of 45 kg =
1
3 × 45 kg = 15 kg
(b) 120% of $40 = $40 1.2 $48
The value of the antique is now $48.
Check:
100% of $40 = $40
20% of $40 = 20 × 40 ÷ 100 = $8
120% of $40 = $40 + $8 = $48
1. Without the use of a calculator find 1
333 % of the following.
(a) 6 kg ____________________________________________________________
(b) 36 litres ____________________________________________________________
(c) 24 hours ____________________________________________________________
(d) 336 km ____________________________________________________________
2. A shop reduced all their prices by 10%. One jacket was originally priced at $60.
(a) Calculate the reduction in price. _________________________________________
(b) Calculate the reduced price. _________________________________________
(c) Calculate 90% of $60. _________________________________________
(d) Explain the relationship between your answers to (b) and (c).
_______________________________________________________________________
_______________________________________________________________________
1 133 %
3 3
Percentages Year 7 Mathematics
Page 30 © Department of Education WA 2012 – MATHSAC018
3. A shop wanted to increase its prices by 15% to increase its profits.
(a) Calculate the price increase on a shirt priced at $80. __________________________
(b) Calculate the price of the shirt after the increase. ____________________________
(c) Calculate 115% of $80. ________________________________________________
(d) Explain the relationship between your answers to (b) and (c).
_______________________________________________________________________
_______________________________________________________________________
4. Another shop discounted all jacket prices by 15% and shoe prices by 22%. Whatpercentage of the original price will jackets and shoes be sold for?
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
5. Due to a new government tax all the books on sale at a book shop had to have the priceincreased by 10% and all other items increased by 5%.
What percentage of the original prices will the new prices be?
_______________________________________________________________________
_______________________________________________________________________
6. Find the new value in each of the following cases.
(a) A person who weighed 90 kg reduced their weight by 15%.
_______________________________________________________________________
(b) During a rainstorm a water tank that had 800 litres in it had a 23% increase in itswater storage.
_______________________________________________________________________
(c) A concert venue had seating for 2600 and then it was extended to increase theseating capacity by 20%.
_______________________________________________________________________
(d) A metal bar that was 15 metres long contracted by 2.4% when placed in a freezer.
_____________________________________________________________________
Check your work before continuing.
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 31
5. Writing one quantity as apercentage of another
When you complete this section you should be able to:
write one quantity as a percentage of another.
Warm-up 5
1. Circle the triangular number. 4, 5, 6, 7, 8
2. 34 + 9 = _________
3. What is the missing number?
q = __________
4.1 2
5 5 __________
5. What is an eighth of 24? ___________
6. 6 m = __________ cm
7. 12 – 3 2 = _________
8. Write 1
10as decimal. ___________
9. Write the next number. 9.3, 8.9, 8.5, 8.1, ________
10.
At what point is the truck? __________
x1 2 3 4 5
y
12345
-6-8 -4 0q
Percentages Year 7 Mathematics
Page 32 © Department of Education WA 2012 – MATHSAC018
Review 5
Example
Write each of the following as a fraction of $20.
(a) $10 (b) $2 (c) 40 cents (d) $17.50
Solution
(a)$10 1
$20 2 (b)
$2 1
$20 10
(c)$0.40 4 1
$20 200 50 (d)
$17.50 175 7
$20 200 8
1. Write each of the following as a simplified fraction of 36 kg.
(a) 18 kg ____________________________________________________________
(b) 4 kg ____________________________________________________________
(c) 22 kg ____________________________________________________________
2. Write 45 cm as a simplified fraction of the following.
(a) 55 cm ____________________________________________________________
(b) 81 cm ____________________________________________________________
(c) 1 metre ____________________________________________________________
Notice that the answer is a fraction and the units are
no longer used.
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 33
Focus problem 5
Lena is a keen tennis player. She has recently taken part in a number of tournaments.
In the first match on Saturday she counted that she served 100 first serves and that she got 53 of them in.
(a) What fraction of her first serves did she get in?
_____________________________________
(b) What percentage of her first serves did she get in during that match?
_____________________________________
In the second match she only got 36 first serves in out of a total number of 66 first serves.
(c) What fraction of her first serves did she get in during the second match?
_____________________________________
(d) What percentage of her first serves did she get in during the second match?
_____________________________________
(e) In which match was she serving better?
_____________________________________
(f) Why was it useful for Lena to know the percentage of first serves in rather than the actualnumber?
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Check your work before continuing.
Percentages Year 7 Mathematics
Page 34 © Department of Education WA 2012 – MATHSAC018
Skills development 5
To write one quantity as a percentage of another first write the relationship as a fraction then convert the fraction to a percentage. The quantities must be in the same units.
Example
What percentage of 4 metres is each of the following?
(a) metres
(b) 40 cm
(c) 5 cm
Solution
(a)3
(3 4 100)% 75%4
(b)40 1
10%400 10
(c)15
(15 400 100)% 3.75%400
1. Write $60 as a percentage of the following.
(a) $100 ____________________________________________________________
(b) $1000 ____________________________________________________________
(c) $200 ____________________________________________________________
(d) 540 ____________________________________________________________
2. The apples in the orchard needed picking. Maria, Kevin and Helen offered to help outafter they finished their homework. Maria picked 38 kg, Kevin picked 27 kg and Helenpicked 15 kg of the apples.
