Chapter 1
Chapter 2
Chapter 3
More about Factorization of Polynomials
1A
p.2
1B
p.9
1C
p.17
1D
p.25
1E
p.32
Laws of Indices
2A
p.39
2B
p.49
2C
p.57
2D
p.68
Percentages (II)
3A
p.74
3B
p.83
3C
p.92
3D
p.99
3E
p.107
3F
p.119
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1
F3A: Chapter 1A
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1A Level 2
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Book 3A Lesson Worksheet 1A
1.1
(Refer to §1.1A)
Factorization Using Identities
1.1A Using the Difference of Two Squares Identity
a2 – b2 ≡ (a + b)(a – b)
Example 1
Factorize
(a) x2 – 32,
(b) 52 – y2.
Sol (a)
(b)
Factorize: convert the
polynomial into
the product of its
factors.
a = x, b = 3
x2 – 32
= (x + 3)(x – 3)
(b)
Sol (a)
52 – y2 a = 5, b = y
= (5 + y)(5 – y)
Example 2
Factorize
(a) x2 – 4,
(b) y2 – 36.
Sol (a)
Instant Drill 1
Factorize
(a) p2 – 42,
(b) 82 – q2.
2
4 = 22
(b)
82 – q2
=(
a = ___, b = ___
)(
)
a = ___, b = ___
)(
)
○→ Ex 1A 1, 2
Instant Drill 2
Factorize
(a) h2 – 9,
(b) k2 – 49.
Do you remember
the square
numbers 1, 4, 9,
16, 25, ?
Sol (a)
x –4
= x2 – 22
= (x + 2)(x – 2)
2
(b)
(b) u2 – 100,
)2
9=(
h –9
= ( )2 – ( )2
=(
)(
36 = 62
y2 – 36
= y2 – 62
= (y + 6)(y – 6)
1. Factorize
(a) x2 – 64,
p2 – 42
=(
k2 – 49
= ( )2
=(
49 = (
(
)
)2
)2
)(
)
(c) w2 – 121.
102 = ___
112 = ___
4
2. Factorize
(a) 25 – y2,
(b) 36 – p2,
(c) 81 – n2.
○→ Ex 1A 3–5
Example 3
Factorize
(a) 4x2 – 1,
(b) 4x2 – y2.
Sol (a) 4x2 – 1
a = 2x
= (2x)2 – 12 b = 1
= (2x + 1)(2x – 1)
(b)
Instant Drill 3
Factorize
(a) 9x2 – 16,
(b) 9x2 – 16y2.
Sol (a) 9x2 – 16
=(
)2 – (
=
4x2 – y2
a = 2x
= (2x)2 – y2 b = y
= (2x + y)(2x – y)
(b)
3. Factorize
(a) 1 – 36x2,
(b) 25p2 – 49,
(c) 64 – 81s2.
9x2 – 16y2
=(
)2 – (
=
a=(
2b = (
)
)
a=(
2b = (
)
)
)
)
4. Factorize
4x2 = (
(a) 4x2 – 25y2,
25y2 = (
(b) 9h2 – 64k2,
2
2
(c) 49m – 100n .
)2
)2
○→ Ex 1A 8–14
5. Factorize
(a) x2y2 – 16,
(b) p2q2 – 81.
6. Factorize
(a) 4x2 – y2z2,
(b) s2 – 25p2q2.
x2y2 = (xy)2
5
4x2 = (
y2z2 = (
)2
)2
○→ Ex 1A 15–18
 Level Up Questions
Factorize the following polynomials. [Nos. 7−8]
7. (a) 49b2 – 36a2c2
(b)
=
–9x2y2 + 16w2
= 16w2 – (
=
)
121h2 – 144m2n2
(c)
=
8. (a)
8x2 – 8
= 8(
=
)
Take out the
common
factor 8
first.
(b)
B
6
6x2 – 6y2
= ( )(
=
The common factor is
)
_________.
New Century Mathematics (2nd Edition) 3A
1
More about Factorization of Polynomials

Consolidation Exercise
1A
Level 1
Factorize the following polynomials. [Nos. 1–18]
1. x2 − 22
2. 42 − y2
3. z2 − 64
4. c2 − 100
5. 36 − x2
6. −y2 + 1
7. 1 − 16u2
8. 25m2 − 49
9. 16a2 − 9
10. −81x2 + 25
11. 36a2 − b2
12. 4p2 − 25q2
13. c2d2 − 9
14. −49 + h2k2
15. 16p2 − q2r2
16. x2 − 100y2z2
17. 11x2 − 11
18. 3m2 − 3n2
Level 2
Factorize the following polynomials. [Nos. 19–33]
19. (3 + x)2 − 1
20. (y − 3)2 − 25
21. 121 − (m − 2n)2
22. (3 + x)2 − (1 + 2x)2
23. (2x + 3y)2 − (x − 2y)2
24. (a − 2b)2 − (2a + b)2
25. 3 − 75x2
26. 18c2 − 72d2
27. 6ab2 − 24ac2
28. 5(p + q)2 − 45
29. 32h2 − 2(k − 3)2
30. 18(x + 2y)2 − 2(x − y)2
31. 16 − a2 + 4b − ab
32. 2x + 7y + 4x2 − 49y2
33. p2 + 3q − 3p − q2
7
Answer
Consolidation Exercise 1A
1. (x + 2)(x − 2)
2. (4 + y)(4 − y)
3. (z + 8)(z − 8)
4. (c + 10)(c − 10)
5. (6 + x)(6 − x)
6. (1 + y)(1 − y)
7. (1 + 4u)(1 − 4u) 8. (5m + 7)(5m − 7)
9. (4a + 3)(4a − 3) 10. (5 + 9x)(5 − 9x)
11. (6a + b)(6a − b) 12. (2p + 5q)(2p − 5q)
13. (cd + 3)(cd − 3) 14. (hk + 7)(hk − 7)
15. (4p + qr)(4p − qr) 16. (x + 10yz)(x − 10yz)
17. 11(x + 1)(x − 1) 18. 3(m + n)(m − n)
19. (4 + x)(2 + x)
20. (y + 2)(y − 8)
21. (11 + m − 2n)(11 − m + 2n)
22. (4 + 3x)(2 − x) 23. (3x + y)(x + 5y)
24. −(3a − b)(a + 3b) 25. 3(1 + 5x)(1 − 5x)
26. 18(c + 2d)(c − 2d)27. 6a(b + 2c)(b − 2c)
28. 5(p + q + 3)(p + q − 3)
29. 2(4h + k − 3)(4h − k + 3)
30. 2(4x + 5y)(2x + 7y)
31. (4 − a)(4 + a + b) 32. (2x + 7y)(1 + 2x − 7y)
33. (p − q)(p + q − 3)
8
F3A: Chapter 1B
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1B Level 1
Maths Corner Exercise
1B Level 2
Maths Corner Exercise
1B Level 3
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1B Multiple Choice
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Book 3A Lesson Worksheet 1B
(Refer to §1.1B)
1.1B Using the Perfect Square Identities
(I)
Square of the Sum of Two Numbers
a2 + 2ab + b2 ≡ (a + b)2
Example 1
Factorize x2 + 2x + 1.
Sol
a = x, b =
x2 + 2x + 1
2
= x + 2(x)(1) + 12 1
= (x + 1)2
1. Factorize
(a) x2 + 14x + 49,
(b) s2 + 20s + 100.
Instant Drill 1
Factorize x2 + 16x + 64.
Sol
x2 + 16x + 64 a = ___, b = ___
= ( )2 + 2( )( ) + ( )2
=(
)2
2. Factorize
(a) 4 + 4m + m2,
(b) 16 + 8x + x2.
○→ Ex 1B 1, 3–5, 11, 12
Example 2
Factorize
(a) 9x2 + 6x + 1,
(b) 4y2 + 12y + 9.
Instant Drill 2
Factorize
(a) 4h2 + 20h + 25,
(b) 9k2 + 24k + 16.
9x2 = (3x)2
Sol (a)
2
9x + 6x + 1
= (3x)2 + 2(3x)(1) + 12
= (3x + 1)2
(b) 4y2 + 12y + 9
= (2y)2 + 2(2y)(3) + 32
= (2y + 3)2
3. Factorize
(a) 81x2 + 18x + 1,
(b) 9n2 + 42n + 49.
4h2 + 20h + 25
=(
)2 + 2(
)(
2
=(
)
(b) 9k2 + 24k + 16
=
Sol (a)
4. Factorize
(a) 25y2 + 60y + 36,
(b) 49m2 + 4 + 28m.
11
)+(
Rearrange the
terms.
)2
○→ Ex 1B 6, 13
(II) Square of the Difference of Two Numbers
a2 – 2ab + b2 ≡ (a – b)2
Example 3
Factorize x2 – 8x + 16.
Sol
x2 – 8x + 16 a = x, b =
4
= x2 – 2(x)(4) + 42
= (x – 4)2
5. Factorize
(a) x2 – 4x + 4,
(b) s2 – 16s + 64.
Instant Drill 3
Factorize x2 – 14x + 49.
Sol
x2 – 14x + 49
= ( )2 – 2( )(
=(
)2
a = ___, b = ___
)+(
)2
6. Factorize
(a) 1 – 2y + y2,
(b) 36 – 12k + k2.
○→ Ex 1B 2, 7–9
Example 4
Factorize
(a) 16x2 – 8x + 1,
(b) 9y2 – 12y + 4.
Sol (a) 16x2 – 8x + 1
= (4x)2 – 2(4x)(1) + 12
= (4x – 1)2
(b) 9y2 – 12y + 4
= (3y)2 – 2(3y)(2) + 22
= (3y – 2)2
Instant Drill 4
Factorize
(a) 25h2 – 30h + 9,
(b) 36k2 – 60k + 25.
Sol (a) 25h2 – 30h + 9
=(
)2 – 2(
)(
=(
)2
(b) 36k2 – 60k + 25
=
7. Factorize
(a) 81x2 – 36x + 4,
(b) 16p2 – 56p + 49.
8. Factorize
(a) 64h2 – 48h + 9,
(b) 49y2 + 36 – 84y.
12
)+(
Rearrange the
terms.
)2
○→ Ex 1B 10, 14
9. Factorize
(a) x2 + 8xy + 16y2,
(b) h2 – 6hk + 9k2.
a2 + 2ab + b2
≡
2 (
a − 2ab + b2
≡
(
10.Factorize
(a) x2 + 4y2 + 4xy,
(b) –10pq + 25p2 + q2.
○→ Ex 1B 15–18
 Level Up Questions
Factorize the following polynomials. [Nos.11–14]
11. 25a2 – 40ab + 16b2
12.100x2 + 140xy + 49y2
13. (a) 49x2 + 4y2 + 28xy
(b) – 48ab + 9b2 + 64a2
13
14. 3x2 – 18x + 27
Take out the common factor
of all the terms first.
14
New Century Mathematics (2nd Edition) 3A
1
More about Factorization of Polynomials

