Nat 5
USAP 1(a)
This booklet contains :



Questions on Topics covered in RHS USAP 1(a)
Exam Type Questions
Answers
Sourced from PEGASYS
USAP 1(a)
EF 1.3
Rounding to a given number of significant figures
APP 3.1
Working with percentages
 Use reverse percentages to calculate an original quantity
 Appreciation including compound interest
 Depreciation
APP 3.2
Working with fractions
 Operations and combinations of vulgar fractions including mixed
numbers
EF 3.3
Working with the volume of a solid sphere, cone, pyramid
2
EF 1.3 SIGNIFICANT FIGURES
1.
(a)
20
(b)
6
(c)
80
(g)
300
(h)
800
(i)
8000 (j)
2000 (k)
8000 (l)
5000
(m)
11
(n)
600
(o)
4
(p)
10000 (q)
1
(r)
90
(s)
0∙9
(t)
600
(a)
8∙7
(b)
93
(c)
0∙19
(d)
680
(e)
2∙1
(f)
6∙5
(g)
31
(h)
26
(i)
24
(j)
19
(k)
6400 (l)
5∙0
(m)
0∙053 (n)
0∙0061(o)
14000 (q)
2∙5
(r)
45000
(s)
29
0∙76
(a)
49∙3
(b)
2∙35
(c)
0∙593
(d)
4770
(e)
6∙08
(f)
24200
(g)
0∙0628
(h)
29∙5
(i)
0∙00947
(j)
56200
(k)
0∙0980
(l)
24∙5
(m)
28∙3
(n)
2460
(o)
3170
(p)
30∙0
(q)
2∙68
(r)
3090
(s)
2∙01
(t)
0∙000318
4.
(a)
248400
(b)
248000
(c)
250000
(d)
200000
5.
(a)
0∙02860
(b)
0∙0286
(c)
0∙029
(d)
0∙03
6.
(a)
120
(b)
4∙0
(c)
250
(d)
41
(e)
49
(f)
0∙49
(g)
3∙8
(h)
0∙084
(i)
250
(j)
17
(k)
500
(l)
65
(a)
133
(b)
4∙78
(c)
56∙5
(d)
988
(e)
8∙78
(f)
334
(g)
19∙8
(h)
26∙3
(i)
0∙965
(j)
326
(k)
2∙07
(l)
0∙0965
2.
3.
7.
8.
(t)
(d)
0∙087 (p)
30
(e)
100
(f)
300
2∙98  108
3
APP 3.1
WORKING with PERCENTAGES
Use reverse percentages to calculate an original quantity.
1.
2.
3.
These amounts have been reduced by 15%. What was the original amount?
(a)
£85
(b)
212∙5 mm
(c)
£63.75
(d)
25∙5 litres
(e)
357 miles
(f)
435∙2 m
(g)
1 275 km
(h)
£4 462.50
(i)
10 200 m
(j)
605∙2 cm
(k)
£658.75
(l)
76∙5 kg
These amounts have been increased by 22%. What was the original amount?
(a)
£26.84
(b)
£54.90
(c)
£87.84
(d)
103∙7 ml
(e)
£21.35
(f)
122 cm
(g)
3 111 m
(h)
10 370 km
(i)
68∙32 m
(j)
£13 664
(k)
118∙95 litres
(l)
£7 564
A shop is having a sale. There is ‘20% OFF’. Calculate the original cost of these items.
(a)
(b)
£52
£32
(d)
(c)
(e)
£20
(f)
£340
£239.20
£5.60
(g)
£1 640
(h)
(i)
£20.80
£21.20
4
4.
A company gave their workers a 7% wage rise. Calculate how much each of these people
were earning each year before the increase.
(a)
Irene
£13 375 per year
(b)
Billy
£19 324.20 per year
(c)
Peter
£26 322 per year
(d)
Isobel
£40 060.80 per year
(e)
Stewart
£481.50 per week
(f)
Jackie
£1 820 per month
(g)
Alan
£75 per week
(h)
Anne
£1 200 per month
EXAM QUESTIONS
1.
A gym’s membership has increased by 17% over the past year. It now has 585 members.
How many members did it have a year ago?
2.
The number of school pupils not wearing school uniform has decreased by 72% since the
start of last year. There are now 42 pupils not wearing school uniform.
How many pupils were not wearing school uniform at the start of last year?
3.
My house has increased in value by 15% in the last two years. It is now worth £230 000.
How much was it worth 2 years ago?
4.
I bought a new car in September of last year. By this September the car had
depreciated by 20% and was now worth £9600.
How much did I pay for the car last September?
5.
Jane bought a painting in an auction. Unfortunately the painting depreciated in value by
7% and is now worth £4185.
How much was the painting worth when it was bought?
6.
An antique chair has increased in value by 34% since it was bought. It is now worth £3
484.
What was it worth when it was bought?
5
APPRECIATION and DEPRECIATION
1.
2.
For each of the investments below , calculate
(i)
the amount due at the end of the term
(ii)
the total interest
Bank/ Building Society
Amount
Invested (£)
Rate of interest
(per year)
Number of
Years
(a)
Hamilton Bank
2000
8%
2
(b)
Allied Friendly
5000
6%
3
(c)
Northern Hill
4800
7%
2
(d)
Highland Bank
3500
7∙5 %
3
(e)
Church National
1600
5∙5 %
4
(f)
Southern Rock
1750
11 %
3
(g)
London Savings Bank
20 000
6%
3
(h)
Bath & Eastern
18 000
8∙5%
2
(i)
Royal Bank of Britain
50 000
9%
3
(j)
Bingford & Bradley
400
4∙8%
2
At the beginning of the year, Mr. Bradford borrows £5000 from the bank. The rate of
compound interest is 8%. He agrees to pay back £108 per month.
Calculate how much he still owes at the end of the second year.
3.
The Smiths buy a house for £60,000. If it appreciates in value at the rate of 9% per year,
how much will it be worth in 5 years time?
4.
Amanda wins some money and decides to spend £200 on some jewellery. If it appreciates
at the rate of 2% per year, how much will the jewellery be worth 3 years from now?
5.
In 1990 the world population was estimated to be 5300 million, and was increasing at the
rate of 1∙7% per annum.
What will the population be in the year 2000? (answer to 2 significant figures)
6
6.
Peter buys a car for £3000. If it depreciates at the rate of 20% per annum, how much will
he be able to sell it for in 3 years time?
7.
Brian buys a new car costing £12600. It depreciates in value by 30% in the first year
and by 20% each year after that.
How much will he be able to trade it in for in 3 years time
8.
Each year a factory’s machinery depreciates by 25% of its value at the beginning of the
year. The initial value of the machinery was £360 000.
(a)
What was the value of the machinery after 1 year
(b)
The machinery was to be scrapped at the end of the year when its value fell below
half its original value. After how many years should the machinery be scrapped?
EXAM QUESTIONS
1.
Joseph invests £4500 in a bank that pays 6∙4% interest per annum.
If Joseph does not touch the money in the bank, how much interest will he have gained
after 3 years?
Give your answer to the nearest penny.
2.