What percentage of the total did each of them pick?
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
(b) was easy becauseyou should already
know that 1
10%.10
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 35
3. A 500 mL drink mix had 380 mLof water, 30 mL of flavouring and the remainder wasskim milk. Find what percentage of the drink each ingredient represents.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
4. The price of a mobile phone case increased from $16 to $22.
(a) What was the actual increase? ________________
(b) Find the increase as a percentage of the original price.
_______________________________________________________________________
_______________________________________________________________________
5. Find the reduction in price of each of the items for sale as a percentage of the originalprice shown below.
$15 caps drastically
reduced to $9
Trainers were $48
now just $32
Model planes
$28
$18
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Percentages Year 7 Mathematics
Page 36 © Department of Education WA 2012 – MATHSAC018
Check your work before continuing.
It would be a really good time to summarise everything you have learned
about using percentages. This will be useful for solving the word problems in
the next two sections.
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 37
6. Problem solvingWhen you complete this section you should be able to:
solve simple problems involving percentages.
Warm-up 6
1. 10.8 10 = _________
2. 47 – 28 = _________
3. The temperature is minus 1 degree.
How much will it need to increase to get to 3 degrees? _________
4.7 3
8 8 _________
5. 1
816 = _________
6. 0.9 g = _________ mg
7. 10 5 2 = _________
8. Write 0.5 as a percentage. __________
9. Write the next number. 1 3 1
4 4 22 , 2, 1 , 1 ________
10. Determine the probability the spinner will land on a 2.
Express your answer as a percentage.
_________
1
23
4
Percentages Year 7 Mathematics
Page 38 © Department of Education WA 2012 – MATHSAC018
Review 6
Example
(a) Write 35% as a simplified fraction and as a decimal.
(b) Find 35% of $40.
(c) If a price was reduced by 35% what percentage of the original price is the new price?
(d) What percentage of $70 is $14?
Solution
(a)35 7
35%100 20
and 35% = 0.35
(b) 35% of $40 = 0.35 × $40 = $14 or1
735% of $40
20 $ 40
2$14
(c) New price is (100 – 35)% or 65% of the original price.
(d) Percentage = 14
( 100)% 20%70
1. (a) Write 60% as a simplified fraction and as a decimal.
_______________________________________________________________________
(b) Find 60% of 900 kg.
_______________________________________________________________________
(c) If a seating capacity was increased by 60% what percentage of the original seating isthe new capacity?
_______________________________________________________________________
(d) What percentage of 60 kg is 42 kg?
_______________________________________________________________________
_______________________________________________________________________
2. Find 1
212 % of the following.
(a) 80 kg ________________________________________________________
(b) 9.6 metres ________________________________________________________
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 39
Focus problem 6
The local shire council decided to increase the rates by 10%.
The Warner family had a rates bill of $1200 before the increase.
What will their new rates bill be?
_________________________________________________________________________
_________________________________________________________________________
Because the Warner family paid their rates within the first week of receiving the notice they got a discount of 10% of the total.
How much will they have to actually pay?
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
You should have found that the second answer is not equal to the $1200 that you started with. Write an explanation of why this has happened.
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
Check your work before continuing.
Percentages Year 7 Mathematics
Page 40 © Department of Education WA 2012 – MATHSAC018
Skills development 6
1. On Friday Jimmy calculated that he had spent 8 hours sleeping, 6 hours at school, 3hours on the computer, 2 hours eating and the remainder at basketball training. Whatpercentage of his day did he spend on each activity?
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
2. (a) The main road builder had completed 45 km of a new section of highway. It isplanned to increase this by 12% next year. How long will the section of highway be altogether after they have extended it?
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
(b) A major veterinary chain has 325 employees. Due to cost cutting they have toreduce that number by 8%. How many employees will they have after the reduction?
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
When solving word problems you need to: Read the question carefully.
Decide on the methods you will need to use.
Show as much working as possible to makeyour calculations clear.
Write only mathematically correct statements.
Include words to explain your answers.
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 41
3. Stamp duty is a government tax that is collected when buying and selling large items likecars, houses and land. In Western Australia in a particular year the stamp duty for buyinga car was listed as 2.75% for a value from $0 to $25 000.
Find the amount of stamp duty payable for cars with the following values.
(a) $2500 ____________________________________________________________
(b) $12 000 ____________________________________________________________
(c) $24 999 ____________________________________________________________
Cars with a value over $50 000 were considered to be in the luxury market and the stamp duty was 6.5% of their value.
Find the amount of stamp duty payable for cars with the following values.
(d) $54 000 ____________________________________________________________
(e) $140 000 ____________________________________________________________
(f) $550 000 ____________________________________________________________
4. The stamp duty calculations on cars in NSW at the same time were 3% for the first
$45 000 of value then a further 5% for any value above $45 000.
