1

Cambridge International Examination

Beaconhouse School System
Boys Campus, Canal Side
Numbers (Revision)

Notes, examples, hints & practice sums from Cambridge Exams for (O1, O2 & O3)

Cambridge International Syllabus

Cambridge O Level Mathematics (4024 & 0580) syllabus for 2025, 2026 and 2027

Numbers Notes and Examples

 natural numbers  convert between numbers and words, e.g. six billion
is 6 000 000 000 10 007 is ten thousand and seven
 integers (positive, zero and negative)
 express 72 as a product of its prime factors
 prime numbers  find the highest common factor (HCF) of two
 square numbers numbers
 find the lowest common multiple (LCM) of two numbers.
 cube numbers  Candidates are expected to be able to write fractions in
 common factors their simplest form.
 common multiples  Recurring decimal notation is required, e.g.

 rational and irrational numbers

 reciprocals.
 Round values to a specified degree of accuracy.
 Includes converting between recurring decimals and
 Make estimates for calculations involving fractions and vice versa, e.g. write as a fraction.
numbers, quantities and measurements.
 Conversion of units  By writing each number correct to 1 significant
figure, estimate the value of

Resources: 1. D-SERIES MATHEMATICS (5th, 6th, 7th & 8th) Edition

By: • Dr Joseph Yeo • Teh Keng Seng • Loh Cheng Ye
• Ivy Chow • Ong Chan Hong • Jacinth Liew

2. Complete Mathematics (Extended) OXFORD UNIVERSITY PRESS

By: • David Rayner • Ian Bettison • Mathew Taylor

3. EDEXCEL INTERNATIONAL MATHEMATICS GCSE

By: • David Turner • Ian Potts
4. CAIE, IGCSE, GCSE & Edexcel Internationsl Exams Sums
 2

 Real Numbers: Each and every number weather it is so small, so big, positive or negative is called
real number. Real numbers are made up of Rational Numbers and Ir-rational Numbers.
e.g.
Real Numbers

Rational Numbers Ir-rational Numbers

1 1 1 3
𝑒. 𝑔 2, −0.7, − , , −3 𝑒. 𝑔 𝜋 , √7 , −√5
2 3 8
Non recurring non terminating decimals
Terminating and recurring decimals 𝒑
𝒑 (cannot be written as 𝒇𝒐𝒓𝒎)
(can be written as 𝒇𝒐𝒓𝒎 𝒘𝒉𝒆𝒓𝒆 𝒒 ≠ 𝟎) 𝒒
𝒒

𝟏 𝟏
 Terminating Decimals: e.g 𝟐, −𝟎. 𝟕, , −𝟑 , 𝟐. 𝟓
𝟐 𝟖
𝟏 𝟏𝟐𝟑 𝟐𝟐
 Recurring Decimals: e.g (𝟎. 𝟑𝟑𝟑𝟑. . ) , − (−𝟎. 𝟐𝟑𝟐𝟑. . . . ), (𝟑. 𝟏𝟒𝟐𝟖𝟓𝟕 𝟏𝟒𝟐𝟖𝟓𝟕 … )
𝟑 𝟗𝟗 𝟕
 Rational Numbers consist of Integers and fractions

.
Rational Number

Integers Fractions

𝒆. 𝒈 𝟎, 𝟏, 𝟓, 𝟖, −𝟑, −𝟏𝟏, . . . . . . . 1 1
𝟎(zero is neither negative nor positive 𝑒. 𝑔 , −2 ,
3 8
integer)

.
Integers

Whole Numbers Negative Integers

e.g 0, 1, 2, 3, 4 ……. 27, ……99, 187……… 𝑒. 𝑔 − 1, −2, −3, … … . , −25, ….

𝟎(zero is whole number but neither
negative nor positive)
Except zero all whole numbers are positive
integers

 A Number is made up of an absolute value with positive or negative sign. e.g − 𝟏𝟐 Absolute Value

Also |−5| = 5 𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑣𝑎𝑙𝑢𝑒 Sign

 Prime Number: A Whole number which has exactly two different factors one (1) and itself.

