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For use only in Whitgift School IGCSE Higher Sheets 1

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IGCSE Higher

Sheet H1-1 1-03a-1 Fractions Sheet H1-2 1-03a-2 Fractions Sheet H1-3 1-04a-b-1 Surds Sheet H1-4 1-04a-b-2 Surds Sheet H1-5 1-04c-1 Indices Sheet H1-6 1-04c-2 Indices Sheet H1-7 1-04c-3 Indices Sheet H1-8 1-04c-4 Indices Sheet H1-9 1-04c-5 Indices Sheet H1-10 1-04c-6 Indices Sheet H1-11 1-04d-1Primes Sheet H1-12 1-04d-2Primes Sheet H1-13 1-05a-1 Sets Sheet H1-14 1-05a-2 Sets Sheet H1-15 1-05a-3 Sets Sheet H1-16 1-05a-4 Sets Sheet H1-17 1-06a-01 Percentages Sheet H1-18 1-06a-02 Percentages Sheet H1-19 1-06a-03 Percentages Sheet H1-20 1-06a-04 Percentages-Non Calculator Sheet H1-21 1-06a-05 Repeated Percentage changes Sheet H1-22 1-06a-06 Percentages



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Sheet H1-1 1-03a-1 Fractions 1. Express the following terminating decimals as fractions in their lowest form:

189.0)h(162.0)g(768.0)f(3125.0)e(15.0)d(04.0)c(2.0)b(25.0)a(

2. Calculate the following as terminating decimals:

1 1(a) (b)8 253 7(c) (d)

40 163 7(e) (f )

200 329 13(g) (h)

50 500

3. Express the following recurring decimals as fractions in their lowest form:

564712.0)h(4235.0)g(

431.0)f(521.0)e(810.0)d(73.0)c(

1.0)b(6.0)a(

4. Calculate the following as recurring decimals:

1 1(a) (b)3 71 3(c) (d)

11 75 2(e) (f )

13 93 4(g) (h)

11 15



Sheet H1-2 1-03a-2 Fractions 1. Express the following as fractions in their lowest form:

(a) 0.36 (b) 0.270(c) 0.38 (d) 0.106(e) 0.2345 (f ) 0.925671

2. Express the following decimals as fractions in their lowest form:

3112.0)f(7376.0)e(

45.0)d(213.0)c(

72.0)b(7.0)a(

3. Express the following fractions as decimals:

2 5(a) (b)7 112 4(c) (d)3 91 3(e) (f )

13 131 5(g) (h)

15 6

4. Express the following numbers as fractions in their lowest form:

(a) 0.62

(b) 0.567(c) 0.12345



Sheet H1-3 1-04a-b-1 Surds 1. Simplify the following as far as possible:

216)r(150)q(135)p(

75)o(200)n(45)m(

12)l(63)k(320)j(

300)i(180)h(18)g(

44)f(245)e(50)d(

343)c(8)b(72)a(

2. Find the following in the form n :

(a) 7 2 (b) 3 3 (c) 2 7

(d) 3 7 (e) 2 2 (f ) 5 5

3. Simplify the following as far as possible, leaving your answer in the form a b :

20180)i(13560)h(20018)g(

5027)f(4555)e(3227)d(

123)c(2327)b(3532)a(

−++

+++

+−+

4. Simplify the following as far as possible:

32128)l(

7283)k(

30012)j(

150610)i(

48312)h(

824)g(

75533)f(28372)e(2250)d(

273)c(2225)b(3532)a(

++

×××

×××



Sheet H1-4 1-04a-b-2 Surds 1. Simplify the following as far as possible:

( ) ( )( ) ( )( )( )( ) ( )3

2

22

25)f(711711)e(

2525)d(37)c(

32)b(25)a(

+−+

−+−

++

2. Write the following, in the simplest possible form:

28)f(

2045)e(

872)d(

48300)c(1275)b(1850)a( −−+

3. Write the following, in the simplest possible form (in the form a b ):

112244)l(

3912)k(

71428)j(

315300)i(

51520)h(

228)g(

1133)f(

515)e(

721)d(

510)c(

33)b(

26)a(

++−

−+−

4. (a) Express 102

in the form ba , where a and b are positive integers.