Consolidation Exercise
1B
Level 1
Factorize the following polynomials. [Nos. 1–18]
1. x2 + 2(x)(2) + 22
2. x2 − 2(x)(7) + 72
3. k2 + 2k + 1
4. r2 + 16r + 64
5. 49c2 + 14c + 1
6. m2 − 6m + 9
7. u2 − 18u + 81
8. 64y2 − 16y + 1
9. 100 + 20t + t2
10. p2 + 16 − 8p
11. 9k2 + 42k + 49
12. 25 − 40x + 16x2
13. 36x2 + 12xy + y2
14. u2 − 22uv + 121v2
15. 144a2 + 24ab + b2
16. 25p2 − 110pq + 121q2
17. 56cd + 49c2 + 16d2
18. 4m2 + 81n2 − 36mn
Level 2
Factorize the following polynomials. [Nos. 19–33]
19. 3x2 + 18x + 27
20. −4k2 − 28k − 49
21. −100t2 + 120t − 36
22. −a2 + 12ab − 36b2
23. 2m2 − 28mn + 98n2
24. −112x2 − 168xy − 63y2
25. x3 − 4x2 + 4x
26. −p3 + 10p2q − 25pq2
27. 12y + 12xy + 3x2y
28. (x − 2)2 + 6(x − 2) + 9
29. 25(m + n)2 + 10(m + n) + 1
30. 16(a + b)2 − 8c(a + b) + c2
31. (a) m2 − 8m + 16
(b) m2 − 8m + 16 − n2
32. (a) p2 + 18pq + 81q2
(b) p2 + 18pq + 81q2 − 25
33. (a) 36a2 − 12ab + b2
(b) 36a2 − 12ab + b2 − 66a + 11b
15
Answer
Consolidation Exercise 1B
1. (x + 2)2
2. (x − 7)2
3. (k + 1)2
4. (r + 8)2
5. (7c + 1)2
6. (m − 3)2
7. (u − 9)2
8. (8y − 1)2
2
9. (10 + t)
10. (p − 4)2
11. (3k + 7)2
12. (5 − 4x)2
13. (6x + y)2
14. (u − 11v)2
15. (12a + b)2
16. (5p − 11q)2
2
17. (7c + 4d)
18. (2m − 9n)2
19. 3(x + 3)2
20. −(2k + 7)2
21. −4(5t − 3)2
22. −(a − 6b)2
23. 2(m − 7n)2
24. −7(4x + 3y)2
2
25. x(x − 2)
26. −p(p − 5q)2
27. 3y(2 + x)2
28. (x + 1)2
29. (5m + 5n + 1)2 30. (4a + 4b − c)2
31. (a) (m − 4)2
(b) (m − 4 + n)(m − 4 − n)
32. (a) (p + 9q)2
(b) (p + 9q + 5)(p + 9q − 5)
33. (a) (6a − b)2
(b) (6a − b)(6a − b − 11)
16
F3A: Chapter 1C
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1C Level 1
Maths Corner Exercise
1C Level 2
Maths Corner Exercise
1C Level 3
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1C Multiple Choice
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Book 3A Lesson Worksheet 1C
1.2
(Refer to §1.2A)
Factorization Using the Cross-method
1.2A Factorization of Polynomials in the Form of x2 + bx + c
Cross-method
Using the fact that
(x + m)(x + n) = x2 + (m + n)x + mn
to factorize polynomials.
x
×)
x2 +
+)
x2 +
+m
x
+n
mx
nx
+ mn
(m + n)x + mn
Example 1
Instant Drill 1
2
Factorize x + 4x + 3.
Factorize x2 + 8x + 7.
Sol [Step 1: Write the constant term +3 as a
Sol [Step 1: Write the constant term +7 as a
product of two factors.
product of two factors.
+3 = (+1)(+3)
+7 = (+1)(
)
+3 = (–1)(–3)
+7 = (–1)(
)
Step 2: Test each possible pair of factors
Step 2: Test each possible pair of factors
by the cross-method.
by the cross-method.
x
+1
x
–1
x
+1
x
–1
x
+3
x
–3
x
x
+x + 3x = +4x –x – 3x = –4x
+x + ___ = ___ –x _____ = ___
]
]
Find the x term.
Which can give +8x?
Can it give
+4x?
x2 + 4x + 3 = (x + 1)(x + 3)
x2 + 8x + 7 = (x
1. Factorize x2 – 3x + 2.
1: +2 = (
)(
+2 = (
)(
2: x
(
)
x
(
)
)(x
)
2. Factorize x2 – 12x + 11.
)
)
x
x
(
(
1: +11 = (
+11 = (
2: x
(
x
(
)
)
)(
)(
)
)
)
)
x
x
(
(
)
)
x2 – 3x + 2 =
○→ Ex 1C 2, 3
Example 2
Factorize x2 + 2x – 3.
Sol [Step 1: Write the constant term –3 as a
product of two factors.
Instant Drill 2
Factorize x2 + 6x – 7.
Sol [Step 1: Write the constant term (
a product of two factors.
19
) as
–3 = (–3)(+1)
–3 = (+3)(–1)
Step 2: Test each possible pair of factors
by the cross-method.
x
–3
x
+3
x
+1
x
–1
–3x + x = –2x +3x – x = +2x
]
This pair can give
(
)=(
)(
)
(
)=(
)(
)
Step 2: Test each possible pair of factors
by the cross-method.
x
(
) x
(
)
x
(
) x
(
)
]
+2x.
x2 + 2x – 3 = (x + 3)(x – 1)
x2 + 6x – 7 = (x
3. Factorize the following polynomials.
(a) x2 + 12x – 13
1: –13 = (
–13 = (
2: x
(
x
(
)(
)(
)
)
)
)
x
x
(
(
(
)
(b) x2 – 12x – 13
)
)
4. Factorize the following polynomials.
(a) x2 – 16x – 17
x (
)
x
)(x
(b) x2 + 18x – 19
)
○→ Ex 1C 4
Example 3
Factorize x2 + 6x + 8.
Sol [Step 1: Write the constant term +8 as a
product of two factors:
+8: (+1)(+8), (+2)(+4),
(–1)(–8), (–2)(– 4)
Instant Drill 3
Factorize x2 + 7x + 6.
Sol [Step 1: +6: (+1)(+6),
(+2)(+3)
We can skip
writing (–1)(–6)
and (–2)(–3).
Step 2: Test each possible pair of factors
by the cross-method.
x
( )
( )
x
( )
( )
Since the coefficient of x is +6, do we
need to test (–1)(–8), (–2)(– 4)?
Why?
Step 2: Test each possible pair of factors
by the cross-method.
x
+1
+2
x
+8
+4
+x + 8x +2x + 4x
= +9x
= +6x
]
x2 + 6x + 8 = (x + 2)(x + 4)
]
x2 + 7x + 6 = (x
20
)(x
)
5. Factorize x2 – 7x + 10.
1: +10: (
2: x
(
x
(
)(
)
)
7. Factorize x2 + 8x + 15.
6. Factorize x2 – 9x + 14.
), (
(
(
)(
)
)
x
x
)
(
(
)
)
8. Factorize x2 + 10x + 24.
○→ Ex 1C 5–9
9. Factorize x2 + 8x – 9.
10.Factorize –15 + x2 + 2x.
11. Factorize x2 – 4x – 21.
12.Factorize x2 – 22 – 9x.
Arrange the terms in
descending order
first.
○→ Ex 1C 10–12, 18, 19
 Level Up Questions
21
13. Factorize 15x + x2 + 26.
14.Factorize x2 – 3x – 18.
22
New Century Mathematics (2nd Edition) 3A
1
More about Factorization of Polynomials

Consolidation Exercise
1C
Level 1
1. (a) List out all the possible ways of writing −10 as a product of two factors.
(The first one is already done as an example for you.)
−10 = (+1)(−10), −10 = (
)(
), −10 = (
)(
), −10 = (
)(
)
(b) Using the result of (a), factorize the following polynomials.
(i) x2 + 9x − 10
(ii) x2 − 9x − 10
(iii) x2 − 3x − 10
Factorize the following polynomials. [Nos. 2–19]
2. x2 + 4x + 3
3. x2 − 3x + 2
4. x2 + 6x − 7
5. r2 + 5r + 4
6. k2 + 13k + 22
7. a2 − 10a + 9
8. m2 − 12m + 35
9. h2 − 7h + 12
10. w2 + 8w − 9
11. b2 + 3b − 10
12. p2 + 7p − 18
13. c2 − c − 20
14. y2 − 12y − 28
15. q2 + 13q + 40
16. −2v + v2 − 15
17. 4n − 21 + n2
18. −10s + 24 + s2
19. 42 + z2 − 13z
Level 2
Factorize the following polynomials. [Nos. 20–34]
20. −x2 + 10x + 11
21. −x2 − 14x − 13
22. −x2 + 5x − 4
23. −x2 − 4x + 32
24. −2a + 35 − a2
25. 11y − y2 + 12
26. −20 + 12z − z2
27. b2 + 24b + 128
28. m2 − 27m − 90
29. u2 + 8u − 84
30. −14q − q2 + 72
31. x2 − 12xy + 11y2
32. r2 + 4rs − 21s2
33. −p2 − 11pq + 26q2
34. −b2 + 48c2 + 8bc
35.(a) Factorize k2 + 10k − 39.
(b) Hence, factorize hk − 3h − k2 − 10k + 39.
23
Answer
Consolidation Exercise 1C
1. (a) −10 = (−1)(+10), −10 = (+2)(−5),
−10 = (−2)(+5)
(b) (i) (x − 1)(x + 10)
(ii) (x + 1)(x − 10)
(iii) (x + 2)(x − 5)
2. (x + 1)(x + 3)
3. (x − 1)(x − 2)
4. (x + 7)(x − 1)
5. (r + 1)(r + 4)
6. (k + 11)(k + 2) 7. (a − 9)(a − 1)
8. (m − 7)(m − 5) 9. (h − 3)(h − 4)
10. (w + 9)(w − 1)
11. (b + 5)(b − 2)
12. (p + 9)(p − 2)
13. (c + 4)(c − 5)
14. (y − 14)(y + 2) 15. (q + 5)(q + 8)
16. (v + 3)(v − 5)
17. (n + 7)(n − 3)
18. (s − 4)(s − 6)
19. (z − 6)(z − 7)
20. −(x + 1)(x − 11) 21. −(x + 1)(x + 13)
22. −(x − 1)(x − 4) 23. −(x + 8)(x − 4)
24. −(a + 7)(a − 5) 25. −(y − 12)(y + 1)
26. −(z − 10)(z − 2) 27. (b + 16)(b + 8)
28. (m + 3)(m − 30) 29. (u + 14)(u − 6)
30. −(q + 18)(q − 4) 31. (x − 11y)(x − y)
32. (r + 7s)(r − 3s) 33. −(p + 13q)(p − 2q)
34. −(b + 4c)(b − 12c)
35. (a) (k − 3)(k + 13)
(b) (k − 3)(h − k − 13)
24
F3A: Chapter 1D
Date
Task
Lesson Worksheet
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Book Example 17
○
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(Video Teaching)
Book Example 18
○
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Book Example 19
○
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○
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Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
1D Level 1
Maths Corner Exercise
1D Level 2
Maths Corner Exercise
1D Level 3
Maths Corner Exercise
1D Multiple Choice
E-Class Multiple Choice
Self-Test
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Complete and Checked
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Complete and Checked
Problems encountered
Skipped
(Full Solution)
25
Teacher’s
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)
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)
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)
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)
Mark:
_________
Book 3A Lesson Worksheet 1D
(Refer to §1.2B)
1.2B Factorization of Polynomials in the Form of ax2 + bx + c
Example 1
Instant Drill 1
Factorize 2x2 + 7x + 5.
Factorize 3x2 + 5x + 2.
Sol [Step 1: Write 2x2 as a product of two
Sol [Step 1: Write 3x2 as a product of two
factors.
factors.
2x2 = (x)(2x)
3x2 = (
)(
)
Step 2: The constant term +5 can be
Step 2: The constant term (
) can be
written as:
written as:
Do not skip
(+1)(+5), (+5)(+1)
(
)(
), (
)(
)
anyone
Step 3: Test each possible pair ofoffactors
Step 3: Test each possible pair of factors
by the cross-method.
by the cross-method.
x
+1
+5
( ) ( )
( )
2x +5
+1
( ) ( )
( )
+2x + 5x
+10x + x
= +7x
= +11x
]
]
2x2 + 7x + 5 = (x + 1)(2x + 5)
3x2 + 5x + 2 = (
)(
)
1. Factorize 5x2 – 34x – 7.
1: 5x2: (
)(
2: –7: (
)(
(
)(
)
3:
( )
( )
(
)
( )
5x2 – 34x – 7 =
2. Factorize 7x2 – 4x – 11.
)
),
(
)
(
)
3. Factorize 2x2 + 3x – 5.
4. Factorize 3x2 – 10x + 3.
○→ Ex 1D 1–7, 18
Example 2
Factorize 6x2 – 11x + 3.
Sol [Step 1: The term 6x2 can be written as:
(x)(6x), (2x)(3x)
Step 2: The constant term +3 can be
written as:
(–1)(–3), (–3)(–1)
Step 3: Test each possible pair of factors
by the cross-method.
Instant Drill 2
Factorize 8x2 + 14x + 5.
Sol [Step 1: The term 8x2 can be written as:
(
)(
), (
)(
)
Step 2: The constant term (
) can be
written as:
(
)(
), (
)(
)
Step 3: Test each possible pair of factors
by the cross-method.
26
x
6x
–1
–3
–3
–1
–6x – 3x
–18x – x
= –9x
= –19x
2x –1
–3
3x –3
–1
–3x – 6x
–9x – 2x
= –9x
= –11x
6x2 – 11x + 3 = (2x – 3)(3x – 1)
]
5. Factorize 4x2 + 4x – 3.
1:
2:
3:
4x2: (
–3: (
)(
)(
]
8x2 + 14x + 5 = (
)(
)
6. Factorize 10x2 – 11x + 3.
), (
), (
)(
)(
)
)
7. Factorize 14x2 – 19x – 3.
Example 3
Factorize 6x2 + 31x + 14.
Sol [Step 1: The term 6x2 can be written as:
(x)(6x), (2x)(3x)
Step 2: The constant term +14 can be
written as:
(+1)(+14), (+14)(+1),
(+2)(+7), (+7)(+2)
Step 3: Test each possible pair of factors
by the cross-method.
x
6x
+1
+14
+6x + 14x
= +20x
+1
+14
+3x + 28x
= +31x
2x
3x
+14
+1
+84x + x
= +85x
+14
+1
○→ Ex 1D 8–11, 17, 19
Instant Drill 3
Factorize 10x2 + 17x – 6.
Sol [Step 1: The term 10x2 can be written as:
(
)(
), (
)(
)
Step 2: The constant term (
) can be
written as:
(
)(
), (
)(
),
(
)(
), (
)(
)
Step 3: Test each possible pair of factors
by the cross-method.
+2
+7
+7
+2
+12x + 7x +42x + 2x
= +19x
= +44x
+2
+7
+7
+2
]
2
6x + 31x + 14 = (2x + 1)(3x + 14)
]
27
8. Factorize 9x2 – 16x – 4.
1:
2:
9x2: (
– 4: (
(
)(
)(
)(
9. Factorize 15x2 – 22x + 8.
), (
), (
), (
)(
)(
)(
)
),
)
3:
10.Factorize 21x2 + 41x + 10.
11. Factorize 13x + 6x2 – 28.
○→ Ex 1D 12–16, 20, 21
 Level Up Questions
12. Factorize 12x2 – 37x + 21.
13.Factorize –7x2 – 6x + 13.
28
–7x2 – 6x + 13
= –(
)
29
New Century Mathematics (2nd Edition) 3A
1
More about Factorization of Polynomials