𝑒. 𝑔 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, … … … ..
(Above all whole numbers are only divisible by 1 or itself)

 Composite Number: A whole number which has more than two different factors

𝑒. 𝑔 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28 … … … ..
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 Practice Sums: [Book & Examination paper sums]

Q1 a) If 5 meter represents 5 𝑚 above the sea level, what does −9 𝑚 represents?

b) If +$40 represents depositing $40 in the bank, what does −$50 represents?

c) If −60𝑜 represents a clockwise rotation of 60𝑜 , what does +30𝑜 represents?

d) Write 2.704 86 correct to 3 decimal places.

e) i) Write down the reciprocal of 9. iii) Find the reciprocal of 0.35

f) Inside room temperature is 22o and outside temperature is -9o, what is the difference of

temperature between inside and outside.

g) The temperature of a village on a particular night is −11 Co. The next morning the

temperature rises by 7 Co. Find the temperature in the morning.

h) At midday the temperature is 8 °C. At midnight the temperature is 12 °C lower.

Find the temperature at midnight

i) Work out the temperature that is 5 ºC colder than −18 ºC.

j) Write down the value of the 5 in the number i) 𝟐𝟓𝟑 𝟔𝟐𝟒. ii) 𝟑𝟏𝟐𝟓𝟔𝟎 iii) 𝟕𝟐𝟎𝟓𝟔

k) The crowd at a sports event is exactly 35 687. Write this number correct to the nearest ten .
l) Write down an irrational number between i) 10 𝑎𝑛𝑑 15. ii) 4.5 𝑎𝑛𝑑 7.5

Q2 Fill in blanks with < , > 𝒐𝒓 =

(i) 16 _____27 (ii) 30______ − 31 (iii) −2 _____0 (iv) −6_______8

4 2
(v) −4 ______ − 6 (vi) 3 _____6 ÷ 2 (vii) −11 _____ − 11.7 (viii) _____
6 3

2 1 2 1 −7 −2
(ix) − 5 _____ − 7 (x) −4 ____ + (−4) (xi) ______ 5 (xii) _____
7 5 3

Q3 Workout the followings:

(i) 23 − 13 (ii) 15 + (−12) (iii) 5 − 9 (iv) 35 + (−40)

(v) −27 + 19 (vi) 28 ÷ 4 (vii) −40 ÷ 4 (viii) −27 ÷ (−9)

(ix) 2 × 23 (x) −0.3 × 0.05 (xi) −12 × (−5) (xii) −21 − 24

(xiii) 3 − (−5) (xiv) −25 − (+12) (xv) −34 − (−14) (xvi) 6 + 4(1 − 0.4)
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Q4 Workout. (Hint: operate from left to right)

(i) 12 ÷ 4 + 3 (ii) 3 ÷ 9 × 3 − 1 (iii) −4 × (3 − 5)2 ÷ 16 (iv) 6 − 7 + 2 × (4 − 3)2

(v) 2 − 14 + (7 − 32 ) (vi) 23 − 12 ÷ 2 − (√25 + 3) (vii) 5 ÷ 30 × 32 (viii) 52 − 30 + 6

3
(ix) −3 × (15 − 7 + 22 ) (x) √32 − 8 × (−2) (xi) √52 − 32 (xii) √102 − 5 × (−5)

(xiii) 12 − 6 ÷ 3 + 4 (xiv) – 8 × 2 + 3 (xv) 12 + 8 ÷ (9 − 5) (xvi) 92 − 90

3 3
(xvii) 600 ÷ 0.2 (xviii) 20 − 12 ÷ (8 − 6) (xix) √1000 (xx) ÷6
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Q5 Workout the followings

(i) 0.3 × 1.5 (ii) 0.01 × 0.27 (iii) (2.05 + 1.4) × 0.2 (iv) 0.4 × 1.3

(v) 0.018 ÷ 0.06 (vi) 0.2 × 0.4 (vii) 0.03 × 0.05 (viii) 0.79 × 0.02

(ix) 0.32 ÷ 0.02 (x) 0.005 × 0.1 (xi) 33 ÷ 0.27 (xii) 0.04 × 0.11

(xiii) 0.52 (xiv) 1234.4 ÷ 8 (xv) 3.25 − 1.73 (xvi) 1.22

Q6 Workout the followings: give your answer in simplest form.