Calculate the stamp duty in NSW for the cars with the following values.
(a) $24 999 ____________________________________________________________
(b) $140 000 ____________________________________________________________
5. Compare your answers from question 4 to (c) and (e) in question 3. Comment onwhich state it was better to buy a car in at that time.
_______________________________________________________________________
_______________________________________________________________________
Check your work before continuing.
Percentages Year 7 Mathematics
Page 42 © Department of Education WA 2012 – MATHSAC018
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 43
7. Solving harder problemsWhen you complete this section you should be able to:
solve harder problems involving percentages.
Warm-up 7
1. 44.5 10 = _________
2. 17 5 = _________
3. The temperature is 8 degrees.
How much will it need to decrease to get to minus 2 degrees? _________
4.6 1
10 2 _________
5. 1
436 = _________
6. 0.4 L = ________ mL
7. (12 – 5) 3 = _________
8. Find 10% of $60. ____________
9. Describe the rule for the following pattern.
5, 10, 15, 20, 25, …
________________________________________________________________
10.
Determine the size of the missing angle.
_________
62°?
Percentages Year 7 Mathematics
Page 44 © Department of Education WA 2012 – MATHSAC018
Focus problem 7
A retailer pays $45 each for some new MP3 players from the manufacturer.
She then adds a 32% profit margin to the price before advertising them.
Raylene comes to buy one and finds that they have been discounted by 10%. How much does Raylene have to pay? Show your working and reasoning in the space below.
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Terry wants to buy one too. He has saved up $48.50. His mum said that if he had saved at least 90% of the sale price she would pay the rest. Will Terry be able to buy the MP3 player?
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Check your work before continuing.
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 45
Skills development 7
1. Increase the prices of the following items by 15% and then find the sale price if adiscount of 12% of the new price is given.
(a) $200 watch
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
(b) $36 backpack
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
(c) $15 book
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
2. Explain why the answer to each of the parts in question 1 cannot be simply found bysaying: 15% – 12% = 3% so find a 3% increase on the original price.
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
3. The price of a music DVD was $25 but it was on sale for $17.50. What was thepercentage discount?
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
Percentages Year 7 Mathematics
Page 46 © Department of Education WA 2012 – MATHSAC018
4. A new chocolate bar is advertised as being 20% bigger for the same price. If the originalbar was 120 g, what is the weight of the new one? Include a check to be sure youranswer is correct.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
5. The Brooks family left for their holidays. They had to travel 948 km altogether.
(a) They hoped to travel 35% of the distance on the first day. How far should that be?
_______________________________________________________________________
_______________________________________________________________________
(b) The second day they wanted to travel a further 237 km. What percentage of the totaldid they want to travel on the second day?
_______________________________________________________________________
_______________________________________________________________________
(c) What percentage of the trip did they plan to cover on the third (and last) day?
_______________________________________________________________________
_______________________________________________________________________
6. Jean had 19 pens and six of them were red. Frank had 25 pens and seven of themwere red. Who had the larger percentage of red pens? Show the working you need tofind the answer.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Check your work before continuing.
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 47
8. Summary A percentage is equivalent to a fraction with a denominator of 100.
1 whole = 100
100%100
To change a decimal to a percentage multiply the decimal by 100 and add a percentsign to your answer.
To change a fraction to a percentage find the decimal equivalent by dividing thenumerator by the denominator then multiply the decimal by 100 and add a percentsign to your answer.
To change a percentage to a fraction write the number with a denominator of 100then simplify.
To change a percentage to a decimal divide the percentage number by 100, that is,move the decimal point two places to the left.
To find a percentage of a quantity choose one of the following.
Multiply the quantity by the percentage number and then divide by 100. This is the same as multiplying by the fraction equivalent with a denominator of 100.
Multiply the quantity by the percentage written as a decimal.
Multiply the quantity by the fractional equivalent of the percentage.
To find a percentage increase of a quantity choose one of the following.
Find the increase and add it to the original value.
Add the percentage to 100% and then calculate the increased percentage of the quantity.
To find a percentage decrease of a quantity choose one of the following.
Find the decrease and take it from the original value.
Take the percentage from 100% and then calculate the decreased percentage of the quantity.
To write one quantity as a percentage of another first write the relationship as afraction then convert the fraction to a percentage. The quantities must be in the sameunits.
When solving word problems you need to do the following.
Read the question carefully.
Decide on the methods you will need to use.
Show as much working as possible to make your calculations clear.
Write only mathematically correct statements.
Include words to explain your answers.
Percentages Year 7 Mathematics
Page 48 © Department of Education WA 2012 – MATHSAC018
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 49
9. Review tasksThe following tasks will assist you to consolidate your learning and understanding of the concepts introduced in this resource, and assist you to prepare for assessments.
Task A
Name: _____________________________ Suggested time: 45 minutes
Actual time taken: __________
Instructions
Complete this work on your own.
You may use a calculator, but show how you got your answer.
Attempt every question. Take as long as you need and record the time in the space provided above after you have finished.