An isosceles right-angled triangle ABC has a right angle at B. The length of its equal sides

is 102

cm.

Diagram NOT accurately drawn

102

cm

102

cm

(b) Find the area of the triangle. Give your answer as an integer. (c) Find also the hypotenuse of the triangle. Give your answer as an integer.

A B

C



Sheet H1-5 1-04c-1 Indices

1. Express the following as powers of 2 (i.e. in the form 2n ):

(a) 4 (b) 16 (c) 641(d) 2 (e) (f ) 0.2521(g) 1 (h) (i) 128

32

2. Express the following as powers of the stated numbers:

(a) 32 as a power of 2 (b) 81 as a power of 31(c) 625 as a power of 5 (d) as a power of 4

161 1(e) as a power of 7 (f ) as a power of 57 251 1(g) as a power of 12 (h) as a power of 2

144 1024

3. Calculate the following:

6 4 3

2 3 2

0 2 3

3 2 4

2 3 4

(a) 2 (b) 3 (c) 5(d) 11 (e) 2 (f ) 10(g) 19 (h) 13 (i) 4

1 2 5( j) (k) (l)2 3 3

1 2 2(m) (n) (o)2 5 3

− −

−

− − −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

4. Find x in the following:

1(a) 7 49 (b) 3 (c) 2 1811 2 27(d) 9 81 (e) 5 (f )

125 3 8

1 3 16(g) 16 (h) (i) 2 10244 4 9

x x x

xx x

x xx

= = =

⎛ ⎞= = =⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

5. Express the following in the form 8n where n is either an integer or a fraction: 1(a) (b) 8 (c) 1

81 1(d) 64 (e) (f )64 512



Sheet H1-6 1-04c-2 Indices 1. Calculate the following without using decimals:

5 2 0

3 1 1

2 1 3

3 2 4

(a) 2 (b) 3 (c) 7(d) 4 (e) 4 (f ) 10(g) 4 (h) 9 (i) 2( j) 10 (k) 8 (l) 2

−

− − −

− −

2. Express the following as powers of 2: 1 1(a) 32 (b) (c)

4 81(d) 128 (e) (f ) 1

16


1(a) 2 64 (b) 3 27 (c)749

1(d) 11 1 (e) 2 (f ) 10 0.0181(g) 2 0.25 (h) 3 (i) 6 3681

1( j) 2 (k) 11 1 (l) 10 100032

x x x

x x x

x x x

x x x

= = =

= = =

= = =

= = =

4. Evaluate the following without using decimals: 2 1 5

2 0 3

1 3 2

(a) 3 (b) 5 (c) 2(d) 7 (e) 13 (f ) 5(g) 9 (h) 10 (i) 6

− − −

−

− − −

5. Express the following as powers of 2:

3 2

2

3

2

3

(a) 8 1664 4(b)

162 8(c)4 16

×

×

××

6. Evaluate the following (without a calculator), leaving your answers as fractions where

necessary: 4 0 4

2 1 3

3 2 4

(a) 2 (b) 11 (c) 3(d) 5 (e) 11 (f ) 3(g) 4 (h) 8 (i) 5

− −

− −



Sheet H1-7 1-04c-3 Indices 1. Find n in the following :

(e.g. 2 1 5 2 1 52 32. We write 32 as 2 . Hence we solve 2 2 . That is 2 1 5. 2n n n n+ += = + = = )

3 1 1

3

2 1 3 7

5 2 1

(a) 2 4 (b) 3 811(c) 7 49 (d) 264

(e) 10 100000 (f ) 5 25(g) 2 64 (h) 3 27

n n

n n

n n

n n

− +

−

+ −

− −

= =

= =

= =

= =

2. Evaluate the following, leaving your answers as fractions where necessary: 5 0 7

3 1 3

2 5 6

(a) 2 (b) 7 (c) 2(d) 5 (e) 9 (f ) 5(g) 11 (h) 10 (i) 2

− −

−

3. Write 2

34

2793 ×

as a power of 3 (that is write it in the form 3n ).

4. Write 25 ×87

164 as a power of 2.