Jane bought a painting in an auction for £32 250.
Unfortunately the painting depreciated in value by 7% each year.
Calculate how much the painting was worth after 2 years.
Give your answer to 3 significant figures.
3.
Non calculator
Last year (2008) a company made a profit of £1 000 000. This year (2009) it expects to
increase its profit by 20% and by 2010 to have increased it by a further 25%.
Calculate the profit the company expects to make in 2010.
4.
A patient in hospital is given 200mg of a drug at 0900. 12% of the amount of the
drug at the beginning of each hour is lost, through natural body processes, by the
end of that hour.
How many mg of the drug will be lost by 1200?
7
5.
Holly buys an antique watch costing £1200. The watch appreciates in value by 3·7% per
annum.
How much will the watch be worth in 4 years time?
Give your answer to the nearest pound.
6.
A local council recycles 28 000 tonnes of glass each year. After a publicity campaign
they expect to increase the amount of glass recycled by 12% each year.
(a)
How much glass do they expect to recycle in 3 years time?
Give your answer correct to 3 significant figures.
(b)
The council aim to double the amount of glass recycled in 6 years.
If this rate is maintained, will the council meet their target?
Give a reason for your answer.
7.
Non calculator
Arthur’s new car cost him £15 000. The value of it will depreciate by 20% each year.
How much will Arthur’s car be worth when he trades it in for a new one in 2 years time?
8.
Barry bought a house last year costing £115 000. This year it is valued at £110 400.
(a)
Calculate the percentage decrease in the value of the house.
(b)
If the value of the house continues to decrease at this rate what will the house
be worth in a further 3 years time?
Give your answer to 3 significant figures.
9.
Marcus invested £3000 in a bank which paid 2∙5% interest per year.
(a)
Calculate how much money Marcus would have in his account after 3 years.
(b)
How long would it take for Marcus’ money to increase by 12%?
8
10.
In 2007 a company made a profit of £45 000. Over the next three years its profit
dropped by 3% each year due to increased manufacturing costs.
Calculate, correct to 3 significant figures, the company's profit in 2010.
11.
The value of an industrial machine is expected to decrease each year by 142% of
its value at the beginning of the year.
If it was valued at £15500 at the beginning of 2011, what will its expected value be at
the end of 2013? Give your answer correct to the nearest pound.
12.
The membership of the ‘Watch your Weight’ slimming club is 40 000 and is increasing at
the rate of 4% per month.
The membership of ‘World of Slimming’ is 70 000 but is decreasing at the rate of 9% per
month.
13.
14.
(a)
Calculate the membership of the ‘Watch your Weight’ club after 3 months,
giving your answer correct to 4 significant figures.
(b)
How many months will it take for the membership of the ‘Watch your Weight’
club to be more than the ‘World of Slimming’?
A woman had a Body Mass Index (BMI) of 30. After following a healthy eating
plan she managed to reduce her BMI to 27∙6 in 1 month.
(a)
Calculate the percentage reduction in her BMI.
(b)
If she managed to continue to reduce her BMI by the same percentage in each of
the next 3 months, what was her BMI then? Give your answer correct to 3
significant figures.
The value of an antique chair increased in value by 12½ % each year.
The chair was bought for £4800. What was its value at the end of 3 years?
15.
Non Calculator
Charlene’s house is valued at £120 000 and is expected to appreciate at the rate of 10% per
annum for the next three years.
If this happens, what will the house be valued at in three years time?
9
16.
Three years ago I bought a new car which cost £10 500. An offer from the garage at the
time stated:
“Keep the car for 3 years, return it to us and we will refund half the original cost”
The car depreciated in value by 20% during the first year and by 15% in subsequent
years.
By calculating the value of the car after 3 years decide whether the garage’s offer, in
this case, was a good one or not. Give a reason for your answer.
17.
A piece of jewellery was bought for £2580 two years ago. Its present value is 65% of its
original price.
(a) What is its present day value?
An expert estimates that it will increase in value at a rate of 12% per annum over the
next few years.
(b)
18.
How many years will it take for the jewellery to regain its original value?
Bill invested £10 000 in the Dodgy Building Society but his money lost 5% per annum
over the first 2 years.
At the end of this time he decided to move his money to the Goody Building Society
which guaranteed that his money would gain 6% per annum over the next 2 years.
How much did Bill gain or lose over the four years?
19.
Chocolate fountains have become very popular at parties.
It takes a minimum of 900g of melted chocolate to operate a fountain properly.
On one occasion 2kg of melted chocolate was added to the fountain.
23% of the remaining chocolate was used every 20 minutes.
Was there still enough chocolate left to operate the fountain properly one hour later?
You must show all working and give a reason for your answer.
20.
In 2008 the Portable Phone Company announced that their profits were £850 000.
In the next 3 years their profits increased by 42% each year. How much profit did the
company make in 2011? Give your answer to the nearest thousand.
10
APP 3.2
WORKING with FRACTIONS
Operations and combinations of vulgar fractions including mixed numbers.
ADDITION and SUBTRACTION
1.
2.
3.
Express each sum as a fraction in its simplest form.
(a)
1 3