1. Complete the following.
(a) Write 0.065 as a percentage.
_____________________________________________________________________
(b) Find the percentage equivalent to 5
.16
_____________________________________________________________________
(c) Write each of the following percentages as a simple fraction.
(i) 25% __________ (ii) 10% __________
(iii) 1
212 % __________ (iv)
1
333 % __________
(d) Find the decimal equivalent to 1.2%. ___________________________________
2. Find the following values.
(a) 14% of 25 km
_____________________________________________________________________
_____________________________________________________________________
Percentages Year 7 Mathematics
Page 50 © Department of Education WA 2012 – MATHSAC018
(b) 1
333 % of 42 litres
_____________________________________________________________________
_____________________________________________________________________
3. (a) The price of a pair of shoes was increased by 5%.
(i) What percentage of the original price is the new price? __________________
(ii) What is the new price of the shoes if the original price was $120?
____________________________________________________________________
____________________________________________________________________
(b) The capacity of a water tank was reduced by 3.5% because of silt in the bottom ofthe tank.
(i) What percentage of the capacity remains? ___________________
(ii) What is the capacity of the tank now if it originally held 6000 litres of water?
____________________________________________________________________
____________________________________________________________________
4. The birth weight of a baby was 3.2 kg. After one month the baby weighed 4 kg. Whatpercentage of its birth weight was the increase in the first month?
_______________________________________________________________________
_______________________________________________________________________
5. Last year the population of the town of Needham was 5650 people. This year thepopulation increased to 6200 due to the opening of a new research farm. Theneighbouring town of Kilton had its population increase from 4225 to 4800 at the sametime.
Which town had the bigger percentage increase in population? Show all necessaryworking.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 51
6. The following table shows the suggested number of grams of protein youngpeople should be consuming per day on a healthy diet.
Ages Grams of protein
Males 9 to 13 35
Males 14 to 18 52
Females 9 to 13 34
Females 14 to 18 45
Edward is a 12 year old male. His protein intake on Saturday was from a bowl of cereal (3 g) with milk (6 g) for breakfast and a burger (28 g) for lunch.
Leith is a 15 year old female. Her protein intake on Saturday was from a boiled egg (6 g) with a slice of toast (2 g) for breakfast and salad with cheese (14 g) for her
lunch.
What percentage of their daily protein requirements have they already met? Approximately what percentage of the suggested protein requirement will they need to eat for dinner to meet the requirement?
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Percentages Year 7 Mathematics
Page 52 © Department of Education WA 2012 – MATHSAC018
Task B
Name: _____________________________ Suggested time: 40 minutes
Actual time taken: __________
Instructions
Complete this work on your own.
You may use a calculator, but show how you got your answer.
Attempt every question. Take as long as you need and record the time in the space provided above after you have finished.
1. Percentages are used by banks to calculate interest. You can get a percentage of yoursavings paid to you in interest for allowing the bank to use your money.
One type of bank account is a term deposit. You can put a sum of money in there and getpaid the interest at the end of a chosen time period.
The interest can be paid to you or put back into the account to increase the amount in theaccount balance.
Annie has put $10 000 into a one year term deposit at an interest rate of 5%.
(a) How much interest will she get at the end of the first year?
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
(b) If she decides to pay the interest back into the account what will her new accountbalance be?
_______________________________________________________________________
Annie continues with this new amount for another one year term deposit at an interest rate of 5%.
(c) How much will the interest be for the second year?
_______________________________________________________________________
_______________________________________________________________________
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 53
(d) If she continues doing the same thing for another two years, how much will heraccount balance be after four years? Show your working.
_______________________________________________________________________
______________________________________________________________________
_______________________________________________________________________
______________________________________________________________________
_______________________________________________________________________
______________________________________________________________________
_______________________________________________________________________
______________________________________________________________________
_______________________________________________________________________
______________________________________________________________________
(e) How much interest has Annie gained altogether over the four years?
_______________________________________________________________________
2. Many items that are bought, especially for businesses, suffer from depreciation. Thismeans that the value at the end of the year has decreased by a certain percentage of thevalue at the beginning of the year.
A company buys a computer for $500 and the depreciation is set at 10%, for taxpurposes. This means that the value of the computer at the end of the year is 10% lessthan the value at the start of the year.
Work through the necessary steps to find out how much the computer is worth after threeyears.
_______________________________________________________________________
______________________________________________________________________
_______________________________________________________________________
______________________________________________________________________
_______________________________________________________________________
______________________________________________________________________
_______________________________________________________________________
______________________________________________________________________
Percentages Year 7 Mathematics
Page 54 © Department of Education WA 2012 – MATHSAC018
_______________________________________________________________________
______________________________________________________________________
_______________________________________________________________________
______________________________________________________________________
_______________________________________________________________________
______________________________________________________________________
_______________________________________________________________________
______________________________________________________________________
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 55
Self-evaluation task Please complete the following.
How well did you manage your own learning using this resource?
Always Usually Rarely Not sure
Each section took approximately 45 minutes to complete.
I needed extra help.