5. Write 54 × 256

1252 as a power of 5.

6. (a) Write 5

34

9273 ×

as a power of 3.

(b) Write 9 8

4

32 162×

as a power of 2.

(c) Write 7 4

2

625 1255×

as a power of 5.

(d) Write 3

52

12525625 × as a power of 5.

(e) Write 2

35

2565121024 × as a power of 2.

7. Given that 3216

6487

3

=×x

find x.

8. Given that 127

9814

2

=×x

find x.



Sheet H1-8 1-04c-4 Indices 1. Calculate the following:

1 113 32

1 116 34

1 1 15 2 2

(a) 4 (b) 8 (c) 125

(d) 64 (e) 81 (f ) 27

(g) 32 (h) 64 (i) 169

−

− − −


(a) 64 (b) 8 (c) 41(d) 2 (e) 1 (f )64

1 1 1(g) (h) (i)8 2 4

3. Calculate the following:

11 132 2

11 154 2

11 132 2

1 1 12 3 3

(a) 9 (b) 64 (c) 81

(d) 10000 (e) 32 (f ) 121

(g) 144 (h) 49 (i) 125

4 1 125( j) (k) (l)9 27 64

−− −

−⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠


(a) 32 2 (b) 81 3 (c) 125 5(d) 49 7 (e) 121 11 (f ) 27 3

1(g) 243 3 (h) 256 16 (i) 33

1 1 1( j) 81 (k) 125 (l) 5123 5 2

x x x

x x x

x x x

x x x

= = =

= = =

= = =

= = =

5. Evaluate the following without using decimals (show all working clearly):

4 3 2

1 3 33 4 2

1 123 4

(a) 3 (b) 5 (c) 7

(d) 8 (e) 16 (f ) 25

3 27 16(g) (h) (i)4 64 625

− − −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠


216)c(8

3216)b(

84)a(

3

2

23

×

×



Sheet H1-9 1-04c-5 Indices 1. Evaluate the following, leaving your answers as fractions where necessary:

11 132 4

1 115 32

11 1102 4

(a) 16 (b) 27 (c) 625

(d) 32 (e) 9 (f ) 1000

(g) 81 (h) 10000 (i) 1024

−−

−−

2. Evaluate the following, leaving your answers as fractions where necessary:

2 233 52

4 2 43 3 5

45 372 4

(a) 8 (b) 9 (c) 32

(d) 1000 (e) 27 (f ) 32

(g) 4 (h) 16 (i) 128

− −

−− −


34

1(a) 4 2 (b) 3 243 (c) 232

(d) 5 625 (e) 11 1 (f ) 2 21(g) 3 3 (h) 3 3 3 (i) 55

x x x

x x x

x x x

= = =

= = =

= = × =


4 1 1 2 1 2 3 1 4 3

1(a) 4 32 (b) 25 125 (c) 644

(d) 2 4 (e) 9 27 (f ) 32 16

x x x

x x x x x x+ + − + − −

= = =

= = =


1 1(a) 4 (b) 49 (c) 1000 0.0132 343

1 2 9 3 27(d) 512 (e) (f )1024 3 4 5 125

2 125 10 49 9 32(g) (h) (i)5 8 7 100 4 243

1 8 81 8 25( j) 1024 (k) (l)2 27 16 125 4

x x x

x xx

x x x

x x x

= = =

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠



Sheet H1-10 1-04c-6 Indices 1. Find x in the following by first of all writing both numbers as the powers of the same

number (e.g. in (b) write ( )2 3 2 31 325 5 5 5 5 2 3125 2

xx x x x− −= ⇒ = ⇒ = ⇒ = − ⇒ = − )

1 1(a) 8 4 (b) 25 (c) 4125 2

(d) 64 32 (e) 9 27 (f ) 121 111(g) 125 (h) 9 3 (i) 100 0.015

9 64( j) 8 0.25 (k) 9 0.3 (l)16 27

x x x

x x x

x x x

xx x

= = =

= = =

= = =

⎛ ⎞= = =⎜ ⎟⎝ ⎠

2. Express the following as powers of the stated numbers:

(a) 32 as a power of 4 (b) 27 as a power of 811(c) 8 as a power of 4 (d) as a power of 48