Consolidation Exercise
1D
Level 1
Factorize the following polynomials. [Nos. 1–18]
1. 2x2 + 3x + 1
2. 3y2 + 7y + 2
3. 5z2 + 16z + 3
4. 3a2 + a − 2
5. 2b2 − 13b − 7
6. 5n2 − 13n + 6
7. 8y2 + 25y + 3
8. 7u2 − 12u − 4
9. 10t2 + 3t − 1
10. 28d2 − 11d + 1
11. 21x2 − 5x − 6
12. 8m2 + 22m + 15
13. 15r2 − 23r + 4
14. 18y2 + 9y − 14
15. 11 + 14c + 3c2
16. 10k2 − 7 − 9k
17. −2 + 25x2 − 5x
18. −z − 35 + 6z2
Level 2
Factorize the following polynomials. [Nos. 19–33]
19. −2x2 + 5x − 2
20. −3y2 − 40y − 13
21. −11t + 3 − 20t2
22. 8x − 4x2 + 21
23. 2p2 − 24p + 22
24. 5k2 + 90 − 45k
25. −35c + 14c2 − 126
26. −12a2 + 9a + 30
27. 58u − 14 − 48u2
28. 11m2 − 32mn − 3n2
29. 5x2 − 18xy − 8y2
30. 18r2 + 45rs − 38s2
31. ab − 63a2 + 12b2
32. 3p2 + 108q2 − 39pq
33. 4h2 + 42k2 − 34hk
34.(a) Factorize 8x2 + 2xy − 3y2.
(b) Hence, factorize 3y2 − 8x2 + 5y − 10x − 2xy.
30
Answer
Consolidation Exercise 1D
1.
3.
5.
7.
9.
11.
13.
15.
17.
19.
21.
23.
25.
27.
29.
31.
32.
34.
(2x + 1)(x + 1) 2. (3y + 1)(y + 2)
(5z + 1)(z + 3)
4. (3a − 2)(a + 1)
(2b + 1)(b − 7) 6. (5n − 3)(n − 2)
(8y + 1)(y + 3) 8. (7u + 2)(u − 2)
(5t − 1)(2t + 1) 10. (7d − 1)(4d − 1)
(7x + 3)(3x − 2) 12. (4m + 5)(2m + 3)
(5r − 1)(3r − 4) 14. (3y − 2)(6y + 7)
(3c + 11)(c + 1) 16. (5k − 7)(2k + 1)
(5x + 1)(5x − 2) 18. (3z + 7)(2z − 5)
−(2x − 1)(x − 2) 20. −(3y + 1)(y + 13)
−(4t + 3)(5t − 1) 22. −(2x + 3)(2x − 7)
2(p − 11)(p − 1) 24. 5(k − 6)(k − 3)
7(2c − 9)(c + 2) 26. −3(4a + 5)(a − 2)
−2(8u − 7)(3u − 1) 28. (11m + n)(m − 3n)
(5x + 2y)(x − 4y) 30. (3r − 2s)(6r + 19s)
−(7a + 3b)(9a − 4b)
3(p − 9q)(p − 4q) 33. 2(h − 7k)(2h − 3k)
(a) (2x − y)(4x + 3y)
(b) (y − 2x)(5 + 4x + 3y)
31
F3A: Chapter 1E
Date
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Book Example 20
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Book Example 21
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Book Example 22
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Book Example 23
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Book Example 24
○
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(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
1E Level 1
Maths Corner Exercise
1E Level 2
Maths Corner Exercise
1E Level 3
Maths Corner Exercise
1E Multiple Choice
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Complete and Checked
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32
Teacher’s
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)
Teacher’s
Signature
___________
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)
Teacher’s
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)
Teacher’s
Signature
___________
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)
E-Class Multiple Choice
Self-Test
○
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○
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33
Mark:
_________
 Book 3A Lesson Worksheet 1E
1.3
(Refer to §1.3)
Factorization Using the Difference and Sum of Two Cubes Identities
(A) Difference of the Two Cubes Identity
a3 − b3 ≡ (a − b)(a2 + ab + b2)
Example 1
Factorize
(a) x3 − 1,
(b) y3 − 27.
Sol (a) x3 − 1
= x3 − 13
= (x − 1)[x2 + x(1) + 12]
= (x − 1)(x2 + x + 1)
(b)
y3 − 27
= y3 − 33
= (y − 3)[y2 + y(3) + 32]
= (y − 3)(y2 + 3y + 9)
1. Factorize
(a) p3 – 216,
(b) h3 – 1 000.
(a)
Instant Drill 1
Factorize
(a) x3 − 64,
(b) y3 − 125.
Sol (a) x3 − 64
= x3 − (
)3
= ( − )[( )2 + (
=
p3 – 216
=(
)3 − (
=
(b)
y3 − 125
= y3 − (
)3
=
43 = ______
)(
)+(
)2]
53 = ______
2. Factorize
(a) 8 – k3,
(b) 343 – n3.
)3
h3 – 1 000
(b)
=
3. Factorize
(a) 27x3 – 1,
(a)
3
(b) 8h – 125.
4. Factorize
(a) x3 – 64y3,
27x3 – 1
=(
)3 – 13
=
34
Try to memorize the
following cube
numbers:
3
1 =1
63 = 216
3
2 = 512p
8
3 73 =3 343
(b)
–3q .
3
3 = 27
8 = 512
43 = 64
93 = 729
53 = 125
103 = 1 000
(b) 8h3 – 125
=
○→ Ex 1E 2, 4, 6, 8, 10, 11, 13
(B)
Sum of the Two Cubes Identity
a3 + b3 ≡ (a + b)(a2 − ab + b2)
Example 2
Factorize
(a) x3 + 1,
(b) y3 + 8.
Sol (a) x3 + 1
= x3 + 13
= (x + 1)[x2 − x(1) + 12]
= (x + 1)(x2 − x + 1)
(b)
Instant Drill 2
Factorize
(a) x3 + 27,
(b) y3 + 125.
Sol (a) x3 + 27
= x3 + (
)3
= ( + )[( )2 − ( )( ) + (
=
y3 + 8
= y3 + 23
= (y + 2)[y2 − y(2) + 22]
= (y + 2)(y2 − 2y + 4)
5. Factorize
(a) p3 + 216,
7. Factorize
(a) 64p3 + 1,
(b) 343h3 + 27.
3
(b) h + 729.
(b)
y3 + 125
=(
)3 + (
=
6. Factorize
(a) 64 + k3,
)2]
Pay attention to the
3 sign!
)
(b) 343 + n3.
8. Factorize
(a) m3 + 512n3,
(b) 1 000h3 + k3.
○→ Ex 1E 1, 3, 5, 7, 9, 12, 14
9. Factorize
35
(a) 27x3 – 125y3,
(b) 8p3 + 729q3.
○→ Ex 1E 15–18
 Level Up Questions
1
10. Factorize x3 + .
8
1 1 
=
=
8 23 



11. Factorize x3y3 − 512z3.
x3y3 = (xy)3
36
3
New Century Mathematics (2nd Edition) 3A
1
More about Factorization of Polynomials


Consolidation Exercise
1E
Level 1
Factorize the following polynomials. [Nos. 1–15]
1. (3k)3 + 1
2. 1 − (5r)3
3. (11x)3 + 1
4. y3 − 27
5. 64 + z3
6. 1 − 8w3
7. 216c3 + 1
8. 125 − x3y3
9. a3 + 343b3
10. 125m3 − 8
11. 343 + 27s3
12. 512x3 − 729
13. 27x3 − 64y3
14. 125a3 + 729b3
15. 1 000p3 − 343q3
Level 2
Factorize the following polynomials. [Nos. 16–27]
16. x3 −
1
64
17. 8y3 +
1
27
18. 27x3 −
y3
125
19. 4k3 + 108
20. −448r3 + 7
21. 686a3 − 54b3
22. 500x3 − 32y3
23. ab4 − a4b
24. 135xy3 − 40x4
25. (x − 2)3 − 729
26. 64x3 + (x − 1)3
27. (1 − 3x)3 − (1 + 3x)3
28.(a) Factorize 9x2 − 4.
(b) Hence, factorize 729x6 − 64.
37
Answer
Consolidation Exercise 1E
(3k + 1)(9k2 − 3k + 1)
(1 − 5r)(1 + 5r + 25r2)
(11x + 1)(121x2 − 11x + 1)
(y − 3)(y2 + 3y + 9)
(4 + z)(16 − 4z + z2)
(1 − 2w)(1 + 2w + 4w2)
(6c + 1)(36c2 − 6c + 1)
(5 − xy)(25 + 5xy + x2y2)
(a + 7b)(a2 − 7ab + 49b2)
(5m − 2)(25m2 + 10m + 4)
(7 + 3s)(49 − 21s + 9s2)
(8x − 9)(64x2 + 72x + 81)
(3x − 4y)(9x2 + 12xy + 16y2)
(5a + 9b)(25a2 − 45ab + 81b2)
(10p − 7q)(100p2 + 70pq + 49q2)
1 
x 1