5 5
(b)
2 1

5 10
(c)
3 1

4 8
(d)
1 2

6 3
(e)
1 2

9 3
(f)
1 1

3 4
(g)
3 1

5 4
(h)
1 1

4 6
(i)
1 5

3 8
(j)
1 2

2 5
(k)
3 1

4 6
(l)
1 3

2 7
(m)
2 1

7 8
(n)
1 3

5 8
(o)
2 3

9 7
(p)
1 3

6 5
Express each sum as a fraction in its simplest form.
(a)
2 4

5 5
(b)
2 9

5 10
(c)
3 5

4 8
(d)
5 2

6 3
(e)
5 2

9 3
(f)
2 3

3 4
(g)
4 1

5 4
(h)
3 5

4 6
(i)
2 7

3 8
(j)
1 4

2 5
(k)
3 5

7 6
(l)
5 5

7 9
(m)
2 7

7 8
(n)
4 3

5 8
(o)
7 3

9 7
(p)
5 4

8 5
(q)
1 1 3
 
2 2 4
(r)
1 3 1
 
2 5 10
(s)
1 5 3
 
2 8 16
(t)
2 1 1
 
3 2 4
(u)
5 1 1
 
6 4 3
(v)
7 1 3
 
12 3 4
(w)
1 1 3
 
6 7 8
(x)
3 2 2
 
4 3 5
Express each difference as a fraction in its simplest form.
(a)
3 1

4 4
(b)
1 1

2 6
(c)
5 2

6 3
(d)
11 5

12 6
(e)
11 2

12 3
(f)
1 1

2 16
(g)
2 1

3 4
(h)
1 2

2 5
11
4.
5.
6.
(i)
7 3

8 16
(j)
4 1

5 2
(k)
3 1

4 2
(l)
7 1

12 3
(m)
5 2

8 5
(n)
5 3

6 5
(o)
7 3

9 7
(p)
7 2

8 3
Express each sum as a fraction in its simplest form.
(a)
1
1
1 1
2
4
(b)
1
3
1 1
2
4
(c)
3 1
2 1
8 4
(d)
1 5
3 1
2 6
(e)
5
1
3 2
8
4
(f)
2
3
5 2
3
4
(g)
3 3
1 1
5 5
(h)
3
5
2 2
8
6
(i)
3
3
5 2
4
8
(j)
1
7
6 2
3
12
(k)
1 5
3 
2 6
(l)
1 3
4 
8 16
(m)
2
7 2

10 5
(n)
2
1
4 3
3 12
(o)
1
11
3
2
16
8
(p)
7 2
5 
9 3
(q)
3
5
3 2
4
12
(r)
2
1
5 2
3
2
(s)
7
1
2 1
8 12
(t)
5
9
5
8
16
8
Express each difference as a fraction in its simplest form.
(a)
3
2
3
(b)
4
7
12
(c)
3
7
8
(d)
1
2
9
(e)
1
1
2
(f)
3
3
4
(g)
2
1
4
(h)
2
5
6
Express each difference as a fraction in its simplest form.
(a)
3 1
3 1
4 2
(b)
7
1
6 4
8
3
(c)
4 1
2 1
5
4
(d)
4
7
1
1
12 3
(e)
4 3
5 1
5 4
(f)
6
11 5
1
12 6
(g)
2 1
4 1
3 7
(h)
3 1
3 1
4 6
(i)
1 1
5 1
3 8
(j)
5
1
5 2
8
2
(k)
8
7
1
4
12
3
(l)
4
(m)
1
1
9 4
3
3
(n)
5
1
8 1
6 12
(o)
2
3
8 1
5 10
(p)
5
1
5 4
9
3
9
4
2
10
5
12
7.
8.
Express each difference as a fraction in its simplest form.
(a)
42
(e)
3
4
4
7
(c)
52
7
5
3
12
6
(g)
4
1
1
5 2
8
2
2 3
2 1
5 4
(b)
73
2
2
5 4
7
5
(f)
7
(i)
1 5
3 1
4 6
(j)
(m)
1
2
9 4
3
3
(n)
2
3
(d)
10  5
2
3
1
1
1
12 3
(h)
3 3
5 1
5 4
(k)
1
1
8 4
4
3
(l)
4
(o)
2
7
8 1
5 10
(p)
1
2
5 4
9
3
3
4
2
10
5
5
3
Tom walked for 1 kilometres and then walked another 2 km.
8
5
How far did he walk in total?
5
2
cm and breadth 1 cm. Calculate its perimeter.
7
5
9.
A rectangle has length 3
0.
Siobhan likes to go to the gym. Last week she went on Monday, Tuesday, Thursday,
Friday, Saturday and Sunday. The table below shows the number of hours she trained on
each of the six days.
MON
1
1
2
TUES
THURS
1
4
3
4
1
FRI
SAT
2
5
4
5
1
SUN
1
3
10
How many hours in total did she spend in the gym last week?
11.
Billy is a long distance lorry driver. One day he drove for 2
drove for another 3
1
hours, had a break and then
2
2
hours.
3
How long did he drive in total?
12.
Peter is walking to school. When he reaches half way he meets Mike. They walk a third of
the way together when they meet Anne. They walk the rest of the way together. What
fraction of the journey is this?
13
13.
A group of friends went to a burger bar.
2
1
of them bought a burger, bought chips and
5
3
the rest bought cola/ What fraction of the group bought cola?
14.
At a school
1
of the time is spent in Mathematics classes and
5
3
is spent in English.
20
(a)
What fraction of the time is spent in Maths and English altogether?
(b)
If
1
of the time is spent in PE, what fraction of the time is spent on all the other
15
subjects apart from the three subjects already mentioned?
MULTIPLICATION and DIVISION
1.
2.
Express each product as a fraction in its simplest form:
(a)
1 4