1 1(e) as a power of 32 (f ) as a power of 416 21 1(g) as a power of 9 (h) as a power of 12527 251(i) as a power of 16 (j)64

1 as a power of 832

3. Evaluate the following, leaving your answers as fractions where necessary:

( ) ( )

( ) ( ) ( )

54 1 2 33 3 3 2

1 43 1 3 33 32 2 2 2

11 12

3 13 28 4

1 1 12 2 2

(a) 27 (b) 8 (c) 8 25

(d) 16 4 (e) 4 125 (f ) 8 25

9(g) (h) 3 (i) 64

( j) 0.25 (k) 0.04 (l) 12.25

−− −

−

− −

⎛ ⎞×⎜ ⎟

⎝ ⎠

× ÷ ÷

⎛ ⎞⎜ ⎟⎝ ⎠

4. Find n in the following :

(e.g. in (a) write ( )2 13 3n −= so 2 1n = − and so 1

2n = − )

( )

( )

3

2

1(a) 9 (b) 8 2 (c) 5 53

(d) 7 49 (e) 11 1 (f ) 3 2 35

1 7(g) 2 16 (h) 6 (i) 74936

n n n

n n n n

nn nn

= = =

= = + =

= = =



Sheet H1-11 1-04d-1 Primes

1. Write the following as the product of prime numbers (e.g. 23 3272 ×= )

(a) 30 (b) 24(c) 18 (d) 28(e) 105 (f ) 64(g) 108 (h) 42(i) 150 ( j) 200(k) 378 (l) 144(m) 385 (n) 126(o) 420 (p) 45(q) 60 (r) 75

2. Use question 1 to find the highest common factor of the following pairs of numbers:

(a) 45 and 105 (b) 28 and 144 (c) 385 and 75 (d) 420 and 126 (e) 108 and 60

3. Use question 1 to find the lowest common multiple of the following pairs of numbers:

(a) 45 and 60 (b) 200 and 420 (c) 108 and 144 (d) 126 and 28 (e) 378 and 75

4. Two cars complete laps of a circuit. One takes 315 seconds per lap, the other takes 525 seconds per lap. They start their circuits of the laps at the same time.

(a) Express 315 and 525 as the product of their primes. (b) Find the lowest common multiple of 315 and 525. (c) Use this to find how many laps the faster car will do before the cars first get to the starting point at the same time.

5. (a) Find the highest common factor and lowest common multiple of 210 and 550. (b) Multiply the two numbers that you found in (a) together. (c) Multiply 210 and 550 together. (d) What do you notice about the answers to (b) and (c)?



Sheet H1-12 1-04d-2 Primes

2. Write the following as the product of prime numbers (e.g. 23 3272 ×= ) (a) 36 (b) 40 (c) 54 (d) 20 (e) 270 (f) 81 (g) 250 (h) 98 (i) 70 (j) 50 (k) 135 (l) 240 (m) 21600 (n) 31360

6. (a) Find the highest common factor of 135 and 50. Let this be h. (b) Find the lowest common multiple of 135 and 50. Let this be l. (c) Calculate h l× and 135 50× . What do you notice? 7. (a) Find the highest common factor of 21600 and 250. Let this be h. (b) Find the lowest common multiple of 21600 and 250. Let this be l. (c) Calculate h l× and 21600 250× . What do you notice? 8. Use question 1 to find the lowest common multiples of the following pairs of numbers: (a) 36 and 31360 (b) 98 and 240 (c) 81 and 270 (d) 240 and 54 9. Use question 1 to find the highest common factor of the following pairs of numbers:

(f) 21600 and 135 (g) 70 and 50 (h) 98 and 31360 (i) 54 and 36 (j) 81 and 70



Sheet H1-13 1-05a-1 Sets

1. In a class of 20 pupils, 9 like tomato sauce but not HP sauce, 6 like HP sauce but not tomato sauce and 3 like neither. Copy and complete the following Venn diagram.

2. In a class of 30 pupils, 20 like football, 12 like rugby and 4 like neither. Suppose n pupils like both football and rugby. (a) Write down an expression for the number of pupils who like football but not rugby. (b) Copy and complete the following Venn diagram using n.