1 4 2 5 1 7 3 2 6 3 90 5 2
(i) 1 3 − 5 (ii) 3 − (iii) 3 ÷ (iv) 1 8 − (v) 7 − (vi) (vii) 7 −
8 9 3 5 0.45 5

1 1 1 2 7 3 1 1 3 1 4 3 2 1
(viii) 1 5 ÷ 2 3 (ix) +3 ÷6 (x) 7 × 3 − (xi) (5 − 3) ÷ 5 (xii) (2 5 − 3) ÷ 15
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 Exercise#1:
i) Iv) vii) x)

ii) v) viii) xi)

iii) vi) ix) xii)

 Exercise#2

Q1 Calculate the value of each of the followings

𝟓 𝟐𝟖 𝟐 𝟏 𝟐 𝟏 𝟏 𝟏
a) − × (− +𝟏 ) b) [− ( ) − (− ) ] ÷ ( − )
𝟕 𝟏𝟓 𝟑 𝟐 𝟑 𝟒 𝟑

𝟏𝟓 𝟑 𝟏 𝟏 𝟑 𝟐 𝟏 𝟏
c) 𝟏𝟎 − × (𝟐 ÷ 𝟒 𝟐) + (− 𝟒) d) (− 𝟐) × (𝟏𝟓 − 𝟐 𝟑)
𝟖
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Q2 Evaluate the followings:

𝟎.𝟏𝟓 𝟎.𝟏𝟔 𝟎.𝟎𝟐𝟕 𝟏.𝟒

a) × (− ) b) × ( )
𝟎.𝟓 𝟏.𝟐 𝟎.𝟎𝟑 −𝟎.𝟏𝟖

27
c)−0.42 × (−1.3) ÷ 0.8 − 0.62 d) (−0.2)3 × 1.6 + 0.105

Q3 a) Insert one set of brackets to make the calculation correct. 𝟑 + 𝟓 × 𝟐 − 𝟕 = 𝟗

b) Insert +, − 𝑎𝑛𝑑 × to make the calculation correct 𝟑 𝟓 𝟐 𝟕 = 𝟐𝟎

c) Put one pair of brackets into this calculation to make it correct. 4 + 4 × 4 − 4 = 4

d) Put one pair of brackets into this calculation to make it correct 𝟐 × 𝟖 ÷ 𝟒 − 𝟕 = −𝟏𝟎

 Exercise #3
Q1 a) Round the following numbers as directed:

# Number Degree of accuracy

1 27.0923 1 significant figure Nearest whole number
2 749.3499 2 decimal places 0.001
3 46.786 One tenths Nearest whole number
4 525267 2 significant figures 4 significant figures
5 0.006729 1 significant figure Nearest thousandths
6 0.0798 2 significant figures 0.01
7 0.00198 One hundredths 2 significant figures
8 725638 4 significant figure Nearest hundred
9 0.70709 4 decimal places 3 significant figures
10 0.427 One hundredths 4 significant figures
11 0.07078 3 significant figures Nearest thousandths
12 41059 1 significant figure Nearest hundred

Q2) Round off 𝟔𝟗𝟖𝟑𝟓𝟐 to the nearest i) 10, ii) 100, iii) 10000, iv) 10,000

Q3) Round off 𝟒𝟓. 𝟕𝟑𝟗𝟓 to i) 1 decimal place, ii) the nearest whole number, iii) 3 decimal places

Q4) Round off

(i) 𝟒. 𝟗𝟏𝟖 𝒎 to the nearest 𝟎. 𝟏𝒎, (ii) 𝟗. 𝟕𝟏 𝒄𝒎 to the nearest cm,

𝟏
(iii) $𝟏𝟎. 𝟗𝟖𝟐 to the nearest ten cents, (iv) 𝟔. 𝟒𝟖𝟗 𝒌𝒈 to the nearest 𝟏𝟎𝟎 𝒌𝒈
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 Exercise#4
Q1 Estimate the followings to one significant figure:

# Number One significant Nearest

1 23.08 10
2 334 100
3 765.90 100
4 123.53 100
5 189.25 100
6 7234 1000
7 8289 1000
8 3386 1000
9 10024 100 or 10000

Q2 Estimate nearest to square or cubic number, hence evaluate the followings:

3 3
a) √11 b) √9.07 c) √125 d) √60
3
e) √220 f) √139 g) √45 h) √164

Q3 By writing each number correct to 1 significant figure, estimate the value of

Example#

19.75 ×2.95 21.75 ×√4.015

a) b) c)
0.269 0.2192

3
40 √127 ×23.95 21.75 ×69.25
Sol: ≈ d) e)
10 ×0.8 0.4769 0.069
40
= 8

58.7 ×4.08 987.65

=5 f) g)
19.72 0.0193

42.32 × √35.7 614.2 ×0.0304 1212.3

h) i) j)
2980 19.88 299.35 ×√24,73

1240 ×3.8 47.5+36.1 3

k) √ l) m) 9.372 − √1046
11.2 64.9 ÷17.7
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 Ratio: A ratio is a way of comparison two or more same kind of quantities. Ratio has no unit or
are measured in the same unit. e.g ( 𝑎 ∶ 𝑏 )
 The ratio notation 𝑎 ∶ 𝑏, where 𝑎, 𝑏 are positive numbers can be expressed in an equivalent form
𝑎 𝑎 5
as a fraction . e.g. 𝑎 ∶ 𝑏 = 5 ∶ 7 is equivalent to =
𝑏 𝑏 7
 Exercise#5
Q1 write the following in ratio form:

a) There are 7 boys and 5 girls in a school badminton team. Find the ratio of

i) Girls to Boys ii) boys to the total number of players in the team.

b) Simplify the following ratios:

(i) 4 ∶ 12 (ii) 15 ∶ 3 (iii) 25 ∶ 65 (iv) 24 ∶ 20 ∶ 8 (v) 39 ∶ 21 ∶ 9

3 9 3
(vi) ∶ (vii) 1 ∶ (viii) 0.5 ∶ 0.3 (ix) 0.35 ∶ 0.07 (x) 1.6 ∶ 4
8 4 7

3 2 5 1 3 5
(xi) ∶ ∶ (xii) ∶ 2.5 ∶ 3 (xiii) 0.75 ∶ 3 (xiv) 0.33 ∶ 0.63 ∶ 1.8
2 3 8 3 4 16

Q2 Simplify the following ratios:

(i) 1.5 𝑚 𝑡𝑜 300 𝑐𝑚 (ii) 600 𝑚𝑙 𝑡𝑜 1.2 𝑙 (iii) 50 𝐶𝑒𝑛𝑡𝑠 𝑡𝑜 $ 1.25 (iv) 2.4 𝑘𝑔 𝑡𝑜 4000 𝑔

3 1 1
(v) 3 4 𝑙 𝑡𝑜 250 𝑚𝑙 (vi) 24% 𝑡𝑜 1 5 (vii) 0.84 𝑡𝑜 0.84% (viii) 7 7 ∶ 4.5

Q3 Find the ratio of

(i) 580 𝑚𝑙 𝑡𝑜 1.12 𝑙 𝑡𝑜 104 𝑚𝑙 (ii) 2.8𝑘𝑔 𝑡𝑜 700 𝑔 𝑡𝑜 1.05 𝑘𝑔

(iii) 32 𝑚 𝑡𝑜 2.4 𝑘𝑚 𝑡𝑜 64.8 𝑚 (iv) $ 7.60 𝑡𝑜 84 𝐶𝑒𝑛𝑡𝑠 𝑡𝑜 $6

𝟐𝒙 𝟑𝒚
Q4 a) Given that = , find the ratio of 𝑥 ∶ 𝑦.
𝟓 𝟖

𝟕𝒙 𝟏𝟒𝒚
b) Given that = , find the ratio of 𝑥 ∶ 𝑦
𝟗 𝟑

Q5 Given that 𝒂 ∶ 𝒃 = 𝟑: 𝟒 𝒂𝑛𝑑 𝒃 ∶ 𝒄 = 𝟑 ∶ 𝟕, find i) 𝒂 ∶ 𝒃 ∶ 𝒄 ii) 𝒂 ∶ 𝒄

𝟑 𝟏 𝟏
Q6 Given that 𝒑 ∶ 𝒒 = 𝟒 ∶ 𝟐 𝑎𝑛𝑑 𝒑 ∶ 𝒓 = 𝟑 ∶ 𝟐 , find i) 𝒑 ∶ 𝒒 ∶ 𝒓 ii) 𝒒 ∶ 𝒓