4 7
(b)
1 3

3 10
(c)
1 4

2 7
(d)
2 1

3 8
(e)
4 1

5 16
(f)
6 2

7 3
(g)
3 10

5 21
(h)
3 4

8 21
(i)
21 4

32 7
(j)
1 12

9 13
(k)
5 6

16 25
(l)
5 14

7 15
(m)
7 12

9 35
(n)
12 39

13 48
(o)
2 5

3 9
(p)
2 3

9 8
(q)
1 3

2 5
(r)
3 2

8 3
(s)
5 8

14 15
(t)
7 5

10 14
Express each product as a fraction in its simplest form:
(a)
1 1
1 1
4 3
(b)
1 2
1 1
4 3
(c)
1
1
2 2
2
2
(d)
3 2
1 1
4 3
(e)
1 1
3 1
4 5
(f)
1
2
1 2
3
3
(g)
1
1
1
2
15
2
(h)
3 1
3 1
4 5
(i)
1
2 5
2
(j)
1
7 4
2
(k)
1 1
2 1
7 3
(l)
5 2
2 3
8 7
14
3.
4.
5.
(m)
4
5
4 2
7
8
(n)
3 1
3 3
5 3
(o)
1 1
1 3
5 3
(p)
1 1
2 3
4 2
(q)
3 1
2 3
4 2
(r)
4
2
1 3
9
3
(s)
3 3
5 3
5 4
(t)
1
4
1 2
7
5
Express as a single fraction:
(a)
1 1

4 3
(b)
2 2

5 7
(c)
4 3

5 4
(d)
3 2

7 5
(e)
5 5

12 3
(f)
5 1

9 3
(g)
2 9

5 10
(h)
3 11

7 14
(i)
4 2

9 3
(j)
2 4

5 5
(k)
24 20

35 21
(l)
6
9

25 20
(m)
8 9

21 14
(n)
10 19

21 14
(o)
20 15

33 44
(p)
3 3

5 4
(q)
8 2

15 3
(r)
11 22

36 24
(s)
10 25

33 36
(t)
4 2

5 15
(b)
1
1
7 2
2
2
(c)
1 3
3 1
2 4
(d)
1
(h)
1
5
3
3
12
16
9 8

35 15
Express as a single fraction:
1
4
1
1
1
10 5
(a)
5 1
(e)
3
1
1 2
8
4
(f)
6
1
2 1
7 14
(g)
2 7
2 1
3 9
(i)
3
1
3 2
5
4
(j)
1
11 14

24 15
(k)
3
11
7
1
15 25
(l)
1
(m)
1
(n)
4
1
4 2
9
12
(o)
2
11
1
3
12
9
(p)
2
1
6 2
3
2
(q)
2
2
5 6
5
5
(r)
1 3
1 1
2 7
(s)
1
1
4 3
5
2
(t)
2
2
1 2
3
9
7
4
4
20
5
Express as a single fraction:
(a)
3 2 3
 