(b) By adding up all four values, find n.

3. In a year of 100 pupils, 70 enjoy Maths, 50 enjoy French and 20 enjoy neither. (a) Set up a Venn diagram showing this information. (b) Use this to find the number of pupils who enjoy both subjects.

4. In a shop there were 120 customers on a certain day. 60 paid using notes, 30 paid using

coins and 50 paid using neither (cheques, cards etc.) (a) Set up a Venn diagram showing this information. (b) Use this to find the number of customers who used both notes and coins.

5. On an Athletics day 150 athletes are running. 60 are in the 100 metres, 50 are in the 200

metres and 80 are in neither. (a) Set up a Venn diagram showing this information. (b) Use this to find the number of athletes who ran in only one race.

6. A group of 200 adults were surveyed about holidays. 150 had been to Spain, 80 had been

to France. Twice as many had been to both countries as had been to neither country. Suppose n adults had been to neither country. (a) Write down an expression for the number of adults who had been to both countries. (b) Set up a Venn diagram using n. (c) Hence find n and set up a new Venn diagram without using n.

Tomato HP

∪

Rugby Football

∪



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1.

In each of the following questions draw the Venn diagram shown above and shade the region.

( ) ( )

(a) (b)(c) (d)(e) (f )

(g) (h)

A B A BA B A BA B A B

A B A B

′∪ ∪′ ′∪ ∩

′ ′ ′∩ ∩

′ ′′∪ ∩

2. In each of the following questions draw the Venn diagram shown below.

( )

( )

(a)(b)

(c)(d)

(e)(f )

A B CA B C

A BA B

A CA C

∩ ∩′ ′∩ ∩

′∪

′ ′∩

′∩

′ ′∪

A B

∪

A B ∪

C



Sheet H1-15 1-05a-3 Sets 1.

{ }{ }{ }{ }{ }

polygons

quadrilaterals

rectangles

parallelograms

triangles

A

B

C

C

=

=

=

=

=

∪

Draw a Venn diagram showing A, B, C and D. 2.

{ }{ }{ }{ }

: is a number under 20

: is an even number under 20

: is a multiple of 3 under 20

: is a prime number under 20, including 1

x x

A x x

B x x

C x x

=

=

=

=

∪

(a) Copy and complete the following: { }6, .....A B∩ =

(b) Find ( )A B ′∪ .

(c) Is it true that ( )A B C′∪ ⊂ ? (d) Is it true that 7 A′∈ ?

(e) Calculate ( )n B′ .

3. { }{ }{ }

: is a postive integer under 15

1, 2, 3, 5, 7, 11, 13

3, 4, 5, 9, 10, 13, 14

x x

A

B

=

=

=

∪

(a) List all the members of A′ . (b) Find ( )n A B′ ∩ .

4.

{ }{ }{ }

: is a postive integer under 13

: is an even number under 13

: is a multiple of 3 under 13

x x

A x x

B x x

=

=

=

∪

(a) List all the members of A′ . (b) Find ( )n B . (c) Write down the members of A B∩ . (d) Express these numbers on a Venn diagram. (e) Express the set { }1, 5, 7, 11 in terms of A and B.




1.

Copy and complete the Venn diagram shown above, drawing on the sets B and C where:

A BC B∩ = ∅⊂

2. A, B and C are sets with the following properties:

( )( )( )

( )

5

1

3

1

n A

n A B

n A BB C Bn CA C

=

∩ =

′∩ =

∪ =

=

∩ = ∅

(a) Draw a Venn diagram indicating the sets A, B and C. (b) Find the following:

( )( )( )

(i)

(ii)

(iii)

n A B

n A C

n B C

∪

∪

∩

3.

{ }{ }1, 3, 5, 7

: is an even number less than 10

A

B x x

=

=

Find the following: (a) ( )n A B∪ (b) A B∩ (c) C such that C A⊂ and ( ) 3n C = .