16.  x −  x 2 + + 
4
4
16 


1 
2y 1 

17.  2 y +  4 y 2 −
+ 
3
3 9


y 
3 xy y 2 


+
18.  3x −  9 x 2 +
5 
5
25 

19. 4(k + 3)(k2 − 3k + 9)
20. 7(1 − 4r)(1 + 4r + 16r2)
21. 2(7a − 3b)(49a2 + 21ab + 9b2)
22. 4(5x − 2y)(25x2 + 10xy + 4y2)
23. ab(b − a)(b2 + ab + a2)
24. 5x(3y − 2x)(9y2 + 6xy + 4x2)
25. (x − 11)(x2 + 5x + 67)
26. (5x − 1)(13x2 + 2x + 1)
27. −18x(3x2 + 1)
28. (a) (3x + 2)(3x − 2)
(b) (3x + 2)(3x − 2)(81x4 + 36x2 + 16)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
38
F3A: Chapter 2A
Date
Task
Lesson Worksheet
Progress
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Book Example 1
○
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Book Example 2
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Problems encountered
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Book Example 3
○
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Problems encountered
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Consolidation Exercise
Maths Corner Exercise
2A Level 1
Maths Corner Exercise
2A Level 2
Maths Corner Exercise
2A Level 3
Maths Corner Exercise
2A Multiple Choice
E-Class Multiple Choice
Self-Test
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Complete and Checked
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Complete and Checked
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Complete and Checked
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Complete and Checked
Problems encountered
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39
Teacher’s
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)
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Book 3A Lesson Worksheet 2A
(Refer to §2.1)
2.1A Zero Index
a0 = 1, where a ≠ 0.
e.g. 30 = 1, (–5)0 = 1, (2x)0 = 1
1. (a) Evaluate the following without using a calculator.
(i) 70
(ii) (–8)0
(iii) –90
(b) Simplify the following expressions, where b, c, d, e ≠ 0.
b
(i)  
4
0
(ii) –5c 0
(iii) (–d 3e4)0
○→ Ex 2A 1, 2, 16, 17
2.1B Negative Integral Indices
a–n =
1
, where a ≠ 0 and n is a positive integer.
an
e.g. 3–1 =
2. (a) Evaluate the following without using a calculator.
(i) 7–1
(ii) 2–3
1 –2 1
1
1
,4 = 2 =
, (6x)–1 =
3
6
x
16
4
(iii) (–8)–2
(b) Simplify the following expressions (where r, s, t ≠ 0) and express the answers with
positive indices.
(i) r– 4
(ii) (–s)–1
(iii) (8t) –2
○→ Ex 2A 3, 4, 21
Without using a calculator, evaluate the following and give the answers in fractions. [Nos. 3–4]
3. 4–1 + 2–1
4. 3–2 × (–9)0
→
○ Ex 2A 5–9
2.1C Laws of Integral Indices
If m and n are integers and a, b ≠ 0, then
(a) am × an = am + n
(b) am ÷ an = am – n
40
(c) (am)n = am × n
(d) (ab)n = anbn
n
n
a a
(e)   = n
b b
Example 1
Instant Drill 1
Without using a calculator, evaluate
Without using a calculator, evaluate
–4
3
–2
–4
(a) 5 × 5 ,
(b) 7 ÷ 7 .
(a) 65 × 6–3,
(b) 9–7 ÷ 9–8.
Sol (a) 5– 4 × 53
Sol (a) 65 × 6–3
(
)
= 5– 4 + 3
am × an = am + n
= 6( )
–1
=5
=
1
1
=
a–1 =
a
5
(b) 7–2 ÷ 7– 4
(b) 9–7 ÷ 9–8
(
)
= 7–2 – (– 4)
am ÷ an = am – n
= 9( )
–2 + 4
=
=7
2
=7
= 49
→
○ Ex 2A 7–11
Simplify the following expressions (where p, q, r, s ≠ 0) and express the answers with positive
indices. [Nos. 5–6]
5. (a) p–8 × p 2
6. (a) q5 ÷ q–3
am ÷ an = am – n
am × an = am + n
7
–5
–4
3
(b) 3r × r
(b) 16s ÷ 2s
→
○ Ex 2A 22–24
Remember to express the
answers with positive
indices!
Example 2
Without using a calculator, evaluate
(a) (2–3)–2,
−1
4
(b)   .
7
Sol (a) (2–3)–2
= 2–3 × (–2)
= 26
= 64
Instant Drill 2
Without using a calculator, evaluate
(a) (4–1)3,
−1
( a m) n = a m × n
5
(b)   .
3
Sol (a) (4–1)3
=(
)(
=
41
)
(
)
−1
4
(b)  
7
4 −1
= −1
7
−1
7
4
71
=
=
−1
4
41
7
−1
5
(b)  
3
(
)(
=
(
)(
n
an
a
  = n
b b
)
)
=
→
○ Ex 2A 12–15
Simplify the following expressions and express the answers with positive indices. [Nos. 7–8]
(All the letters in the expressions represent non-zero numbers.)
7. (a) (k2)–5
(b) (y–3)–6
m n
(a ) = a
m×n
2
8. (a)  
h
−1
 p
(b)  
q
→
○ Ex 2A 25–27
Example 3
Simplify the following expressions (where
x, y ≠ 0) and express the answers with positive
indices.
(a) (4x)–2
(b) (–5y–1)3
Sol (a) (4x)–2
= 4–2x–2
(ab)n = anbn
1
1
=
a–1 =
2
a
16 x
n
n
a a
  = n
b b
−2
→
○ Ex 2A 30
Instant Drill 3
Simplify the following expressions (where
r, s ≠ 0) and express the answers with positive
indices.
(a) (–3r–1) 4
(b) (2s2)–5
Sol (a) (–3r–1) 4
=(
)( ) (
)( )
=
42
(b)
(–5y–1)3
= (–5)3y–1 × 3
= (–5)3y–3
125
=− 3
y
(b)
(2s2)–5
=
Simplify the following expressions and express the answers with positive indices. [Nos. 9–14]
(All the letters in the expressions represent non-zero numbers.)
9. (a) (7g4)–3
10.(a) (–hk)–9
(b) (–6t –5)2
(b) (xy–2)–4
(ab)n = anbn
→
○ Ex 2A 31
11. (a)
24
(−2s) −3
(b)
(−e −1 ) 5
e6
(3d 3 ) −2
9d
5f 3
(b)
(5 f − 4 ) 2
12.(a)
43
→
○ Ex 2A 28, 29
13.(a) (4h–1k0)3
 − 3 p0 
(b)  − 4 
 q 
−2
14.(a) (6rt –2)2
(b) (–5ab4)–3
→
○ Ex 2A 32, 33
 Level Up Questions
44
15. Evaluate the following without using a calculator.
(a) 4–1 ÷ 2–3 = (2( ))–1 ÷ 2–3
=
(b) 272 × (–3)– 4
Convert 27 into the powers of
3.
i.e. 27 = 3( )
16.Simplify the following expressions (where x, y ≠ 0) and express the answers with positive
indices.
( x 5 y −4 ) −1
(a) (–x–3y2)– 4
(b)
x −6
x 0 y −7
(c)
(− x − 2 y −1 ) −5
45
New Century Mathematics (2nd Edition) 3A
2
Laws of Indices

Consolidation Exercise
2A
Level 1
Without using a calculator, find the values of the following expressions and give the answers in
integers or fractions. [Nos. 1−12]
1. −70
2. (−7)0
3. (190)−2
4.
1
8−1
7. 80 ÷ (−5)−2
 2
10. −  
 3
−1
5. 6−1 + (3 + 1)0
6. 3−2 × (−2−1)
8. 97 × 9–5
9. 7−3 ÷ 7−2
 3
11.  − 
 5
−2
12. (2−2)−2
Simplify the following expressions and express the answers with positive indices. [Nos. 13−27]
(All the letters in the expressions represent non-zero numbers.)
13. (8p)0
14. 5q0
15. −(r−9)0
16. (a3b−3)0
17. (−x0)−6
18. (−y)−9
19. c−4 × c5
20. g ÷ g−4
21. (s−3)5
22. (−k −3)−4
23.
 u4 
25.  
 2 
 1
5 
z 
4
(5b) − 2
−2
24. 
−3
1 
27.  −

26. (3c−2d)2
 2mn 
46
−3
Level 2
Without using a calculator, find the values of the following expressions and give the answers in
integers or fractions. [Nos. 28−33]
 1
 2
−4
28. 25−2 ÷ 5−3
29. (−4) −1 ×  − 
30. 5−4 ÷ 125−4 × 25−5
31. 24 ÷ 6−2 × 2−4
32. (2−4 × 1250) ÷ 12−2
33. 62 − 6–11 ÷ 6−12
Simplify the following expressions and express the answers with positive indices. [Nos. 34−45]
(All the letters in the expressions represent non-zero numbers.)
34. (x −3y −2)−3
35. (7−1a−5b3)−1
(m 2 n −3 ) −1
37.
n −4
 5c 0 
38.  − 6 
 d 
36. (−5−1r3s−2)−2
−2
 s 0t − 2 
39.  −2 4 
s t 
−1
− 35 x −9
3 2 x −7
41. (mn3)−2(nm−4)−1
42.
(−2 x 3 y 4 ) 2
43.
− 4( x −5 y −1 ) − 2
(a 6 ) 2
44. (6a ) ×
( 2a ) −6
 a −5b −1 
 − b −5 
 ×  2 −1 
45. 