I marked and corrected my work at the end of each section.
I made the journal entries and summaries when asked.
I have kept to my work schedule.
How much mathematics have you learnt using this resource?
Always Usually Rarely Not sure
Understanding I can recognise the equivalent fractions, decimals and percentages.
I understand how to calculate a percentage of a quantity.
Fluency
I can write one quantity as a percentage of another.
I can calculate a percentage of a quantity
Problem Solving I can write the correct calculations needed to solve a word problem.
I can solve word problems involving percentages.
Reasoning
I can choose the easiest method to solve a problem.
I can compare percentage changes and decide on the best.
Percentages Year 7 Mathematics
Page 56 © Department of Education WA 2012 – MATHSAC018
Write a list of topics for which you need additional assistance. Discuss these with your teacher.
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 57
Solutions
1. What is a percentage?
Solutions to Warm-up 1
1. Yes, 4 is a factor of 20.
2. 12
3. a = (-6)
4.6
6 should be circled.
5. 9
6. 1.7
7. 27.0 or 27
8.1
3
9. 25
10. (3, 4)
Solutions to Review 1
1. (a) 9
100
(b)88
100
(c)20
100
(d)24
100
(e)6
100
2. (a) 5 87 92
100 100
(b)13 8 29 50
100 100
(c)51 15 36
100 100
(d)97
0.97100
Percentages Year 7 Mathematics
Page 58 © Department of Education WA 2012 – MATHSAC018
(e)57
0.20 0.37 0.57100
(f)3 4 30 8 38
10 50 100 100 100
Solution to Focus problem 1
You were asked to complete the table of percentages for categories of basketball players.
Female players 40%
Male players 60%
Blond haired players 64%
Players without blond hair 36%
Players over 185 cm tall 15%
Players not over 185 cm tall 85%
Players wearing green uniforms 23%
Players wearing other colour uniforms apart from green 77%
All the players 100%
Solutions to Skills development 1
1. (a) 26%
(b) 100 – 18 – 26 – 41 = 15. 15% of the squares are uncoloured.
(c) 100 – 18 = 82. 82% of the squares are not red.
(d) 100 – (18 + 41) = 41. This means that 41% of the squares are not red or yellow.
(e) There are no purple squares so the percentage of purple squares is 0%.
2. (a)
(b) (i) Percentage shaded = 39%
(ii) Percentage unshaded = 61%
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 59
3. (a) 50 squares
(b) One square is 1
50 of the grid so it is 2%.
(c) (i) Percentage unshaded = 24 × 2% = 48%
(ii) Ten squares are shaded with \\\\, this is 20% of the grid.
(iii) Four squares are shaded with a solid colour. This is 8% of the grid.
(iv) Twelve squares are crosshatched which is 24% of the grid.
4. (a) 20
(b) 10
(c) 5
(d) 40
(e) 200
2. Fraction and decimal equivalents
Solutions to Warm-up 2
1. Yes, 4 is a common factor.
2. 8
3. (-3) degrees
4.1 2
2 4
5. 5
6. 7
7. 2.1
8. 1.5
9. 3.5
10.3 1
6 2
Solutions to Review 2
1. (a) (i) 94
100
(ii)19
100
(iii)30
100
Percentages Year 7 Mathematics
Page 60 © Department of Education WA 2012 – MATHSAC018
(iv)176 4 44
400 4 100
(v)7 4 28
25 4 100
(b) (i) 0.65
(ii) 0.34
(iii) 0.5625
(iv) 0.625
Solution to Focus problem 2
(a) (i) 1 25
25%4 100 and
270.27 27%
100
(ii) Smallest to largest percentages are 22%, 25% then 27%.
(ii) Hugh had the largest percentage.
(b) (i) 11
0.27540
(ii) To change 0.275 to a percentage you move the decimal point two places right(or multiply by 100). You get 27.5%.
(c)
Pocket money Amount spent on music Calculation Percentage
$10 $2.70 (2.7 ÷ 10 × 100) 27%
$25 $7.50 (7.5 ÷ 25 × 100) 30%
$32 $28.00 (28 ÷ 32 × 100) 87.5%
Solutions to Skills development 2
1. (a) 0.61 = (0.61 × 100)% = 61%
(b) 0.04 = (0.04 × 100)% = 4%
(c) 1.13 = (1.13 × 100)% = 113%
(d)2
5 = (2 ÷ 5 × 100)% = 40%
(e)5
16 = (5 ÷ 16 × 100)% = 31.25%
(f)2
75 = (2 ÷ 75 × 100)% = 2.666…%
2. (a) 45% = 45 9
100 20
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 61
(b) 0.5% = 0.5 1
100 200
(c) 125% = 125 25 1
1 1100 100 4
3. (a) 23% = 23 ÷ 100 = 0.23
(b) 1.5% = 1.5 ÷ 100 = 0.015
(c) 2
366 % 66.666... 100 0.666...
4. (a) All written as decimals gives 0.47, 0.45, 0.48.
In order the values are 0.45, 47%, 12
.25
(b) All written as percentages as follows.