A

∪



Sheet H1-16 1-05a-4 Sets (cont.) 4. Describe the following shaded areas:

(a)

(b) (c) (d)

A B

A B

A B

A B



Sheet H1-17 1-06a-01 Percentages

1. The numbers of customers per day in a certain town increases by 25% at the weekend. (a) If there were 120 customers on a Friday in one shop then how many would there be

on the Saturday? (b) If there were 195 customers on a Saturday in another shop then how many would

there have been on the previous Friday? 2. A filing cabinet is selling at £121 in a magazine with an advert next to it saying What was the retail price? 3. A fan club had a membership of 15,930 in 1999 which was, approximately, an 11%

decrease from the year before. What was the membership (to the nearest ten) in 1998? 4. A picture is reduced by 20% so that the copy measures 14cm by 20cm.

(a) What were the dimensions of the original? (b) If the copy is now enlarged by 15% what will its new dimensions be?

5. 4,200 copies of a certain book were sold in March which was an decrease of ¼ on the

number sold in February. In April there was an increase of 52 on the number sold in

March. (a) How many were sold in February? (b) How many were sold in April?

6. In a summer sale a shop offered 20% off all goods.

(a) What would be the sale price of a pair of shoes if they originally cost £40? (b) If the sale price of a jacket was £64 how much did it cost before the sale? 7. (a) How much would a computer cost including VAT (of 17.5%) if it cost £1200 before

the VAT had been added on? (b) Another computer cost £2115 including VAT (of 17.5%) - how much would it cost without the VAT?

PTO

Only £121 – that’s an amazing 45%

off the retail price!



Sheet H1-17 1-06a-01 Percentages (cont.)

8. A woman sold a vase for £250. This was a profit of 25%. How much did she buy the vase for?

9. An employee’s salary is increased by 5%. She now earns £27,090. What did she earn

before her rise? 10. A garage sold a car at a loss of 4%. If it sold the car for £11100 then what did the garage

pay for the car? 11. The audience of a certain TV show increased by ¾ over 1998. If 21 million people

watched it at the end of 1998 how many watched it at the beginning of that year? 12. The monthly profits of a certain company fell by 5

1 over the summer period and their monthly profits were £20,000 in the Autumn then what were they in the previous spring?

13. If a boy sells a CD for £13.20 he makes a 20% profit. How much did he buy he CD for? 14. A man buys a computer. He sells it a year later for £1200 but by doing so he makes a loss

of 25%. What was the original price of the computer?



Sheet H1-18 1-06a-02 Percentages 1. A document was photocopied so that the lengths of the copy were 70% of the original

lengths. If the copy measured 12.6cm by 17.5cm what are the dimensions of the original document?

2. The attendance at a football match one week increased by 2

3 from the previous week. If the new attendance was 45000 what the attendance the week before?

3. A boy’s height increased by 1

5 over a year. If his height is now 1.68m what was his height a year ago?

4. The number at a school in 2006 was 85% of the number at the school in 2005. In 2006 the

number of pupils was 1020. How many pupils were in the school in 2005? 5. Find the original price of a car which was sold at £1200 at a loss of 4%. 6. Find the original price of an antique which was sold at £545 at a profit of 9%. 7. The profit of a company in 2004 was £1,500,000. In 2005 the profit was 25% higher than

it was in 2004 but in 2006 the profit fell by 40%. (a) Show that the profit made in 2005 was £1,875,000. (b) What was the profit made in 2006?

8. A travel agents offers a 15% reduction off all holidays.

(a) What would be the new price of a holiday which originally cost £480? (b) The reduced price of another holiday is £765. What was its original price?

9. A supermarket has the following advert “We are selling wine without VAT – this is a

reduction of 17.5% off the retail price”. By considering the price of a bottle of wine which cost £11.75 after VAT (at 17.5%) comment on the accuracy of the advert.

10. If a boy buys a calculator for £32 which included a 20% discount on the retail price. Find

the retail price. 11. A house cost £180,000 in 1999. The price rose by 7% over the next year. Find the price of

the house in 2000. 12. A man buys a computer from as shop which was making a 10% profit. If the man paid

£1320 then find the price which the shop paid. 13. A boy’s height increased by ¼ over a year. If his height is now 1.5m what was his height

a year ago? 14. The number of teachers at a school in 1999 was 95% of the number of the teachers at the

school in 1998. In 1998 the number of teachers was 80. How many teachers were in the school in 1999?