𝟗 𝟏𝟓 𝟏
Q7 Given that 𝒙 ∶ 𝟑 ∶ = ∶ 𝟒 ∶ 𝒚, find the value of 𝑥 𝑎𝑛𝑑 𝑦.
𝟐 𝟒 𝟐

Q8 Find the number that must be added to 3 and 8 so that the ratio of the first number to the second
number becomes 𝟐 ∶ 𝟑.
 8

 Percentage: Exercise#6
Q1 Fill in the blanks, show your working.
# Percentage Decimal Fraction
1 𝟐𝟒% 𝟐𝟒 𝟐𝟒 ÷ 𝟒 𝟔
= 𝟎. 𝟐𝟒 =
𝟏𝟎𝟎 𝟏𝟎𝟎 ÷ 𝟒 𝟐𝟓
2 𝟒𝟖%
3 𝟎. 𝟎𝟒𝟓
4 𝟗
𝟐𝟎
5 𝟕𝟓. 𝟓%
6 𝟎. 𝟑𝟖
7 𝟔𝟒%
8 𝟐
𝟑
9 𝟗𝟏. 𝟒%
10 𝟎. 𝟕𝟕𝟔
11 𝟐
𝟖𝟏
𝟑

Q2 For each of the followings, express the first quantity as a percentage of the second quantity.
( a) 𝟒𝟓 𝒎𝒊𝒏𝒖𝒕𝒆𝒔, 𝑰 𝒉𝒐𝒖𝒓 (b) 𝟐𝟓 𝒔𝒆𝒄𝒐𝒏𝒅𝒔, 𝟑. 𝟓 𝒎𝒊𝒏𝒖𝒕𝒆𝒔 (c) 𝟏 𝒚𝒆𝒂𝒓, 𝟒 𝒎𝒐𝒏𝒕𝒉𝒔

(d) 𝟏𝟓 𝒎𝒎 , 𝟏𝒎 (e) 𝟑𝟑𝟓𝒄𝒎, 𝟓𝒎 (f) 𝟏 𝒌𝒈, 𝟖𝟎𝟎 𝒈

(g) 60o , 360o (h) 𝟔𝟑 𝒄𝒆𝒏𝒕𝒔, $𝟐. 𝟏𝟎

Q3 Find the value of each of the followings:

(a) 𝟐𝟒% 𝒐𝒇 𝟓𝟎𝟎 (b) 𝟎. 𝟖% 𝒐𝒇 𝟒. 𝟓 𝒎 (c) 𝟏𝟐𝟎% 𝒐𝒇 𝟕𝟎

𝟒 𝟏
(d) 𝟏𝟏𝟏 𝟓 % 𝒐𝒇 𝟐𝟒 𝒌𝒈 (e) 𝟑𝟏𝟐. 𝟓% 𝒐𝒇 𝟕𝟎 (f) 𝟏𝟐 % 𝒐𝒇 𝟐𝟒𝒎

Q4 A class has 40 students, of which 25% are boys. Given that 70% of the boys passed a mathematics
test, calculate the number of boys who passed the test.

Q5 There are 120 cars in a multi-story car park. Given that 30% of them are blue, find the number of cars
which are not blue.

Q6 There are 600 pages in a novel. Imran reads 150 pages of the novel on Friday and 40% of the
remaining pages on Sunday. Express the number of pages that remains to be read as a percentage of
the total number of pages in the novel.
 9

Q7 Yasir was asked to find the value of 7.5 % of 10,000 m and his answer was 75,000 m. Without
performing any calculation, explain to Yasir why his answer is incorrect.