 4a b 
 8a 


40.
3 −3
47
(a 2b −1 ) 2
(a −1b) 3
−1
3
Answer
Consolidation Exercise 2A
27. −8m3n3
1
28.
5
29. −4
1
30.
25
31. 36
32. 9
33. 30
34. x9y6
−1
1
1
8
7
5.
6
1.
2.
3.
4.
1
18
7. 25
8. 81
1
9.
7
3
10. −
2
25
11.
9
12. 16
13. 1
14. 5
15. −1
16. 1
17. 1
1
18. − 9
y
6. −
7a 5
b3
25s 4
36. 6
r
35.
37.
38.
39.
40.
41.
19. c
20. g5
1
21. 15
s
22. k12
42.
43.
23. 100b2
44.
24. z10
8
25. 12
u
9d 2
26. 4
c
45.
48
n7
m2
d 12
25
t6
s2
27
− 2
x
m2
n7
a7
b5
y6
− 4
x
8a 9
27
1
− 11
8b
F3A: Chapter 2B
Date
Task
Lesson Worksheet
Progress
○
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(Full Solution)
Book Example 4
○
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○
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(Video Teaching)
Book Example 5
○
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○
Complete
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(Video Teaching)
Book Example 6
○
○
○
Complete
Problems encountered
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(Video Teaching)
Book Example 7
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 8
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
2B Level 1
Maths Corner Exercise
2B Level 2
Maths Corner Exercise
2B Level 3
Maths Corner Exercise
2B Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
Complete and Checked
Problems encountered
Skipped
○
○
○
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○
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Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
(Full Solution)
49
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 3A Lesson Worksheet 2B
2.2
(Refer to §2.2)
Scientific Notation
2.2A Introduction
A positive number expressed in scientific notation is in the form of
a × 10n,
where 1 ≤ a < 10, and n is an integer.
Example 1
Express each of the following numbers in
scientific notation.
(a) 5 000
(b) 10 500 000
Sol (a) 5 000
5 000.
= 5 × 103 Move to the left
Instant Drill 1
Express each of the following numbers in
scientific notation.
(a) 60 000
(b) 4 020 000
Sol (a) 60 000
= 6 × 10( ) 60 000.
for 3 digits
(b)
(b)
10 500 000
= 1.05 × 107 10 500 000.
4 020 000
___ digits
=
4 020 000.
7 digits
___ digits
1. Express each of the following numbers in scientific notation.
(a) 300 000
(b) 87 000 000
(c) 923.1
○→ Ex 2B 2(a)–(c)
Example 2
Express each of the following numbers in
scientific notation.
(a) 0.004
(b) 0.000 062
0.004
Sol (a) 0.004
Move to the right
= 4 × 10–3
for 3 digits
Instant Drill 2
Express each of the following numbers in
scientific notation.
(a) 0.000 3
(b) 0.000 008 4
Sol (a) 0.000 3
= 3 × 10( ) 0.000 3
___ digits
(b)
(b)
0.000 062
= 6.2 × 10–5 0.000 062
0.000 008 4
=
5 digits
2. Express each of the following numbers in scientific notation.
(a) 0.000 07
(b) 0.001 34
50
0.000 008 4
___ digits
(c) 0.000 000 269
(d) 0.000 000 049
○→ Ex 2B 2(d)–(f)
Example 3
Express the following numbers as integers or
decimals.
(a) 7 × 103
(b) 6 × 10–5
(c) 2.1 × 10–6
Sol (a) 7 × 103
7.000
= 7 × 1 000
Move to the right
= 7 000
Instant Drill 3
Express the following numbers as integers or
decimals.
(a) 9.2 × 105
(b) 2 × 10–7
(c) 3.8 × 10–4
Sol (a) 9.2 × 105
= 9.2 × (
)
9.2_______
=
for 3 digits
(b)
_
6 × 10–5
0000 06.
= 6 × 0.000 01
Move to the left
= 0.000 06
(b)
2.1 × 10–6
= 2.1 × 0.000 001
= 0.000 002 1
(c)
2 × 10–7
=
__________2.
for 5 digits
(c)
___ digits
3.8 × 10–4
___ digits
=
0000
002.1
6 digits
________3.8
___ digits
3. Express the following numbers as integers or decimals.
(a) 8 × 104
(b) 7.3 × 106
(c) 4 × 10–8
(d) 5.06 × 10–5
○→ Ex 2B 4
2.2B Applications of Scientific Notation
I. Simplifying Operations
Example 4
Instant Drill 4
51
Without using a calculator, evaluate
5 × 106 + 3.8 × 107 and express the answer in
scientific notation.
Sol
5 × 10 + 3.8 × 10
= 0.5 × 107 + 3.8 × 107
= (0.5 + 3.8) × 107
= 4.3 × 107
6
7
5 × 106
= (0.5 × 101) × 106
= 0.5 × 107
Example 5
Without using a calculator, evaluate
(4 × 102) × (3 × 106) and express the answer in
scientific notation.
Sol
(4 × 102) × (3 × 106)
= (4 × 3) × 102 + 6 am × an = am +
n
= 12 × 108
9
= 1.2 × 10 12 = 1.2 × 101
Without using a calculator, evaluate
6.3 × 108 + 2 × 107 and express the answer in
scientific notation.
6.3 × 10 + 2 × 10
8
Sol
7
=
2 × 107
= [(
) × 101] ×
107
=(
) × 108
Instant Drill 5
Without using a calculator, evaluate
(6 × 10–5) × (7 × 109) and express the answer in
scientific notation.
(6 × 10–5) × (7 × 109)
Sol
=
10–5 × 109
= 10( ) +
(
)
Without using a calculator, evaluate the following expressions and express the answers in scientific
notation. [Nos. 4–7]
4. 7.2 × 106 – 4.2 × 105
5. 2 000 000 + 40 000 000
Express each
term in
scientific
notation
→
○ Ex 2B 5(a), (b)
6. (6.2 × 109) × (2 × 10–13)
7. (8.1 × 10–4) ÷ (3 × 105)
am ÷ an = am –
n
→
○ Ex 2B 5(c), (d)
II. Practical Applications
8. Express the following data in scientific notation.
(a) The radius of moon is about 17 381 000 m.
(b) The length of an Amoeba is about 0.000 22 m.
(c) The world population in 2017 is about 7 510 000 000.
(d) The diameter of a human red blood cell is about 0.000 006 2 m.
‘Explain Your Answer’ Question
52
→
○ Ex 2B 7
9.
The diameter of the Earth is about 1.27 × 104 km. The distance between the Jupiter and the
Earth is about 629 000 000 km. Susan claims that the distance between the Jupiter and the
Earth is more than 50 000 times the diameter of the Earth. Do you agree? Explain your answer.
Distance between the Jupiter and the Earth = 629 000 000 km = (
The required number of times =
∵ ______________ ( > / = / < ) 50 000
∴ The claim is (agreed / disagreed).
) × 10(
)
km
Express this
data in
scientific
notation
 Level Up Questions
10. Round off the following numbers to 3 significant figures and express the results in scientific
notation.
(a) 2 468.3 = (
), cor. to 3 sig. fig.
Round off the number
i.e. 2 468.3 =
, cor. to 3 sig. fig.
first.
(b) 0.000 517 29
11. Without using a calculator, evaluate 7.4 × 106 – 3.62 × 105 + 8 × 104 and express the answer
in scientific notation.
53
New Century Mathematics (2nd Edition) 3A
2
Laws of Indices
Consolidation Exercise
2B

Level 1
1. Determine whether each of the following is expressed in scientific notation. If not, express the
number in scientific notation.
(a) −3.14 × 100
(c) 8.73 × 10−8
(b) 4.27 × (−10)5
(d) −65.3 × 10−5
2. Express each of the following numbers in scientific notation.
(a) 8 000 000 000
(b) 2 296.03
(c) 9 580 000
(e) 0.000 401
(d) 0.003 109
(f) 0.000 098
3. Round off the following numbers to 3 significant figures and express the results in scientific
notation.
(a) 9 753.1
(c) 0.040 742
(e) 360
27
51
(b) 907 684.27
(d) 0.000 246 89
(f) 73
5
11
4. Express each of the following numbers as an integer or a decimal.
(a) 5 × 103
(c) −6.19 × 105
(e) 6.1 × 10–1
(b) 9.53 × 106
(d) −10−3
(f) 3.93 × 10−5
5. Without using a calculator, evaluate the following expressions and express the answers in
scientific notation.
(a) 256 000 000 000 + 8 300 000 000
(c) (5 × 1036) × (2 × 10−24)
(b) 0.000 000 045 − 0.000 000 144
(d) (6 × 10–4) ÷ (3 × 102)
6. Use a calculator to evaluate the following expressions and express the answers in scientific
notation.
(a) 8.5 × 106 + 7.3 × 107
(c) (5.4 × 103) × (9.5 × 10−6)
(b) 2.3 × 1015 − 6.9 × 1014
(d) (2.43 × 10−12) ÷ (9 × 10−7)
54
7. Express the following data in scientific notation.
(a) The surface area of the moon is about 38 000 000 km2.
(b) Hong Kong citizens dispose about 15 000 000 000 g of garbage every day.
(c) The average diameter of human hair is about 0.000 05 m.
(d) The diameter of a water molecule is about 0.000 000 29 mm.
Level 2
8. Without using a calculator, evaluate the following expressions and express the answers in
scientific notation.
(a) 4 × 102 + 0.7 × 104 − 2 × 103
(b) 6.6 × 10−6 + 20 × 10−8 + 1 × 10−7
(c) 34 × 103 + 5.8 × 104 + 0.72 × 105
(d) –840 × 10−2 + 98 × 10−1 + 7 × 101
9. Without using a calculator, find the values of the following expressions and express the answers
in scientific notation.
(a)
(b)
25 000 000 × 5 000 000
0.000 000 000 5
3 000 000 2 − 8 300 000 000 000
2 800 000 000 −1
10.Use a calculator to find the values of the following expressions correct to 2 significant figures,
and express the answers in scientific notation.
 13 000 32 000 
(a) 
+

9 
 7
(c)
11.
2 331
4.6 × 10−6
−3
(b) 83 ÷ (4 001−3 × 4 0078)