15. Find the original price of a motorbike which was sold at £2720 at a loss of 15%.



Sheet H1-19 1-06a-03 Percentages 1. Find the original price of a painting which was sold at £1680 at a profit of 12%. 2. Find the selling price of an antique which cost £1500 and was sold at a profit of 75%. 3. The profits of a company in 1998 were £7,000,000. In 1999 the profits rose by 15%.

What were the profits in 1999? 4. An electrical shop offers a 35% reduction off all its products.

(a) What would be the new price of a camera which originally cost £150? (b) The reduced price of a television is £390. What was its original price?

5. A petrol station increased all its prices by 3% from October to November.

(a) In October a litre of unleaded cost 80p. What did it cost in November? (b) In November a litre of diesel cost 80.34p. What did it cost in October?

6. A man buys a desk for £120 in a sale in which all desks were reduced by 25%. What was

the price before the sale? 7. A girl sees a bike for sale with the notice ‘Price £105.75 (inc. VAT 17.5%)’. What was the

price before VAT? 8. A doctor’s pay before tax was £30,000. If tax was at 31% what will he actually receive

each year? 9. The number of cases of a certain disease fell by 5

2 in 1999 to 3240. How many cases were there in 1998?

10. A photocopier is set to increase the sides of the original by 3

1 . A picture measuring 12cm by 18cm is photocopied. (a) What will the dimensions of the copy be? (b) If that copy is then later reduced to get back to the size of the original then by what

factor will its sides have been reduced? 11. An internet company claims to offer holidays at prices which are 15% cheaper than in the

high street. (a) How much would it charge for a holiday which cost £450 in the high street (b) If the internet company charged £544 for a holiday then how much would you expect

to pay for it in the high street? 12. The cost of a new car fell by 5% per month for the first three months of the year 2000. The

car cost £16,800 at the start of the year. (a) What did is cost after three months? (b) By what factor was its price reduced each month? (c) By what factor was its price reduced over these three months? (d) What is the connection between the answers to (b) and (c).

13. The profits of a certain factory rose by 21% over two years.

(a) By what factor did the profits rise over this time? (b) On average, what was the average annual rise in the profits over this time?



Sheet H1-20 1-06a-04 Percentages-Non Calculator 1. Find the original price of a car which was sold at £3600 at a profit of 20%. 2. Find the selling price of a camera which cost £160 and was sold at a loss of 25%. 3. A man buys a set of garden furniture for £64 in a sale in which all garden furniture had

been reduced by 20%. What was the price before the sale? 4. A policeman’s pay after tax was £28,000. If tax was at 30% what was his annual salary

before tax? 5. The number of cases of an infection fell by 3

1 in the 1990s to 600,000 per year. How many cases were there per year in the 1980s?

6. A high street stationery shop offers 10% off the retail price of all its pens.

(c) How much would it charge for a pen with a retail price of £30? (d) What was the retail price of a pen which was selling for £45 in the sale?

7. The average attendance at a football ground rose by ¼ over the 1998-9 season to 45,000.

(a) Find the average attendance in the 1997-8 season. (b) The ground expected a further rise of 3

1 over the 1999-2000 season. What would its new attendance be?

8. The value of a company rose by 60% to £32million from 1997 to 1998. What was the

value in 1997? 9. A boy’s height increased by 10% over a six month period. At the end of that period he was

99cm tall. What was he before those six months? 10. The number of visitors of a castle fell by 25% over a year. At the end of the year on

average 150 people visited it per day. What was the average attendance at the start of that year?

11. The number of candidates from a certain school getting an A* in French fell by 10% from

1999 to 2000. If 120 got A*s in French in 1999 then how many got A*s in 2000? 12. A picture is reduced by 15%. The original size of the picture was 120mm by 160mm.

What were the dimensions of the reduced copy?



BLANK PAGE



Sheet H1-21 1-06a-05 Repeated Percentage changes 1. The shares in a certain company increased by 20% per year for three years.