19(19 20 100)% 95%
20 92% and 94.8% stay the same
15(15 16 100)% 93.75%
16 0.975 = (0.975 × 100)% = 97.5%
Arranged in order from the smallest to the largest: 92%, 15
16, 94.8%,
19
20, 0.975.
3. Finding a percentage of a quantity
Solutions to Warm-up 3
1. 9 should be circled.
2. 63
3. a = (-2)
4.
5. 5
6. 9 – 4 = 5
7. 4.8
8. 20%
9. 7
81
10. Rectangle
0 1
Percentages Year 7 Mathematics
Page 62 © Department of Education WA 2012 – MATHSAC018
Solutions to Review 3
1. (a) 1
8of 72 litres = (72 × 1 ÷ 8) litres = 9 litres
(b)3
4of 72 litres = (72 × 3 ÷ 4) litres = 54 litres
(c) 0.6 of 72 litres = (72 × 0.6) litres = 43.2 litres
(d) 0.09 of 72 litres = (72 × 0.09) litres = 6.48 litres
2. (a) 0.15 of $70 = $(70 × 0.15) = $10.50
(b)2
3of 51 kg = (51 × 2 ÷ 3) kg = 34 kg
(c) 0.06 of 150 min = (150 × 0.06) min = 9 minutes
(d)11
25of 94.5 km = (94.5 × 11 ÷ 25) km = 41.58 km
Solution to Focus problem 3
1. (a) 5
(b) 5 × 2 =10
(c) 5 × 3 =15
(d)5
100× 260 = 13
(e)5
100 × 480 = 24
2. (a) 2
(b) 4
(c) 1
(d) 11
3. (a) Multiply the number of trees by 5 then divide by 100 or divide the number of treesby 100 then multiply by 5.
(b) Multiply the number of trees by 2 then divide by 100 or divide the number of treesby 100 then multiply by 2.
Solutions to Skills development 3
1 (a) 20% = 0.2. 20% of $35 = $(0.2 × 35) = $7
(b) 20% of $35 = 20
$35 $(35 20 100) $7100
2. (a) (27 × 0.15) litres = 4.05 litres
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 63
(b)1
4724 people = 118 people
(c) (15 × 9 ÷ 100) km = 1.35 km
(d) $(2300 × 1.2 ÷ 100) = $27.60
3. House and sheds: 1.1% of 540 hectares = 5.94 hectares
Cattle grazing: 28 % of 540 hectares = 151.2 hectares
Rare orchids: 2% of 540 hectares = 10.8 hectares
Uncleared scrub: 17% of 540 hectares = 91.8 hectares
4. Finding a percentage of a quantity again
Solutions to Warm-up 4
1. 17
2. 6
3. 2 degrees
4.2
3w
5. 10
6. 5.201
7. 2.06
8.1
2
9. 42
10. 140°
Solutions to Review 4
1. (a) (75 × 3 ÷ 5) kg = 45 kg
(b) (105 × 3 ÷ 5) mins = 63 mins
(c) $(13 × 3 ÷ 5) = $7.80
2. (a) (500 × 0.07) ha = 35 ha (b) (56 × 0.07) kg = 3.92 kg
(c) $(4.20 × 0.07) = $0.29 (to the nearest cent)
Solution to Focus problem 4
1. Answers will vary. Check with your teacher.
2. In Skills development 2 the percentage 1
333 % is written as a decimal.
Percentages Year 7 Mathematics
Page 64 © Department of Education WA 2012 – MATHSAC018
3. 1 1
3 333 %
4.
Percentage Fraction
100% 1
10% 1
10
5% 1
20
25% 1
4
1
212 %
1
8
1
333 %
1
3
5. (a)1
2
312.5% 3 37.5% or 37 %
8
(b) 1
2
512.5% 5 62.5% or 62 %
8
6. (a)2 2
3 366 %
(b) 4 1 1
3 3 31 133 %
7. Deposit = 1
3$264 $88
Solutions to Skills development 4
1. (a) 1
3 of 6 kg = 2 kg
(b)1
3 of 36 litres = 12 litres
(c)1
3 of 24 hours = 8 hours
(d)1
3 of 336 km = 112 km
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 65
2. (a) 10% of $60 = 1
$60 $610
(b) Reduced price = $60 – $6 = $54
(c) 90% of $60 = 0.9 $60 $54
(d) The original price would be 100% of the price so when you reduce it by 10% it isthe same as finding 90% of the original price.
3. (a) 15% of $80 = 15
$80 $12100
(b) Increased price = $80 + $12 = $92
(c) 115% of $80 = 1.15 $80 $92
(d) The original price would be 100% of the price so when you increase it by 15% it isthe same as finding 115% of the original price.
4. Jackets will be sold for 85% of the original price because 100% – 15% = 85%.Shoes will be sold for 78% of the price because 100% – 22% = 78%.