4 9 8
(b)
2
1 1
of 5 
11
2 6
(c)
1 3
1
 2
3 5
7
(d)
1 1 1
3 1  8
3 8 3
(e)
3 3
16  4 
8 5
(f)
3
1 2
6 2 
4
9 19
(g)
1 1
1
(3  1 )  7
3 2
2
(h)
11
4 3
( of 2 ) 
14
5 4
(i)
1
1
2
( of 5 )  6
11
2
3
15
6.
7.
(j)
9
2
18  ( of 4 )
10
7
(k)
1
1
5
6  (7 of )
4
5
18
(l)
4
3
2
 (1 of 6 )
13
11
7
(m)
5
3
1
(5  3 ) 
8
4 12
(n)
13 26
8
(  )
16 49
21
(o)
5
3
7
 ( of )
24 8
9
(p)
1
1
4
9  (17 of 10 )
4
2
7
(q)
1 1
3
(8  1 ) 
3 2
4
(r)
(
13
7
8
3 ) 
30 15 9
A sack of potatoes weighs 11 kgs.
(a)
3
How many bags each weighing 1 kgs can be filled from the bag?
4
(b)
What weight of potatoes would be left over?
A twenty – one metre length of fabric is cut into
(a)
How many pieces can be cut?
(b)
What length of fabric would be left over?
5
metre pieces.
8
3
2
cm and height 3 cm. Calculate its area.
4
5
8.
A triangle has base 2
9.
1
2
A rectangle measures 5 metres by 5 metres. Calculate its area.
4
3
MIXED QUESTIONS on FRACTIONS
1.
2
1
Anne mixed 1 kgs of flour with 1 kgs of sugar.
3
4
What is the total of flour and sugar?
2.
3.
3
Brendan ran 6 km of a 10 km run. How far did he still have to run?
4
3
7
At a Christmas party, David drank 1 litres of fruit punch. Simon drank of a litre and
8
4
1
John drank 1 litres.
3
(a)
How much fruit punch did they drink altogether?
(b)
If the bowl held 8 litres altogether, how much was left over?
16
4.
5.
6.
A garden is rectangular in shape and measures 7
(a)
Calculate the perimeter of the garden.
(b)
Find its area.
4
2
metres by 10 metres.
5
3
Billy is a long distance lorry driver. One day he had to drive to Birmingham. He drove for
1
2
2 hours at a speed of 76km/h and then for 3 hours at a speed of 81km/h before
2
3
arriving at his destination.
(a)
How far did he drive during the first part of his journey?
(b)
How far did he drive during the second part?
(c)
How far did he travel altogether?
(d)
How many hours did it take him in total?
(e)
Calculate Billy’s average speed for the whole journey.
Laura has applied to join the RAF and has to sit an ‘Entrance Test’. Part of it includes
some problems with fractions. Here are the answers that Laura worked out.
(a)
5 3

6 7
(b)
3 5
3 
8 16
8
13
(e)
A plank of wood 3
3
1
16
(c)
11 3

15 22
(d)
1
10
17
27
3
metres long is cut up into 5 equal pieces.
4
How long is each piece?
(f)
34 85