6.4 × 10 − 3 

(d) 10 − 2 −
2 × 10 2 

2
A jet has travelled for 39.7 hours. If the average speed of the jet is 2.8 × 102 m/s, find the
distance travelled in m.
(Give the answer correct to 3 significant figures and express the result in scientific notation.)
12. △ABC is a right-angled triangle with ∠ABC = 90°. If AB = 1.3 × 1010 m and
BC = 9.6 × 109 m, find the length of AC in m.
(Give the answer correct to 2 significant figures and express the result in scientific notation.)
55
Answer
Consolidation Exercise 2B
(a) yes
(b) no, −4.27 × 105
(c) yes
(d) no, −6.53 × 10−4
2. (a) 8 × 109
(b) 2.296 03 × 103
(c) 9.58 × 106 (d) 3.109 × 10−3
(e) 4.01 × 10−4 (f) 9.8 × 10−5
3. (a) 9.75 × 103 (b) 9.08 × 105
(c) 4.07 × 10−2 (d) 2.47 × 10−4
(e) 3.61 × 102 (f) 7.35 × 101
4. (a) 5 000
(b) 9 530 000
(c) −619 000
(d) −0.001
(e) 0.61
(f) 0.000 039 3
11
5. (a) 2.643 × 10
(b) −9.9 × 10−8
13
(c) 1 × 10
(d) 2 × 10−6
6. (a) 8.15 × 107 (b) 1.61 × 1015
(c) 5.13 × 10−2 (d) 2.7 × 10−6
7. (a) 3.8 × 107 km2 (b) 1.5 × 1010 g
(c) 5 × 10−5 m (d) 2.9 × 10−7 mm
8. (a) 5.4 × 103
(b) 6.9 × 10−6
(c) 1.64 × 105 (d) 7.14 × 101
9. (a) 2.5 × 1023
(b) 1.96 × 1021
10. (a) 6.3 × 10−12 (b) 8.0 × 10−17
(c) 2.3 × 104
(d) 9.9 × 10−5
11. 4.00 × 107 m
12. 1.6 × 1010 m
1.
56
F3A: Chapter 2C
Date
Task
Lesson Worksheet
Progress
○
○
○
Complete and Checked
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Book Example 9
○
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Book Example 10
○
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Book Example 11
○
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Book Example 12
○
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○
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Problems encountered
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(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
2C Level 1
Maths Corner Exercise
2C Level 2
Maths Corner Exercise
2C Level 3
Maths Corner Exercise
2C Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
Complete and Checked
Problems encountered
Skipped
○
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Complete and Checked
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Complete and Checked
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Complete and Checked
Problems encountered
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Complete and Checked
Problems encountered
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(Full Solution)
57
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
 Book 3A Lesson Worksheet 2C
(Refer to §2.3)
2.3A Denary System and Denary Numbers
(a) Denary system consists of ten numerals:
0, 1, 2, 3, 4, 5, 6, 7, 8 and 9
e.g. 2310 and 48910 are denary numbers.
(b) Taking 8 54210 as an example,
8
5
Digit
3
Place value
10
102
× 10
× 10
4
10
‘10’ indicates
that they are
denary numbers.
2
1
× 10
∴ 8 54210 = 8 × 103 + 5 × 102 + 4 × 10 + 2 × 1
The expression is called the expanded form of 8 54210.
Example 1
Consider the denary number 7 05810.
(a) Write down the place value of each digit in
the number.
(b) Hence, express 7 05810 in the expanded
form.
Sol (a) The place value of each digit is listed
as follows:
7
0
5
8
Digit
Place value 103 102 10
1
(b) 7 05810
= 7 × 103 + 0 × 102 + 5 × 10 + 8 × 1
Instant Drill 1
Consider the denary number 6 90410.
(a) Write down the place value of each digit in
the number.
(b) Hence, express 6 90410 in the expanded
form.
Sol (a) The place value of each digit is listed
as follows:
6
9
0
4
Digit
10
1
Place value
(b) 6 90410
= 6 × 10( ) + 9 × (
)+
1. Consider the denary number 29810.
(a) Write down the place value of each
digit in the number.
(b) Hence, express 29810 in the expanded
form.
2. Express the denary number 18 30710 in the
expanded form.
Digit
Place
v
a
(a) The place value of each digit is listed
as follows:
Digit
Place value
(b)
1
Represent each of the following expressions as a denary number. [Nos. 3–4]
58
8
3
0
7
→
○ Ex 2C 1, 6, 9
3. 5 × 102 + 6 × 10 + 7 × 1
4. 8 × 103 + 0 × 102 + 5 × 10 + 9 × 1
Digit
Place value
102
10
1
→
○ Ex 2C 12
2.3B Binary System and Binary Numbers
(a) Binary system consists of two numerals: 0 and 1
e.g. 1012 and 1001102 are binary numbers.
(b) Taking 11102 as an example,
1
Digit
Place value
23
1
22
1
2
×2
×2
0
1
‘2’ indicates
that they are
binary numbers.
×2
∴ 11102 = 1 × 23 + 1 × 22 + 1 × 2 + 0 × 1
The expression is called the expanded form of 11102.
Example 2
Consider the binary number 10112.
(a) Write down the place value of each digit in
the number.
(b) Hence, express 10112 in the expanded
form.
Sol (a) The place value of each digit is listed
as follows:
1
0
1
1
Digit
Place value
23
22
2
1
(b) 10112
= 1 × 23 + 0 × 22 + 1 × 2 + 1 × 1
Instant Drill 2
Consider the binary number 11112.
(a) Write down the place value of each digit in
the number.
(b) Hence, express 11112 in the expanded
form.
Sol (a) The place value of each digit is listed
as follows:
1
1
1
1
Digit
2
1
Place value
(b) 11112
=1×(
)+1×(
)+
5. Consider the binary number 1002.
(a) Write down the place value of each
digit in the number.
(b) Hence, express 1002 in the expanded
form.
6. Consider the binary number 110012.
(a) Write down the place value of each
digit in the number.
(b) Hence, express 110012 in the
expanded form.
(a) The place value of each digit is listed
as follows:
Digit
59
Place value
(b)
→
○ Ex 2C 2, 7
Express the following binary numbers in the expanded form. [Nos. 7–8]
7. 10012
8. 101012
Digit
Place
v
a
1
0
0
1
Digit
Place
v
a
1
0
1
0
1
○→ Ex 2C 10
Represent each of the following expressions as a binary number. [Nos. 9–10]
9. 1 × 22 + 1 × 2 + 0 × 1
10.1 × 24 + 0 × 23 + 0 × 22 + 1 × 2 + 1 × 1
Digit
Place value
22
2
1
→
○ Ex 2C 13
2.3C Hexadecimal System and Hexadecimal Numbers
(a) Hexadecimal system consists of sixteen numerals:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F,
where the values of A to F are as follows:
A
B
C
D
E
10
11
12
13
14
e.g. 1A2B16 and 56CD8916 are hexadecimal numbers.
(b) Taking C50D16 as an example,
C
5
Digit
3
Place value
16
162
× 16
0
16
× 16
D
1
× 16
∴ C50D16 = 12 ×
+5×
+ 0 × 16 + 13 × 1
The expression is called the expanded form of C50D16.
163
162
60
F
15
‘16’ indicates
that they are
hexadecimal
numbers.
Example 3
Consider the hexadecimal number 34AB16.
(a) Write down the place value of each digit in
the number.
(b) Hence, express 34AB16 in the expanded
form.
Sol (a) The place value of each digit is listed
as follows:
3
4
A
B
Digit
Place value 163 162 16
1
(b) 34AB16
= 3 × 163 + 4 × 162 + 10 × 16 + 11 × 1
Instant Drill 3
Consider the hexadecimal number 2CE16.
(a) Write down the place value of each digit in
the number.
(b) Hence, express 2CE16 in the expanded
form.
Sol (a) The place value of each digit is listed
as follows:
2
C
E
Digit
1
Place value
(b) 2CE16
=2×(
)+(
)×(
)+
11. Consider the hexadecimal number F0516.
(a) Write down the place value of each
digit in the number.
(b) Hence, express F0516 in the expanded
form.
12.Consider the hexadecimal number D7A816.
(a) Write down the place value of each
digit in the number.
(b) Hence, express D7A816 in the
expanded form.
(a) The place value of each digit is listed
as follows:
Digit
Place value
(b)
→
○ Ex 2C 3, 8
Express the following hexadecimal numbers in the expanded form. [Nos. 13–14]
13.36916
14.B657016
Digit
Place value
3
6
9
Digit
Place
v
a
61
B
6
5
7
0
○→ Ex 2C 11
Represent each of the following expressions as a hexadecimal number. [Nos. 15–16]
15.8 × 162 + 14 × 16 + 12 × 1
16.13 × 163 + 0 × 162 + 15 × 16 + 11 × 1
Digit
Place value
162
16
1
→
○ Ex 2C 14
 Level Up Questions
17. Represent 5 × 104 + 3 × 102 + 2 × 10 as a denary number.
Digit
Place value
5 × 104 + 3 × 102 + 2 × 10
= 5 × 104 + (
) × 103 + 3 × 102 + 2 × 10 + (
=
5
104
)×(
18.Represent 25 + 23 + 1 as a binary number.
19. Represent 165 + 162 + 16 + 1 as a hexadecimal number.
62
0
103 102
10
1
Add the place holder ‘0’
in suitable places.
)
63
New Century Mathematics (2nd Edition) 3A
2
Laws of Indices


Consolidation Exercise
2C
Level 1
1. Write down the place value of each digit in 3 04510.
Digit
3
0
4
Place value
5
2. Write down the place value of each digit in 101102.
Digit
1
0
1
Place value
1
3. Write down the place value of each digit in A4CF16.
Digit
A
4
C
Place value
F
0
4. Write down the place value of the underlined digit in each of the following numbers.
Number
Place value
4610
(a)
100112
(b)
357916
(c)
110001012
(d)
D24B16
(e)
5. Consider the denary number 3 57910.
(a) Write down the place value of each digit in the number.
(b) Hence, express 3 57910 in the expanded form.
6. Consider the binary number 10112.
(a) Write down the place value of each digit in the number.
(b) Hence, express 10112 in the expanded form.
7. Consider the hexadecimal number E4B216.
(a) Write down the place value of each digit in the number.
(b) Hence, express E4B216 in the expanded form.
64
8. Express the following denary numbers in the expanded form.
(a) 2310
(b) 15310
(c) 2 34510
9. Express the following binary numbers in the expanded form.
(a) 1102
(b) 11012
(c) 110012
10. Express the following hexadecimal numbers in the expanded form.
(a) B516
(b) C3F16
(c) A5DF16
11.Represent each of the following expressions as a denary number.
(a) 4 × 10 + 7 × 1
(b) 7 × 102 + 6 × 10 + 0 × 1
(c) 8 × 102 + 0 × 10 + 1 × 1
12.Represent each of the following expressions as a binary number.
(a) 1 × 22 + 0 × 2 + 1 × 1
(b) 1 × 23 + 0 × 22 + 1 × 2 + 1 × 1
13.Represent each of the following expressions as a hexadecimal number.
(a) 9 × 162 + 5 × 16 + 3 × 1
(b) 11 × 162 + 10 × 16 + 2 × 1
Level 2
14.Represent each of the following expressions as a denary number.
(a) 5 × 102 + 3 × 10
(b) 2 × 105 + 5 × 102 + 4 × 10
(c) 8 × 100 + 9 × 100 000 + 3 × 1 000 + 4
15.Represent each of the following expressions as a binary number.
(a) 24 + 23
(b) 1 × 16 + 0 × 8 + 1 × 2 + 0 × 1 + 0 × 4
(c) 1 × 32 + 3 × 16 + 1 × 8 + 1 × 4 + 0 × 2 + 2 × 1
16.Represent each of the following expressions as a hexadecimal number.
(a) 2 × 165 + 10 × 164 + 256
(b) 15 × 16 + 12 × 163 + 2 × 164 + 4 × 162 + 13
(c) 18 × 16 + 14 × 163 + 3 × 164 + 2 × 162 + 19
65
17. Use each of the digits 0, 2, 3 and 6 once only to form the smallest 4-digit denary number and the
largest 4-digit denary number respectively. Write these two numbers in the expanded form.
18. (a)
Find the smallest 4-digit binary number in which only one digit is 0. Write the number in
the expanded form.
(b) Find the largest 4-digit binary number in which only one digit is 0. Write the number in
the expanded form.
19. Evaluate each of the following expressions and express the answer as a denary number.
(a) 5216 + 1610 + 102
(b) (34516 − 67810) ÷ 112
20. If x is a digit between 0 and 9 inclusive such that x5A16 = 2 39410, find the value of x.
21. If y is a digit between 0 and 9 inclusive such that 1yAB16 + 1012 = 5 04010, find the value of y.
66
Answer
Consolidation Exercise 2C
1.
3
Digit
2.
4
5
10
1
10
Digit
1
0
1
1
0
4
3
2
2
1
2
2
A
Digit
Place value
4. (a) 1
(d) 25
10
2
Place value
Place value
3.
0
3
16
(b) 22
(e) 163
2
4
3
16
2
C
F
16
1
(c) 16
7
9
10
1
(b) 3 × 103 + 5 × 102 + 7 × 10 + 9 × 1
1
0
1
Digit in 10112
6. (a)
1
5. (a)
Digit in 3 57910
Place value
3
10
3
10
2
1
(b) 1 × 23 + 0 × 22 + 1 × 2 + 1 × 1
4
B
Digit in E4B216 E
7. (a)
2
Place value
163
2
2
2
Place value
2
5
3
162
16
1
(b) 14 × 163 + 4 × 162 + 11 × 16 + 2 × 1
8. (a) 2 × 10 + 3 × 1
(b) 1 × 102 + 5 × 10 + 3 × 1
(c) 2 × 103 + 3 × 102 + 4 × 10 + 5 × 1
9. (a) 1 × 22 + 1 × 2 + 0 × 1
(b) 1 × 23 + 1 × 22 + 0 × 2 + 1 × 1
(c) 1 × 24 + 1 × 23 + 0 × 22 + 0 × 2 + 1 × 1
10. (a) 11 × 16 + 5 × 1
(b) 12 × 162 + 3 × 16 + 15 × 1
(c) 10 × 163 + 5 × 162 + 13 × 16 + 15 × 1
11. (a) 47
(b) 760
(c) 801
12. (a) 1012
(b) 10112
13. (a) 95316
(b) BA216
14. (a) 530
(b) 200 540 (c) 903 804
15. (a) 110002
(b) 100102 (c) 10111102
16. (a) 2A010016 (b) 2C4FD16 (c) 3E33316
17. smallest: 2 036,
2 × 103 + 0 × 102 + 3 × 10 + 6 × 1
largest: 6 320,
6 × 103 + 3 × 102 + 2 × 10 + 0 × 1
18. (a) 10112, 1 × 23 + 0 × 22 + 1 × 2 + 1 × 1
(b) 11102, 1 × 23 + 1 × 22 + 1 × 2 + 0 × 1
19. (a) 10010
(b) 5310
20. 9
21. 3
67
F3A: Chapter 2D
Date
Task
Lesson Worksheet
Progress
○
○
○
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(Full Solution)
Book Example 13
○
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(Video Teaching)
Book Example 14
○
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○
Complete
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Skipped
(Video Teaching)
Book Example 15
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 16
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
2D Level 1
Maths Corner Exercise
2D Level 2
Maths Corner Exercise
2D Level 3
Maths Corner Exercise
2D Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
Complete and Checked
Problems encountered
Skipped
○
○
○
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○
○
○
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Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
(Full Solution)
68
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
 Book 3A Lesson Worksheet 2D
(Refer to §2.4A)
2.4A Conversion of Binary or Hexadecimal Numbers into Denary Numbers
Example 1
Convert 1102 into a denary number.
Instant Drill 1
Convert 10112 into a denary number.
Express 1102 in the expanded
form first.
Sol
Digit
Place value
1102
= 1 × 22 + 1 × 2 + 0 × 1
=4+2+0
=6
1
22
1
2
0
1
Sol
1. Convert 100012 into a denary number.
Digit
Place
v
a
1
0
0
Digit
Place
val
) + 0ue
×(
1
0
1
1
10112
=1×(
) + __________
_______________________________
=
2. Convert 110012 into a denary number.
0
Digit
Place
v
a
1
1
1
0
0
1
→
○ Ex 2D 1–6
Example 2
Convert 12B16 into a denary number.
Instant Drill 2
Convert 1A516 into a denary number.
Express 12B16 in the expanded
form first.
Sol
Digit
1
12B16
162
Place value
= 1 × 162 + 2 × 16 + 11 × 1
= 256 + 32 + 11
= 299
2
16
B
1
Sol
3. Convert CD16 into a denary number.
Digit
Place value
C
Digit
Place
value
1
A
5
1A516
=1×(
)+(
)×(
)+
_________________________
=
4. Convert 1F016 into a denary number.
D
Digit
Place value
A = 10, B =
11, …
69
1
F
0
→
○ Ex 2D 7–15
‘Explain Your Answer’ Question
5. Paul claims that 1111102 must be greater than BE16 because 1111102 has more digits than BE16.
Do you agree? Explain your answer.
1111102 = 1 × (
) + ___________________________________
______________________________________________
=
First, convert 1111102 and
BE16 into denary
numbers. Then, do
comparison.
BE16 =
∵
i.e. 1111102 (> / = / <) BE16
∴ The claim is (agreed / disagreed).
 Level Up Questions
6. Convert 11101112 into a denary number.
Digit
Place
v
a
7. Convert DEC16 into a denary number.
70
1
1
1
0
1
1
1
New Century Mathematics (2nd Edition) 3A
2
Laws of Indices