Q8 99 boys and 1 girl are in a lectur theatre. How many boys ust lave the theatre so that the perentage
of boys becones 98%?

 Prime factors: Exercise#7

Q1 Write down the followings as a product of prime factor, hence write your answer in index notation.
(a) 𝟕𝟐 (b) 𝟏𝟕𝟖 (c) 𝟔𝟑𝟎 (d) 𝟔𝟖𝟎𝟒 (e) 𝟖𝟔𝟐𝟒 (f) 196000

(g) Write 325 as a product of its prime factor.

Q2 If 𝑝 𝑎𝑛𝑑 𝑞 are whole numbers such that 𝑝 × 𝑞 = 37, find the value of 𝑝 + 𝑞. Explain your answer.

Q3 If 𝑛 is a whole number such that 𝑛 × (𝑛 + 42) is a prime number, find the prime number. Explain
your answer.

Q4 If 𝑎 𝑎𝑛𝑑 𝑏 are whole numbers such that 𝑎 × 𝑏 = 2027, find the value of 𝑎 + 𝑏. Explain your answer.

Q5 Given that 𝑥 𝑎𝑛𝑑 𝑦 are the whole numbers such that 𝑥is less than 𝑦 and 𝑥 × 𝑦 = 24. Write down all
the possible pairs (𝑥 , 𝑦)

 HCF (highest common factor) and LCM (least common multiple)

 Exercise#8
Q1 a) Find the highest common factor of each of the following sets of numbers:

(i) 12 𝑎𝑛𝑑 30 (ii) 13 𝑎𝑛𝑑 91 (iii) 126 𝑎𝑛𝑑 240 (iv) 180 𝑎𝑛𝑑 450
(v) 11 𝑎𝑛𝑑 31 (vi) 90, 135 𝑎𝑛𝑑 270 (vii) 15, 60 𝑎𝑛𝑑 75 (viii) 12, 18 and 24

b) Find the smallest number of 𝑚 such that 24𝑚 is a perfect square number.

c) Find the smallest number of 𝑘 such that 32𝑘 is a cube number.

d) Given that 729 = 23 × 32 × 11, find the smallest number 𝑝, such that 729𝑝 is a perfect square
number.
e) Given that 2700 = 22 × 33 × 52 , find the smallest value of 𝑘, such that 2700𝑘 is a cube number.

f) 𝟑𝟏𝟓 = 𝟑𝟐 × 𝟓 × 𝟕, Use this information to find the smallest integer value of n, such that 315n is a
square number.
 10

Q2 a) Given that 729 = 23 × 32 × 11 and 2700 = 22 × 33 × 52 , find HCF of 729 𝑎𝑛𝑑 2700.

[ Hint: HCF is the product of only common prime numbers with smallest power]

Example Solution: 𝑯𝑪𝑭 = 𝟐𝟐 × 𝟑𝟐 = 𝟒 × 𝟗 = 𝟑𝟔

b) Given that 729 = 23 × 32 × 11 and 2700 = 22 × 33 × 52 , find LCM as a product of its prime

factors in index form. [Hint: LCM is product of common prime numbers with largest power and all

the non-common prime numbers)

Example Solution: 𝑳𝑪𝑴 = 𝟐𝟑 × 𝟑𝟑 × 𝟓𝟐 × 𝟏𝟏

c) i) Write 420 as the product of its prime factors.

ii) Given that 𝟏𝟓𝟏𝟐 = 𝟐𝟑 × 𝟑𝟑 × 𝟕, find the highest common factor of 𝟒𝟐𝟎 𝑎𝑛𝑑 𝟏𝟓𝟏𝟐

d) 𝑃 = 𝑥 𝑛 𝑦 2 𝑎𝑛𝑑 𝑄 = 𝑥 𝑛−1 𝑦 4 , where x and y are prime.