45 102
Each cow in a herd of 25 produces 4
2
litres of milk.
3
116
How much milk is this in total?
(i)
How many did Laura get correct out of the six questions?
(ii)
Write down the correct answers for the ones that Laura got wrong.
3
4
metre
2
3
litres
17
EF WORKING with VOLUME of a CYLINDER
This topic is not mentioned in National 5 arrangements but worth covering at this stage
Circular – based prism (cylinder)
1.
Find the volume of a circular-based prism for the values of r and h given.
r
h
2.
(a)
r = 6 cm
h = 15 cm
(b)
r = 8 cm
h = 24 cm
(c)
r = 4 cm
h = 12 cm
(d)
r = 10 cm
h = 8 cm
(e)
r = 20 cm
h = 60 cm
(f)
r = 7 cm
h = 20 cm
(g)
r = 15 cm
h = 40 cm
(h)
r = 11 cm
h = 35 cm
(i)
r = 44 cm
h = 125 cm
(j)
r = 8.8 cm
h = 30 cm
A milk dispenser is cylindrical in shape with diameter 30cm.
(a)
If 14 litres of milk are poured into it, calculate the depth of the milk
in the cylinder.
30cm
25cm
(b)
The height of the cylinder is 25cm.
How many more litres of milk are needed to completely fill it?
18
3.
8cm
Calculate the volume of a cylinder with diameter 12cm and height
8cm.
12cm
4.
This paint tin has diameter 20 cm and height 30 cm as shown in the diagram.
30 cm
10 litres
20 cm
It is claimed that it can hold 10 litres of paint. Is this claim correct?
You must show all working and give a reason for your answer.
19
EF 3.3 WORKING with the VOLUME of a SOLID SPHERE, CONE, PYRAMID
1.
Calculate the volume of each sphere described below, rounding your answer to 1 decimal
place.
r
2.
(a)
r = 6cm
(b)
r = 2m
(c)
r = 9mm
(d)
r = 3cm
Find the volume of a sphere for the following values of r and d.
(give your answers correct to 3 significant figures)
r
3.
(a)
r = 10cm
(f)
d = 18cm
(b)
r = 25cm
(g)
r = 80mm
(c)
d = 2m
(h)
d = 55cm
(d)
r = 200mm
(i)
r = 3·5m
(e)
d = 11cm
(j)
d = 48cm
A sphere has a diameter of 8cm.
Calculate its volume giving your answer correct to 3 significant figures.
4.
Find the volume of a cone for the following values of r and h.
(give your answers correct to 3 significant figures)
(a)
r = 5cm
h = 14cm
(b)
r = 7cm
h = 25cm
(c)
r = 3cm
h = 22cm
(d)
r = 12cm
h = 7cm
h
r
20
5.
Find the volume of a cone for the following values of d and h.
(give your answers correct to 3 significant figures)
6.
(a)
d = 15cm
h = 40cm
(b)
d = 11cm
h = 37cm
(c)
d = 22cm
h = 125cm
(d)
d = 8ˑ8cm
h = 30cm
Calculate the volume of each cone described below, rounding your answers to 1 decimal
place.
h
r
(a)
r = 3cm and h = 6cm
(b)
r = 8mm and h = 12mm
(c)
r = 3cm and h = 5cm
(d)
r = 2m and h = 6m
7.
A cone has a base diameter of 8cm and a height of 5cm. Calculate the volume of this cone.
8.
A cone has a base diameter of 10cm and a slant height of 13cm.
13 cm
Calculate the volume of the cone.
5cm
9.
A cone has a base radius of 9cm and a slant height of 15cm.
Calculate the volume of the cone.
10.
A pyramid has a square base of side 4cm and a vertical height of 7cm.
Calculate the volume of the pyramid correct to 2 significant figures.
11.
A pyramid has a rectangular base measuring 16mm by 12mm and a vertical height of
10mm. Calculate the volume of the pyramid.
WORKING with the VOLUME of a SOLID SPHERE, CONE, PYRAMID & CYLINDER
21
EXAM QUESTIONS
1.
The Stockholm Globe Arena is the largest hemispherical building in the world.
The radius of the building is 110 m.
Calculate the volume of the building in cubic metres,
giving your answer in scientific notation correct to 3
significant figures.
2.
A metal bottle stopper is made up from a cone topped with a sphere.
The sphere has diameter 15cm.
The cone has radius 09cm.
Radius = 09cm
The overall length of the stopper is 65cm.
65cm
Calculate the volume of metal required to make the stopper.
Give your answer correct to 3 significant figures.
3.
The volume of this sphere is 524cm3.
Calculate the diameter, d cm.
dcm
4.
Non Calculator!
Calculate the volume of this sphere which has radius 3m.
[Take   314 ]
3m
22
5.
Sherbet in a sweet shop is stored in a cylindrical container like the one shown
in diagram 1.
20cm
32cm
Diagram 1
The volume of the cylinder, correct to the nearest 1000cm3, is 10 000 cm3.
5 cm
The sherbet is sold in conical containers with diameter 5 cm
as shown in diagram 2.
250 of these cones can be filled from the contents of the cylinder.
Diagram 2
Calculate the depth, d cm, of a sherbet cone.
6.
d cm
Non Calculator!
The diagram shows a cone with radius 10
centimetres and height 30 centimetres.
30 cm