(a) By what factor do they increase each year? (b) If the shares were worth 750p three years ago then what are they worth now? (c) By what factor have they increased over the three years? (d) What is the connection between the two factors in (b) and (c)?

2. A bank offered 5% per annum (compound interest). If he invested £4000 then:

(a) How much would the money be worth in one year? (b) By what single number (to 3sf) would the £4000 be multiplied by in order to

calculate the amount of money in the account after 4 years? (c) Hence calculate how much money (to the nearest pound) would be in the account

after 4 years. 3. Peter invests a sum of money for 20 years at 6% annum compound interest.

Write down the letter of the graph which best shows how the value of Peter’s investment changes over the 30 years. 4. A bank offered 8% per annum (compound interest). A man invested £70,000. By how

much would his money have increased after two years? 5. The shares of a certain company fell by 10% per month for three consecutive months. If

they were 780p before the fall then what were they after the fall? 6. A bank offers 5% interest on investments. A man invests £2,000 in the bank.

(a) What is it worth after 2 years? (b) What is the total percentage increase in his money?

7. An investment fund claims that it has increased by a total of 21% over the last two years.

(a) If a man invested £1000 in the fund two years ago then what would it be worth now? (b) If you asssume a constant rate of interest per year then find that rate of interest.

8. The number of burglaries in a certain country rose by 8% for two years in a row from 2001

to 2003. If there were 20,000 burglaries in 2001 then: (a) How many burglaries were there in 2002? (b) How many burglaries were there in 2003? (c) What was the overall percentage increase in burglaries over the two years? (NB it is

not 16%).

O A

Value

Years

O D

Value

YearsO C

Value

Years

O B

Value

Years



BLANK PAGE



Sheet H1-22 1-06a-06 Percentages

1. In April a lawnmower is selling for £265. In the September sale it was only £225.25.

What was the percentage discount? 2. A garden table is selling for £48 with a sign saying “Reduction of 20%”. What was the

price of the table before the sale? 3. A bleach bottle says “900ml for the price of 750ml – x% extra free” where x cannot be

read. What is x? 4. A boy’s weight has increased by 3

1 over two years. He now weighs 84kg. What was his weight two years ago?

5. A school claims that the pupils’ average mark in an exam has increased by 15% over 5

years. Two boys are told that the average mark is now 85.1. George thinks that the average mark five years ago was 72.335 but James thinks it was 74. Who is right and how is the right answer obtained?

6. In sale all items are reduced by 15%. A carpet is selling for £15.30 per square metre. What

was is before the sale? 7. A table measures 80cm by 130cm. A second table is said to be similar to this table but

20% bigger. (a) By what factor have the dimensions increased? (b) Find the dimensions of the new table. (c) By what factor has the area increased?

8. A TV station says that the audience for a particular programme has increased five fold over

ten years (i.e the audience is five times what it was). What percentage increase is this? 9. A drill costs £66.27 after VAT of 17.5%. How much would it cost a builder who didn’t

have to pay VAT? 10. A man receives a cheque for £497.87 after tax has been taken off. If he would have

received £630.21 before tax then what rate (to the nearest %) was the tax? 11. A certain college states that the numbers in the college have increased by 3

2 over two years. If there are now 245 people in the college then how many were there two years ago?

12. A boy is told that his height will increase by ¾ over the next eight years. If he is now

90cm tall then how tall will he be in eight years? 13. The shares in a certain company increased by 30% per year for three years.

(e) If the shares were worth 1600p three years ago then what are they worth now. (f) By what factor do they increase each year? (g) By what factor have they increased over the three years? (h) What is the connection between the two factors in (b) and (c)?

PTO



Sheet H1-22 1-06a-06 Percentages (cont.)

14. An investment fund claims that it has increased by 2450% over the last ten years. (c) If a man invested £1000 in the fund ten years ago then what would it be worth now? (d) What is the average increase (to the nearest %) over a year of this fund?

15. What is the “factor” for the following: (a) An increase of 21% (b) A fall of 4% (c) A rise of 4.8% (d) A decrease by 3

1 (e) An decrease by 3.5% (f) An increase by 150% (g) A fall by 5

3 (h) An increase of p%

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