Find the highest common factor (HCF) of 𝑃 𝑎𝑛𝑑 𝑄. Give your answer in terms of 𝑥, 𝑦 𝑎𝑛𝑑 𝑛.
e) i) Write 180 as the product of its prime factors.
ii) Expressed as the product of their prime factors, 36 = 22 × 32 𝑎𝑛𝑑 𝑁 = 22 × 3𝑘 , 𝑤ℎ𝑒𝑟𝑒 𝑘 > 3.
180 is the lowest common multiple (LCM) of 36 and N. Find the value of k.
Q3 a) A number has exactly 8 factors, two of which are 4 𝑎𝑛𝑑 20. List all the factors of the number.

b)A number has exactly 8 factors, two of which are 6 𝑎𝑛𝑑 27. List all the factors of the number.

c) A number has exactly 8 factors, two of which are 27 𝑎𝑛𝑑 45. List all the factors of the number.

Q4 The lights on two lightships flash on regular intervals. The first light flashes every 18 second and the
second every 40 seconds. The two lights flash together at 10:00pm. At what time will they next flash
together?

Q5 A number has exactly 12 factors, two of which are 40 𝑎𝑛𝑑 100. List all the factors of the number.

Q6 The two bells toll at regular intervals of 15 minutes and 36 minutes respectively. If they toll together at
2:00 pm, what time will they next toll together?

Q7 Li ting has two pieces of rope measuring 140 cm and 168 cm. She wishes to cut the two pieces of rope
equally into smaller without any leftover rope.
a) What is the greatest possible length of each of the smaller pieces of rope?
b) How many smaller pieces of rope can she cut altogether?

Q8 Ahmad needs to pack 171 pens and 60 pencils equally into identical gift bag. Find
a) the largest number of gift pack that can be packed.
b) The number of each item in the gift bag.
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 Fractions and recurring decimals: The decimals, which contain digits that repeat
indefinitely, are called recurring (or repeating) decimals.
𝟏 𝟏
e.g = 𝟎. 𝟑𝟑𝟑 … = 𝟎. 𝟑̇ or = 𝟎. 𝟏𝟔𝟔𝟔 … = 𝟎. 𝟏𝟔̇
𝟑 𝟔

or
𝟏 ̇ ̇
= 𝟎. 𝟏𝟒𝟐𝟖𝟓𝟕 𝟏𝟒𝟐𝟖𝟓𝟕 … . = 𝟎. 𝟏𝟒𝟐𝟖𝟓𝟕
𝟕

 Exercise #8
Q1 Write the followings as a fraction in its simplest form.
a) 𝟎 . 𝟕̇ b) 𝟎. 𝟑̇𝟔̇
Example Solution: Example Solution:
𝒙 = 𝟎 . 𝟕̇ 𝒙 = 𝟎. 𝟑̇𝟔̇
𝒙 = 𝟎 .𝟕 𝟕 𝟕 𝟕… 𝒙 = 𝟎. 𝟑𝟔 𝟑𝟔 𝟑𝟔 ….
𝟏𝟎𝒙 = 𝟕. 𝟕𝟕𝟕 … 𝟏𝟎𝟎𝒙 = 𝟑𝟔. 𝟑𝟔 𝟑𝟔….
𝟏𝟎𝒙 − 𝒙 = (𝟕. 𝟕𝟕. . ) − (𝟎. 𝟕𝟕. . ) 𝟏𝟎𝟎𝒙 − 𝒙 = (𝟑𝟔. 𝟑𝟔𝟑𝟔. . ) − (𝟎. 𝟑𝟔𝟑𝟔. . )
𝟗𝒙 = 𝟕 𝟗𝟗𝒙 = 𝟑𝟔
𝟕
𝒙=𝟗 𝟑𝟔 𝟒
𝒙= =
𝟗𝟗 𝟏𝟏

Q2 Write the followings as a fraction in its simplest form.

a) 𝟎. 𝟐𝟑̇ b) 𝟎. 𝟎̇ 𝟓̇

c) 𝟎. 𝟏̇ 𝟔 𝟕̇ d) 𝟎. 𝟕 𝟑̇

e) 𝟎. 𝟕 𝟑̇ 𝟏̇ f) 𝟏. 𝟓̇ 𝟔̇

g) 𝟐. 𝟔 𝟕̇ h) 𝟐. 𝟒 𝟑̇ 𝟐̇

i) 𝟑. 𝟐 𝟒̇ 𝟖 𝟒̇ j) 𝟏. 𝟓 𝟕̇