Taking π = 3∙14, calculate the volume of the
cone.
10 cm
A children’s wobbly toy is made from a cone, 21 cm
high, on top of a hemispherical base of diameter 20 cm.
7.
21 cm
The toy has to be filled with liquid foam.
Calculate the volume of foam which will be required.
20 cm
23
8.
The lamp cover in a street lamp is in the
shape of a cone with the bottom cut off.
The height of the cone is 50cm and its
radius is 25cm. The height of the lamp is 30cm
and the base of the lamp has a radius of 18cm
Calculate the volume of the lamp cover. [Answer to 3 significant figures.]
9.
A glass candle holder is in the shape of a cuboid with a cone
removed. The cuboid measures 4cm by 4cm by 6cm.
The cone has a diameter of 3cm and a height of 5cm.
6 cm
Calculate the volume of glass in the candle holder.
4 cm
4 cm
10.
For the Christmas market a confectioner has created a chocolate Santa.
It consists of a solid hemisphere topped by a solid cone.
Both have diameter 5cm and the height of the cone is 4cm as
shown in the diagram.
4cm
5cm
Calculate the volume of chocolate required to make one chocolate
Santa, giving your answer correct to 3 significant figures.
24
11.
The diameter of an ordinary snooker ball is 525cm.
Calculate the volume of a snooker ball giving your answer correct to 3
significant figures.
12.
A dessert is in the shape of a truncated cone [a cone with
a ‘slice’ taken from the top].
The radius of the base is 4∙1cm and is 1∙6 cm at the top.
The other dimensions are shown in the diagram.
5∙6cm
3∙7cm
Calculate the volume of the dessert.
13.
A young child was given a slab of moulding clay. It was a cuboid and measured 15∙2cm by
4∙8cm by 3∙4cm.
(a)
Calculate the volume of the cuboid rounding your answer to 2 significant figures.
The clay was made into 25 identical spheres.
(b) Using your answer from part (a), calculate the radius of one of the spheres.
25
14.
6∙6 cm
An ice cream is shaped like the one in the diagram.
The overall height of is 11∙7 cm.
3∙2 cm
The height of the cylinder is 3∙2 cm.
The diameter of the cone and cylinder is 6∙6 cm.
11∙7 cm
Calculate the volume of ice cream.
15.
A company that produces bins uses the design of a cylindrical base with a hemispherical
lid.
If the total height of the bin is 60cm and the
radius of the bin is 14cm, calculate the
60cm
total volume of the bin in litres correct to 3 significant
figures.
(Volume of cylinder = r2h;
Volume of sphere = 4/3r3)
14cm
16.
A Christmas bauble is made from a sphere of perspex with
a coloured cylinder in the middle. The volume round the
cylinder is filled with a thick liquid.
The sphere has a diameter of 8 cm. The cylinder has
a radius of 2∙6 cm with a height of 6 cm.
Calculate the volume of liquid needed to fill the sphere, giving your answer correct to 2
significant figures.
Revise for USAP 1(a)
26
Answers for USAP 1(a)
EF 1.3 SIGNIFICANT FIGURES
1.
(a)
(g)
(m)
(s)
20
300
11
0∙9
2.
(a)
(g)
(m)
(s)
(a)
(e)
(i)
(m)
(q)
(a)
(a)
(a)
(e)
(i)
(a)
(e)
(i)
8∙7
(b)
31
(h)
0∙053 (n)
29
(t)
49∙3
6∙08
0∙00947
28∙3
2∙68
248400
0∙02860
120
49
250
133
8∙78
0∙965
3.
4.
5.
6.
7.
8.
(b)
(h)
(n)
(t)
6
800
600
600
(c)
(i)
(o)
80
(d)
8000 (j)
4
(p)
93
(c)
0∙19
26
(i)
24
0∙0061(o)
0∙087
0∙76
(b)
2∙35
(f)
24200
(j)
56200
(n)
2460
(r)
3090
(b)
248000
(b)
0∙0286
(b)
4∙0
(f)
0∙49
(j)
17
(b)
4∙78
(f)
334
(j)
326
30
(e)
2000 (k)
10000 (q)
100 (f)
8000 (l)
1
(r)
300
5000
90
(d)
(j)
(p)
680 (e)
19
(k)
14000 (q)
2∙1
(f)
6400 (l)
2∙5
(r)
6∙5
5∙0
45000
(c)
(g)
(k)
(o)
(s)
(c)
(c)
(c)
(g)
(k)
(c)
(g)
(k)
0∙593
0∙0628
0∙0980
3170
2∙01
250000
0∙029
250
3∙8
500
56∙5
19∙8
2∙07
(d)
(h)
(l)
(p)
(t)
(d)
(d)
(d)
(h)
(l)
(d)
(h)
(l)
4770
29∙5
24∙5
30∙0
0∙000318
200000
0∙03
41
0∙084
65
988
26∙3
0∙0965
2∙98  108
APP 3.1
WORKING with PERCENTAGES
Use reverse percentages to calculate an original quantity.
1.
(a)
(d)
(g)
(j)
£100
30 litres
1 500 km
712 cm
(b)
(e)
(h)
£775
250mm
420 miles
£5 250
2.
(a)
(d)
(g)
(j)
£22
85 ml
2 550 m
£11 200
(b)
(e)
(h)
(k)
£45
£17.50
8 500 km
97∙5 litres
(c)
(f)
(i)
(l)
£72
100 cm
56 m
£6 200
3.
(a)
(d)
(g)
£40
£425
£2 050
(b)
(e)
(h)
£65
£7
£26
(c)
(f)
(i)
£25
£299
£26.50
4.
(a)
(d)
(g)
£12 500
£37 440
£3 644.86
(b)
(e)
(h)
£18 060
£23 400
£13 457.94
(c)
(f)
£24 600
£20 411.22
(k)
(l)
(c)
£75
(f)
512 m
(i)
12 000m
90 kg
27
EXAM QUESTIONS
1.
4.
500
£12 000
2.
5.
150
£4 500
3.
6.
£200 000
£2 600
£5955.08, £955.08
£1982.12, £382.12
£21190.05, 3190.05
(c)
(f)
(i)
£5495.52, £695.52
2393.35, 643.35