Consolidation Exercise
2D
Level 1
Convert the following binary numbers into denary numbers. [Nos. 1−5]
1. (a) 102
(b) 1002
2. (a) 1112
(b) 10112
3. (a) 110002
(b) 110012
4. (a) 100112
(b) 111112
5. (a) 1100102
(b) 1001112
Convert the following hexadecimal numbers into denary numbers. [Nos. 6−12]
6. (a) 2916
(b) D716
7. (a) 2F16
(b) E016
8. (a) 35716
(b) 60E16
9. (a) B3016
(b) C0F16
10. (a) 201716
(b) 6C5B16
11. (a) DBA16
(b) DFAC16
12. (a) A3BD16
(b) BDFC16
71

Level 2
Convert the following denary numbers into binary numbers. [Nos. 13−15]
13. (a) 3010
(b) 3210
14. (a) 11210
(b) 23710
15. (a) 37710
(b) 39310
Convert the following denary numbers into hexadecimal numbers. [Nos. 16−18]
16. (a) 6410
(b) 24310
17. (a) 42610
(b) 62510
18. (a) 2 57510
(b) 3 66410
19. Convert the following binary numbers into hexadecimal numbers.
(a) 11001002
(b) 100110112
20. Convert the following hexadecimal numbers into binary numbers.
(a) 4216
(b) CD16
21. Consider the following three numbers:
111111112, 24010, F416
Arrange them in descending order.
72
Answer
Consolidation Exercise 2D
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
(a) 2
(b) 4
(a) 7
(b) 11
(a) 24
(b) 25
(a) 19
(b) 31
(a) 50
(b) 39
(a) 41
(b) 215
(a) 47
(b) 224
(a) 855
(b) 1 550
(a) 2 864
(b) 3 087
(a) 8 215
(b) 27 739
(a) 3 514
(b) 57 260
(a) 41 917
(b) 48 636
(a) 111102
(b) 1000002
(a) 11100002
(b) 111011012
(a) 1011110012 (b) 1100010012
(a) 4016
(b) F316
(a) 1AA16
(b) 27116
(a) A0F16
(b) E5016
(a) 6416
(b) 9B16
(a) 10000102
(b) 110011012
111111112 > F416 > 24010
73
F3A: Chapter 3A
Date
Task
Lesson Worksheet
Progress
○
○
○
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(Full Solution)
Book Example 1
○
○
○
Complete
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(Video Teaching)
Book Example 2
○
○
○
Complete
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(Video Teaching)
Book Example 3
○
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○
Complete
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Skipped
(Video Teaching)
Book Example 4
○
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○
Complete
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Skipped
(Video Teaching)
Book Example 5
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
3A Level 1
Maths Corner Exercise
3A Level 2
Maths Corner Exercise
3A Level 3
Maths Corner Exercise
3A Multiple Choice
○
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○
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Skipped
○
○
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○
○
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○
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Skipped
Complete and Checked
Problems encountered
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Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
(Full Solution)
74
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
E-Class Multiple Choice
Self-Test
○
○
○
Complete and Checked
Problems encountered
Skipped
75
Mark:
_________
Book 3A Lesson Worksheet 3A
3.1
(Refer to §3.1)
Simple Interest
If the principal is $P and the annual interest rate is R%, then
the simple interest ($I) received after T years is given by:
I = P × R% × T
Example 1
$4 000 is deposited in a bank at an annual
interest rate of 3%. Find the simple interest
received after
(a) 2 years,
(b) 5 years.
Sol (a) Simple interest received
= $4 000 × 3% × 2
= $240
P = 4 000, R = 3,
(b) Simple interest received
T = 2. Find I.
= $4 000 × 3% × 5
= $600
Instant Drill 1
$6 000 is deposited in a bank at an annual
interest rate of 4%. Find the simple interest
received after
(a) 3 years,
(b) 8 years.
Sol (a) Simple interest received
= $(
)×(
)% × (
)
=$
P=?
(b) Simple interest received
R
=?
= $(
)×(
)% × (
)
T
=$
1. $30 000 is deposited in a bank at an
interest rate of 10% p.a. Find the simple
interest received after
(a) 7 years,
(b) 2.5 years.
2. $15 000 is deposited in a bank at an interest
rate of 8% p.a. Find the simple interest
received after
(a) 6 months,
(b) 9 months.
6 months
(a) Simple interest received
‘p.a.’ means
= $(
)×(
) ‘per
×(
)
=
year’.
(a) Simple interest received
= $(
)×(
=
(
)
( ( ))
)×
( )
years
=
(b)
(b)
9 months
( )
=
years
( )
○→ Ex 3A 1
76
Example 2
Mr Chan deposits $2 000 in a bank at an
interest rate of 5% p.a. Find the time required
to receive a simple interest of $600.
Sol Let T years be the time required.
600 = 2 000 × 5% × T
600 = 100T
Set up an equation
T=6
to find T.
∴ The time required is 6 years.
Instant Drill 2
Miss Wong deposits $5 000 in a bank at an
interest rate of 6% p.a. Find the time required
to receive a simple interest of $1 200.
Sol Let T years be the time required.
(
)=(
)×(
)×T
=
3. Gloria deposits $9 000 in a bank at an
interest rate of 3% p.a. How long will it
take to receive a simple interest of $945?
4. A sum of money is deposited in a bank at
an interest rate of 2% p.a. The simple
interest received after 4 years will be $260.
Find the sum of money deposited.
∴ The time required is
years.
Let $P be the sum of money deposited.
(
)=(
)×(
)×(
)
=
∴ The sum of money deposited is
.
○→ Ex 3A 4, 5
5. Ben borrows a sum of money from a bank.
The interest rate is 9% p.a. If he has to pay
a simple interest of $10 800 after 5 years,
find the sum of money borrowed.
6. $75 000 is deposited in a bank. The simple
interest received after 8 years will be
$42 000. Find the interest rate per annum.
○→ Ex 3A 6
○→ Ex 3A 7, 8
77
(a) Amount ($A) = principal ($P) + interest ($I)
(b) Since I = P × R% × T, we have
A = P(1 + R% × T)
i.e. A = P + I
A=P+I
= P + P × R% × T
= P(1 + R% × T)
Example 3
Edward deposits $700 in a bank at a simple
interest rate of 4% p.a. Find the amount
received after 2 years.
Sol Interest = $700 × 4% × 2 I = P × R% × T
= $56
Amount = $(700 + 56)
A=P+I
= $756
Alternative
A = P(1 + R% × T)
Amount = $700 × (1 + 4% × 2)
= $756
Instant Drill 3
Joey deposits $3 000 in a bank at a simple
interest rate of 6% p.a. Find the amount
received after 5 years.
Sol Interest = $(
)×(
)×(
)
=
Amount = $[(
)+(
)]
=
Alternative
Amount = $(
) × [1 + (
)×(
=
7. Mr Poon deposits $5 000 in a bank at a
simple interest rate of 3% p.a. Find the
amount received after 4.5 years.
8. Teresa borrows $32 000 from a bank at a
simple interest rate of 10% p.a. How much
will she repay after 3 months?
A=P+I
or
A = P(1 + R% ×
T)
9. $1 800 is deposited in a bank at a simple
interest rate of 5% p.a. How long will it
take to receive an amount of $2 160?
)]
○→ Ex 3A 2, 3
10. Samuel deposits a sum of money in a bank
at a simple interest rate of 2% p.a. If he
receives an amount of $51 300 after
7 years, find the sum of money deposited.
○→ Ex 3A 10, 11
○→ Ex 3A 12
78
11. Flora deposits $6 000 in a bank at a certain simple interest rate. If she receives an amount of $9
240 after 9 years, what is the interest rate per annum?
Method 1
Method 2
A = P(1 + R% × T)
Interest = $[(
)–(
)]
Let R% be the interest rate per annum.
=
(
)=(
)×(
)
Let R% be the interest rate per annum.
=
(
)=(
) × R% × (
)
=
○→ Ex 3A 13
‘Explain Your Answer’ Question
12. Mr Hung invests $20 000 in a bond which offers simple interest at 4.7% p.a. Will the simple
interest received after 20 years be greater than his original principal? Explain your answer.
Simple interest received =
∵ $ ________ ( < / = / > ) $20 000
∴ The simple interest received (will be / will not be) greater than his original principal.
 Level Up Question
13. A sum of money $P is deposited in a bank at a simple interest rate of 5% p.a. How long will it
take to receive an amount which is 3 times the original principal?
Amount = $______
Let T years be the time required.
(
)=(
) × [(
)+(
Express the amount in terms of
P.
Then set up an equation to find
)×(
=
79
)]
New Century Mathematics (2nd Edition) 3A
3
Percentages (II)