5. New price for books = 110% of the old price.New price for other items = 105% of the old price.
6. (a) 15% reduction gives 85% of the original. 85% of 90 kg = 0.85 × 90 kg = 76.5 kg
(b) 123% of 800 litres = 1.23 × 800 L = 984 L
(c) 120% of 2600 seats = 1.2 × 2600 seats = 3120 seats
(d) 100% – 2.4% = 97.6%97.6% of 15 metres = 0.976 × 15 metres = 14.64 metres
5. Writing one quantity as a percentage of another
Solutions to Warm-up 5
1. 6 is a triangular number (1 + 2 + 3 = 6)
2. 43
3. q = (-2)
4.3
5
5. 3
6. 600
7. 6
8. 0.1
9. 7.7
10. (5, 2)
Percentages Year 7 Mathematics
Page 66 © Department of Education WA 2012 – MATHSAC018
Solutions to Review 5
1 (a) 18 1
36 2 (b)
4 1
36 9 (c)
22 11
36 18
2. (a) 45 9
55 11 (b)
45 5
81 9 (c)
45 9
100 20
Solution to Focus problem 5
(a)53
100
(b) 53%
(c)36 6
66 11
(d) (6 ÷ 11 × 100)% = 54.5454…% = 54.5% (1dp)
(e) She was serving better in the second match.
(f) With the percentages she could easily compare howwell she served in each match.
Solutions to Skills development 5
1 (a) 60
60%100
(b)6 0
100 06%
(c)60 30
30%200 100
(d)60
(6 54 100)% 11.11...% 11.1% (1dp)540
2. Total weight picked = (38 + 27 + 15) kg = 80 kg
Percentage Maria picked = (38 ÷ 80 × 100)% = 47.5%
Percentage Kevin picked = (27 ÷ 80 × 100)% = 33.75%
Percentage Helen picked = (15 ÷ 80 × 100)% = 18.75%
3. Percentage of water = 380
76%500
Percentage of flavouring = 30
6%500
Percentage of skim milk = 100% 76% 6% 18%
Note that a check is often a good idea in a question like this.
Note that in 2(c) the metre was changed to
100 cm.
Notice that if the percentage has a repeating decimal in
the answer it is usually rounded.
Unless the answers are really easy to find you should be using your calculator for these
answers.
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 67
Check: Skim milk is 90 mL because (500 – 380 – 30) mL = 90 mL.
Percentage of skim milk = 90
18%500
4. (a) Increase = $(22 – 16) = $6
(b) Increase as a percentage = 6
15100 % 37.5%
5. Reduction in price of caps as a percentage = 6
15100 % 40%
Reduction in price of trainers as a percentage = 1
3
16
48100 % 33 %
Reduction in price of model planes as a percentage = 10
28100 % 35.7% (1dp)
6. Problem solving
Solutions to Warm-up 6
1. 108
2. 19
3. 4 degrees
4.4 1
8 2
5. 2
6. 900
7. 4
8. 50%
9. 1
41
10. 25%
Solutions to Review 6.
1. (a) 60% = 60 3
100 5 and 60% = 0.6
(b) 60% of 900 kg = 0.6 × 900 kg = 540 kg
(c) The new seating capacity is 160% of the old seating.
(d)42 7
70%60 10
Percentages Year 7 Mathematics
Page 68 © Department of Education WA 2012 – MATHSAC018
2. (a) 1
2
112 %
8 and
1
8of 80 kg = 10 kg.
(b)1
8 of 9.6 metres = 1.2 metres
Solution to Focus problem 6
New rates bill: 110% of $1200 = 1.1 × $1200 = $1320
With a 10% discount the family will pay 90% of $1320.
90% of $1320 = 0.9 × $1320 = $1188
The 10% reduction for paying quickly was 10% of the larger amount, $1320, so it was more than 10% of $1200.
Solutions to Skills development 6
1. There are 24 hours in a day
Activity Working Percentage
Sleeping 8 1
24 3
1%
333
School 6 1
24 4 25%
Computer 3 1
24 8
1
212 %
Eating 2 1(1 12 100)%
24 12
1
38.333...% 8 %
Basketball training 5(5 24 100)%
24 20.8% (1dp)
2. (a) 12% increase mean that the total will be 112% of 45 km. 112% of 45 km = 1.12 × 45 km = 50.4 km
(b) After they have the reduction they will have 92% of the original number ofemployees.92% of 325 = 0.92 × 325 = 299 employees
3. (a) 2.75% of $2500 = 0.0275 × $2500 = $68.75
(b) 2.75% of $12 000 = 0.0275 × $12 000 = $330
(c) 2.75% of $24 999 = 0.0275 × $24 999 = $687.47 (to the nearest cent)
(d) 6.5% of $54 000 = 0.065 × $54 000 = $3510
(e) 6.5% of $140 000 = 0.065 × $140 000 = $9100
(f) 6.5% of $550 000 = 0.065 × $550 000 = $35 750
4. (a) 3% of $24 999 = $749.97
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 69
(b) 3% of $45 000 + 5% of $(140 000 – 45 000)
= 0.03 × $45 000 + 0.05 × $95 000
= $1350 + $4750
= $6100
5. Possible comment: Cheaper cars have lower stamp duty in Western Australia but it isbetter to buy expensive cars in NSW because the stamp duty is lower.