£64751.45, £14751.45
4.
5.
6300 million
APPRECIATION and DEPRECIATION
1.
(a)
(d)
(g)
(j)
2.
6.
£3240
£1536
8.
(a)
£2332.80, £332.80
£4348.04, £848.04
£23820.32, 3820.32
£439.32, £39.32
3.
7.
(b)
(e)
(h)
£92317
£5644.80
£270 000
(b)
£212.24
after 3 years
EXAM QUESTIONS APPRECIATION and DEPRECIATION
1.
4.
6.
7.
9.
11.
13.
16.
17.
18.
£920.48
2.
£27 900
3.
£1 500 000
136∙3mg
5.
£1 388
(a)
39 300 tonnes
(b)
just falls short of doubling
£9 600
8.
(a)
4%
(b)
£97 700
(a)
£3 230.67
(b)
5 years
10.
£41 100
£9 790
12.
(a)
44 990 (b) 5 months
(a)
8%
(b)
21∙5
14.
£6 834.38
15.
£159 720
Car is valued at more than half the original value.
(a)
£1 677
(b)
4 years
Gained £140.49
19.
Yes, since 913g > 900g
20.
£962 000
APP 3.2
WORKING with FRACTIONS
Operations and combinations of vulgar fractions including mixed numbers.
ADDITION and SUBTRACTION
1.
(a)
(g)
(m)
2.
(a)
(g)
(m)
(s)
4
5
17
20
23
56
1
1
5
1
1
20
9
1
56
5
1
16
(b)
(h)
(n)
(b)
(h)
(n)
(t)
1
2
5
12
23
40
3
1
10
7
1
12
7
1
40
5
1
12
(c)
(i)
(o)
(c)
(i)
(o)
(u)
7
8
23
24
41
63
3
1
8
13
1
24
13
1
63
5
1
12
(d)
(j)
(p)
(d)
(j)
(p)
(v)
5
6
9
10
23
30
1
1
2
3
1
10
17
1
40
2
1
3
(e)
(k)
(e)
(k)
(q)
(w)
7
9
11
12
2
9
11
1
42
3
1
4
5
1
12
1
(f)
(l)
(f)
(l)
(r)
(x)
7
12
13
14
5
12
17
1
80
1
1
5
49
1
60
1
28
3.
(a)
(g)
(m)
4.
(a)
(g)
(m)
(s)
5.
6.
2
(e)
1
2
(a)
11.
14.
1
4
11
3
21
5
(a)
1
(m)
1
3
2
(m)
(g)
8.
3
4
1
3
5
1
3
10
23
3
24
2
(a)
(g)
7.
1
2
5
12
9
40
1
4
3
2
4
2
4
3
9
km
40
1
6 hours
6
7
(a)
20
4
(b)
(h)
(n)
(b)
(h)
(n)
(t)
(b)
(f)
(b)
(h)
(n)
(b)
(h)
(n)
9.
12.
(b)
1
3
1
10
7
30
(c)
(i)
(o)
1
6
11
16
22
63
1
(c)
4
5
5
(i)
24
3
7
(o)
4
3
14
16
5
8
1
8
8
1
4
16
5
12
1
2
4
13
2
24
7
2
12
3
7
4
3
3
7
17
3
20
13
20
8
10
35
1
6
7
12
1
8
3
1
4
11
1
20
5
4
24
1
7
10
1
2
3
5
1
12
7
6
10
3
3
(c)
(g)
(c)
(i)
(o)
(c)
(i)
(o)
3
2
cm
10.
13.
4
15
(d)
(j)
(p)
(d)
(j)
(p)
(d)
(h)
(d)
(j)
(p)
(d)
(j)
(p)
1
12
3
10
5
24
1
3
11
8
12
4
6
9
5
7
9
1
1
6
1
3
4
1
3
8
2
1
9
1
4
3
5
2
8
4
9
(e)
(k)
(e)
(k)
(q)
(e)
(k)
(e)
(k)
1
4
1
4
7
8
1
4
3
1
6
6
5
1
20
1
4
4
4
31
35
11
3
12
(f)
(l)
(f)
(l)
(r)
(f)
(l)
(f)
(l)
7
16
1
4
5
12
5
4
16
1
8
6
8
1
12
1
2
10
5
3
4
1
1
2
3
7 hours
MULTIPLICATION and DIVISION
1.
(a)
(g)
1
7
2
7
(b)
(h)
1
10
1
14
(c)
(i)
2
7
3
8
(d)
(j)
1
12
4
39
(e)
(k)
1
20
3
40
(f)
(l)
4
7
2
3
29
(m)
(s)
2.
(a)
(g)
3.
(t)
(b)
(h)
12
(n)
(s)
21
(t)
(a)
(m)
(s)
3
4
4
9
16
27
24
55
(a)
4
(g)
1
(m)
(s)
5.
2
3
2
2
3
1
(n)
(m)
(g)
4.
4
15
4
21
(a)
(g)
(m)
6.
(a)
8.
4
1
2
9
32
1
1
5
1
16
7
10
1
8
6
27
cm²
40
(b)
(h)
(n)
3
4
1
4
1
12
1
4
2
2
14
1
5
2
1
5
6
11
20
57
(o)
(c)
(i)
(o)
10
27
1
4
1
12
2
6
4
(c)
(i)
(o)
1
15
2
3
7
1
9
1
6
(b)
3
(c)
2
(h)
4
9
(i)
1
(t)
(b)
(h)
(n)
(b)
9.
1
12
(d)
2
(j)
30
(k)
(p)
7
7
8
(q)
(d)
1
11
12
(q)
3
10
(e)
3
9
10
6
2
7
5
9
8
(r)
(f)
(l)
(r)
1
4
5
9
5
8
8
8
5
27
3
3
(t)
(n)
(p)
2
(o)
15
3
4
1
(c)
6
14
2
(i)
15
7
(o)
12
1
kg
2
3
29 cm²
4
2
3
5
15
16
3
7
3
40
5
7
(j)
(p)
(d)
(j)
(p)
(d)
(j)
(p)
1
14
1
2
4
5
11
12
9
1
16
1
1
3
1
4
2
4
3
1
20
31
7.
(a)
33
3.
(a)
3
(e)
(k)
(q)
(e)
(k)
(q)
(e)
(k)
(q)
(b)
1
4
18
25
4
5
11
18
11
2
12
27
32
42
(f)
(l)
(r)
(f)
(l)
(r)
(f)
1
(l)
8
2
16
(r)
3
3
metre
8
3
2
3
8
15
1
3
1
2
3
5
2
14
9
20
2
1
2
1
26
1
9
1
MIXED QUESTIONS on FRACTIONS
11
kgs
12
1.
2
2.
4.
(a)
36
5.
(a)
190 km
1
3 km
4
14
metres
15
(b)
(b)
23
litres
24
(b)
4
1
litres
24
1
83 m²
5
297 km
(c)
487 km
30
6.
(d)
1
6 hours
6
(i)
4 questions correct
(e)
78
36
km/h
37
(ii)
(a)
WORKING with VOLUME of a CYLINDER
1.
2.
4.
1
11
42
(d)
68
75
(a) 1696·5 cm3
(b) 4825·5 cm3 (c) 603·2 cm3
(d) 2513·3 cm3
(e) 75398·2 cm3
(f) 3078·8 cm3
(g) 28274·3 cm3 (h) 13304·6 cm3 (i) 760265 cm3
(j) 7298·5 cm3
(a)
19·8cm
(b)
No; volume is 9·72 litres
3·7 litres
3.
904cm³
EF 3.3 WORKING with the VOLUME of a SOLID SPHERE, CONE, PYRAMID
1.
2.
3.
4.
5.
6.
7.
11.
(a) 904·3cm³
(a)
4190cm³
(e)
697cm³
(i)
180m³
268cm³
(a)
366cm³
(a)
2369cm³
(a)
56∙5cm³
83·7cm³
640mm³
(b)
(b)
(f)
(j)
33·5m³
65400cm³
3050cm³
57900cm³
(c)
(c)
(g)
3052·1mm³
4·19m³
2140000mm³
(d)
(d)
(h)
113·0cm³
33500000mm³
87100cm³
(b)
(b)
(b)
8.
1280cm³
1170cm³
803·8mm³
314cm³
(c)
(c)
(c)
9.
207cm³
15800cm³
47·1cm³
1020cm³
(d)
(d)
(d)
10.
1060cm³
608cm³
25·1cm³
37cm³
WORKING with the VOLUME of a SOLID SPHERE, CONE, PYRAMID and CYLINDER
EXAM QUESTIONS
1.
5.
9.
13.
15.
2·79 × 106 m³ 2.
6·11cm
6.
84·225cm³ 10.
(a) 250cm³
34·1 litres
6·01cm³
3.
3140cm³
7.
58·9cm³
11.
(b)
1·3cm
16.
140 cm3
10cm
4.
4291cm³
8.
75·7cm³
12.
14.
206cm³
113·04m³
25900cm³
93·4cm³
31