7. Solving harder problems
Solutions to Warm-up 7
1. 4.45
2. 85
3. 10 degrees
4.6 5 1
10 10 10
5. 9
6. 400 mL
7. 21
8. $6
9. Add five to get the next number.
10. 62°
Solution to Focus problem 7
Advertised price for the MP3 player = 132% of $45
= 1.32 × $45
= $59.40
Raylene's price = 90% of $59.40 = $53.46
There are several ways of finding if Terry can buy one too. Get your working checked if it differs from the answer below.
The easiest way to find the answer is to find 90% of $53.46 and see if Terry's $48.50 is enough.
90% of $53.46 = 0.9 × $53.46 = $48.11 (to nearest cent). Terry will be able to buy the player as he has saved up at least 90% of the sale price.
Solutions to Skills development 7
1. (a) 115% of $200 = 1.15 × $200 = $23088% of $230 = 0.88 × $230 = $202.40
Percentages Year 7 Mathematics
Page 70 © Department of Education WA 2012 – MATHSAC018
(b) 1.15 × $36 = $41.400.88 × $41.40 = $36.43 (nearest cent)
(c) 1.15 × $15= $17.250.88 × $17.25 = $15.18
2. The percentage discount is on a different amount than the percentage increase so theyhave to be calculated separately.
3. Discount = $25 – $17.50 = $7.50
Discount percentage = 7.5
25100 % 30%
4. The new chocolate bar will be 120% of the original.120% of 120 g = 1.2 ×120 g = 144 gThe new bar weighs 144 g
Check: 20% of 120 g = 1
120 g5 = 24 g and 120 g + 24 g = 144 g
5. (a) 35% of 948 km = 0.35 × 948 km = 331.8 km
(b) 237
948100 % 25%
They wanted to travel 25% of the trip on the second day.
(c) Percentage of the trip on the third day = (100 – 35 – 25)% = 40%
6. Percentage of red pens for Jean = 6
19100 % 31.6% (1dp)
Percentage of red pens for Frank = 7
25100 % 28%
Jean had the larger percentage of red pens.
Solutions to Review tasks
Solutions to Task A
1. (a) 6.5%
(b) 31.25% or 1
431 %
(c) (i) 1
4
(ii)1
10
(iii)1
8
(iv)1
3
Year 7 Mathematics Percentages
© Department of Education WA 2012 – MATHSAC018 Page 71
(d)1.2
0.012100
2. (a) 14% of 25 km = 0.14 × 25 km = 3.5 km
(b) 1
333 % of 42 litres =
1
3 × 42 litres = 14 litres
3. (a) (i) 105%
(ii) 105% of $120 = 1.05 × $120 = $126
(b) (i) 96.5%
(ii) 96.5% of 6000 litres = 0.965 × 6000 litres = 5790 litres
4. Change in weight of baby = (4 – 3.2) kg = 0.8 kg
Percentage change = 0.8
3.2100 % 25%
The weight of the baby increased by 25% in the first month.
5. Change in population for Needham = 6200 – 5650 = 550
Percentage change = 550
5650100 % 9.7%
(1dp)
Change in population for Kilton = 4800 – 4225 = 575
Percentage change = 575
4225100 % 13.6%
(1dp)
Kilton had the bigger percentage increase in population. 6. Total of protein for Edward = (3 + 6 + 28) g = 37 g
Percentage of daily requirements = 37
35100 % 105.7%
(1dp)
Total of protein for Leith = (6 + 2 + 14) g = 22 g
Percentage of daily requirements = 22
45100 % 48.9%
(1dp)
Edward has already consumed more protein than the daily requirement. Leith will need to consume approximately 51.1% for dinner to meet the daily
requirement.
Solutions to Task B
1. (a) 5% of $10 000 = 0.05 × $10 000
= $500
Annie will get $500 interest at the end of the first year.
(b) New balance = $10 500
(c) 5% of $10 500 = 0.05 × $10 500
= $525
The interest for the second year is $525.
Percentages Year 7 Mathematics
Page 72 © Department of Education WA 2012 – MATHSAC018
(d) New balance = $10 500 + $525 = $11 025
5% of $11 025 = 0.05 × $11 025
= $551.25
Balance for third year = $11 025 + $551.25 = $11 576.25
5% of $11 576.25 = 0.05 × $11 576.25
= $578.81 (nearest cent)
Final balance = $12 155.06 (nearest cent)
(e) Interest over four years = $12 155.06 – $10 000 = $2155.06
2. Year 1: Value at start of year = $500
10% of $500 = 1
10× $500
= $50
Value at the end of the year = $500 – $50
= $450
Year 2: Value at start of year = $450
10% of $450 = 1
10× $450
= $45
Value at the end of the year = $450 – $45
= $405
Year 3: Value at start of year = $405
10% of $405 = 1
10× $405
= $40.50
Value at the end of the year = $364.50
The computer will have a value of $364.50 after three years.
MATHSAC018 PERCENTAGES
ISBN: 9780730744382