Consolidation Exercise
3A
Level 1
1. Complete the following table.
Principal
Interest rate (p.a.)
$2 000
5%
(a)
$3 000
6.25%
(b)
$47 000
3%
(c)
$96 000
8%
(d)
Time
4 years
8 years
2.5 years
13 months
Simple interest
2. $175 000 is deposited in a bank at a simple interest rate of 4% p.a. Find the amount received
after 9 months.
3. $8 000 is deposited in a bank at an interest rate of 3.5% p.a. How long will it take to receive a
simple interest of $560?
4. $96 000 is deposited in a bank at an interest rate of 1% p.a. How many months will it take to
receive a simple interest of $4 320?
5. A sum of money is deposited in a bank at an interest rate of 2% p.a. The simple interest
received after 7.5 years will be $4 119. Find the sum of money deposited.
6. $30 000 is deposited in a bank. The simple interest received after 6 years will be $14 400. Find
the interest rate per annum.
7. A sum of money is deposited in a bank at an interest rate of 3% p.a. The simple interest
received after 9 months will be $1 800. Find the amount received.
8. $24 000 is deposited in a bank at a simple interest rate of 7% p.a. How long will it take to
receive an amount of $29 040?
9. Sam invests $60 000 at a simple interest rate of 4.8% p.a. How long will it take to receive an
amount of $75 840?
10. Mandy deposits a sum of money in a bank at a simple interest rate of 4% p.a. If she receives an
amount of $7 410 after 42 months, find the sum of money deposited.
11. Samson plans to deposit $50 000 in either bank H or bank K. The table below shows the simple
interest rates per annum offered by the two banks.
Bank H
Bank K
5.4% p.a.
3.7% p.a.
Interest rate
Samson will take out the amount after 4 years.
(a) Which bank should Samson choose in order to earn more interest? Explain your answer.
(b) If Samson deposits the money in the bank in (a), find the interest he will earn.
80
Level 2
12. A railway company plans to borrow $5 000 000 from a bank. The amount will be repaid after 6
years. Bank M charges simple interest at 4% p.a. while bank N charges simple interest at 5.5% p.a.
How much more interest will the company pay if it borrows the money from bank N
rather than bank M?
13.Susan deposits $36 000 in bank A at a simple interest rate of 2% p.a. and $5 000 in bank B at a
simple interest rate of 1.8% p.a. Find the total amount she will receive after 7 years.
14.Annie deposits $380 000 in a bank at a simple interest rate of 6% p.a.
(a) Find the amount received after 9 months.
(b) If she deposits the amount in (a) in another bank which offers a simple interest rate of
4% p.a., how many months will it take to receive an amount of $401 071?
15. When a sum of money is deposited in a bank at a simple interest rate of 5% p.a., how long will
it take to triple the original sum of money?
16.Jessica borrows $30 000 from a bank at a simple interest rate of R% p.a. Find the value of R in
each of the following.
(a) The interest is 21% of the original principal after 3.5 years.
(b) The amount is 2 times the original principal after 20 years.
17. William deposits $18 000 in a bank at a simple interest rate of R% p.a. for T years. If he
deposits the money for further 6 months, he will receive $360 more. Find the value of R.
18.Rick wants to borrow $41 000 from a bank for 3 years. The bank offers the following loan
scheme.
For a loan term of more than 4 years, the interest rate is 9% p.a. Otherwise,
the interest rate is 7% p.a.
Note: Simple interest is charged for all loans.
(a) Find the interest that Rick will pay after 3 years.
(b) If Rick saves a fixed amount of money every month during these 3 years to repay the loan
and the interest found in (a), can the amount of money saved each month be less than
$1 300? Explain your answer.
19.Mrs Chan deposits $100 000 in a bank at a simple interest rate of 3% p.a. for 2 months.
Meanwhile, she makes an investment of $30 000 and loses 1.25%. On the whole, does she
make a profit or a loss? Explain your answer.
81
Answer
Consolidation Exercise 3A
1. (a) $400
(b) $1 500
(c) $3 525
(d) $8 320
2. $180 250 3. 2 years
4. 54
5. $27 460
6. 8%
7. $81 800
8. 3 years
9. 5.5 years
10. $6 500
11. (a) bank H
(b) $10 800
12. $450 000
13. $46 670
14. (a) $397 100
(b) 3 months
15. 40 years
16. (a) 6
(b) 5
17. 4
18. (a) $8 610
(b) no
19. a profit
82
F3A: Chapter 3B
Date
Task
Lesson Worksheet
Progress
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Book Example 6
○
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Complete
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(Video Teaching)
Book Example 7
○
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Complete
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(Video Teaching)
Book Example 8
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 9
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
3B Level 1
Maths Corner Exercise
3B Level 2
Maths Corner Exercise
3B Level 3
Maths Corner Exercise
3B Multiple Choice
E-Class Multiple Choice
Self-Test
○
○
○
Complete and Checked
Problems encountered
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○
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Complete and Checked
Problems encountered
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Complete and Checked
Problems encountered
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Complete and Checked
Problems encountered
Skipped
(Full Solution)
83
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 3A Lesson Worksheet 3B
(Refer to §3.2)
[In this worksheet, give the answers correct to the nearest dollar if necessary.]
3.2A Formula for Compound Interest
If the principal is $P and the interest rate per period is R%, then
the amount ($A) received after n periods is given by:
A = P(1 + R%)n
Example 1
If $500 is deposited in a bank at an interest rate
of 4% p.a. compounded yearly, find the amount
received after
(a) 5 years,
(b) 10 years.
Sol (a) Amount
= $500 × (1 + 4%)5
= 500, R = 4,
= $608, cor. to the nearestP dollar
n = 5. Find
(b) Amount
A.
= $500 × (1 + 4%)10
= $740, cor. to the nearest dollar
Instant Drill 1
If $1 000 is deposited in a bank at an interest
rate of 7% p.a. compounded yearly, find the
amount received after
(a) 4 years,
(b) 8 years.
Sol (a) Amount
= $(
) × [1 + (
)%]( )
P=?
=$
, cor. to the nearest dollar R
=?
(b) Amount
n
= $(
) × [1 + (
)%]( )
=$
, cor. to the nearest dollar
1. If $20 000 is deposited in a bank at an
interest rate of 5% p.a. compounded
yearly, find the amount received after
(a) 6 years,
(b) 4.5 years.
2. Grace deposits $P in a bank at an interest
rate of 3% p.a. compounded yearly. If the
amount received after 2 years is $63 654,
find the value of P.
(a) Amount
= $(
=
(
)×(
(
)
)=P×(
)
=
)(
)
Set up an
equat
ion
to
(b)
○→ Ex 3B 8
The compound interest ($I) is given by:
I = P[(1 + R%)n – 1]
84
∵
∴
A= P + I
I= A – P
= P(1 + R%)n – P
= P[(1 + R%)n – 1]
Example 2
If $2 000 is deposited in a bank at an interest
rate of 6% p.a. compounded yearly, find the
compound interest received after 8 years.
Sol Amount
= $2 000 × (1 + 6%)8
A = P(1 + R%)n
= $3 187.70, cor. to the nearest $0.01
Compound interest
= $(3 187.70 – 2 000)
= $1 188, cor. to the nearest dollar
I=A–P
Alternative
Compound interest
= $2 000 × [(1 + 6%)8 – 1]
I = Pdollar
[(1 + R%)n –
= $1 188, cor. to the nearest
Instant Drill 2
If $8 000 is deposited in a bank at an interest
rate of 2% p.a. compounded yearly, find the
compound interest received after 5 years.
Sol Amount
= $(
)×(
)( )
= $(
), cor. to the nearest $0.01
Compound interest
= $[(
)–(
)]
=$
, cor. to the nearest dollar
Alternative
Compound interest
= $(
) × [(
)( ) – 1]
=$
, cor. to the nearest dollar
3. If $50 000 is deposited in a bank at an
interest rate of 8% p.a. compounded
yearly, find the compound interest
received after 6 years.
4. Leo deposits a sum of money in a bank at
an interest rate of 10% p.a. compounded
yearly. If he receives a compound interest
of $5 296 after 3 years, find the sum of
money deposited.
1]
Let $P be the sum of money deposited.
(
) = P × [(
)( ) – 1]
=
∴ The sum of money deposited is
.
○→ Ex 3B 9
○→ Ex 3B 1–3
5. Complete the following table. (The interest is compounded yearly.)
Interest rate
Time
Amount
Principal
(p.a.)
$7 000
3%
4 years
(a)
$15 000
7%
5 years
(b)
5%
3 years
$46 305
(c)
4%
2 years
(d)
85
Compound
interest
$16 320
3.2B Comparison between Compound Interest and Simple Interest
Example 3
$4 000 is deposited in a bank at an interest rate
of 3% p.a. for 10 years. Find the amount
received in each of the following situations.
(a) Simple interest is calculated.
(b) The interest is compounded yearly.
Sol
(a) Amount
For simple
= $4 000 × (1 + 3% × 10)
interest,
= $5 200
A
=
P
(1 + R% × T)
(b) Amount
10
For compound
= $4 000 × (1 + 3%)
interest,
= $5 376, cor. to the nearest
dollar
A = P(1 + R%)n
Instant Drill 3
$6 500 is deposited in a bank at an interest rate
of 4% p.a. for 15 years. Find the amount
received in each of the following situations.
(a) Simple interest is calculated.
(b) The interest is compounded yearly.
Sol
(a) Amount
=
(b) Amount
=
6. Nancy deposits $30 000 in a bank at an interest rate of 2.5% p.a. for 4 years. Find the interest
received in each of the following situations.
(a) Simple interest is calculated.
(b) The interest is compounded yearly.
I = P[(1 + R%)n –
1]
I = P × R% × T
○→ Ex 3B 10, 11
3.2C Interest Compounded at Different Periods
Suppose $P is deposited in a bank at an interest rate of R% per period, and the
compound interest is calculated once per period. Then the amount ($A) after n
periods is given by:
total number of
A = P(1 + R%)n
periods
interest rate
per
period
86
Example 4
$1 000 is deposited in a bank at an interest rate
of 6% p.a. for 5 years. Find the amount
received if the interest is compounded
half-yearly.
Sol
P = 1 000
Number of periods per year = 2
∵
∴
Interest rate per period (R%) =
Instant Drill 4
$5 000 is deposited in a bank at an interest rate
of 8% p.a. for 3 years. Find the amount
received if the interest is compounded
quarterly.
Sol
P = 5 000
∵
∴
6%
2
Total number of periods (n) = 5 × 2
Interest rate for half a year
6%
=
2
= 3%
Taking half a year as a period,
number of periods in 5 years
=5×2
= 10
amount
= $1 000 × (1 + 3%)10
= $1 344, cor. to the nearest dollar
7. $9 000 is deposited in a bank at an interest
rate of 10% p.a. compounded half-yearly.
Find the amount after 4 years.
Number of periods per year = (
R% =
)
8%
( )
n=3×( )
Interest rate per quarter
=
Taking a quarter as a period,
number of periods in (
) years
=
amount
=
8. Hubert borrows $25 000 from a bank at an
interest rate of 14% p.a compounded
quarterly. Find the amount to be repaid
after 6 years.
○→ Ex 3B 12, 13
87
‘Explain Your Answer’ Question
9. Anna wants to deposit $100 000 in either bank A or bank B for 3 years. Bank A offers an
interest rate of 4.5% p.a. compounded yearly while bank B offers an interest rate of 4% p.a.
compounded half-yearly. Which bank pays a higher interest? Explain your answer.
 Level Up Questions
10. At the beginning of 2017, Miss Lau deposits $4 000 in a bank at an interest rate of 8% p.a.
compounded yearly. Find the amount she will receive at the beginning of 2024.Find the time for
Time for deposit (in years) = 2024 – (
=
Amount =
)
deposit first.
11. Winnie borrows a sum of money at an interest rate of 12% p.a. compounded monthly. If she
has to pay an interest of $482.4 after 2 months, find the sum of money borrowed.
88
New Century Mathematics (2nd Edition) 3A
3
Percentages (II)

Consolidation Exercise
3B
[In this exercise, give the answers correct to the nearest dollar if necessary.]
Level 1
Complete the following table. [Nos. 1–3]
(The interest is compounded yearly.)
Interest rate
Principal
Time
(p.a.)
$2 000
3%
4 years
1.
$8 000
5%
6 years
2.
$15 000
7%
8 years
3.
Amount
Compound
interest
Complete the following table, given that the interests are compounded half-yearly. [Nos. 4–6]
Interest rate Interest rate
Number of
Principal
Time
Amount
(p.a.)
per period
periods
$1 000
2%
1%
3 years
4.
$300 000
4%
5 years
10
5.
$60 000
8%
42 months
6.
7. Mr Wong deposits $P in a bank at an interest rate of 6% p.a. compounded yearly. If the
amount received after 5 years is $70 000, find the value of P, correct to the nearest integer.
8. $4 000 is deposited in a bank at an interest rate of 3% p.a. for 6 years. Find the interest received
in each of the following situations.
(a) Simple interest is calculated.
(b) The interest is compounded yearly.
9. $90 000 is deposited in a bank at an interest rate of 7% p.a. for 5 years. Find the difference
between the interests calculated on the bases of simple interest and compound interest
(compounded yearly).
10.$36 000 is borrowed from a bank at an interest rate of 4% p.a. compounded quarterly. Find the
amount to be repaid after 2 years.
11.$68 000 is deposited in a bank at an interest rate of 5% p.a. compounded monthly. Find the
amount after 36 months.
12.Jackie borrows $10 000 from a bank at an interest rate of 16% p.a. compounded every 4
months. Find the interest he will pay after 5 years.
89
Level 2
13.Ann deposits $14 000 in a bank for 2 years. The interest is compounded yearly.
(a) Find the compound interest received if the interest rate is
(i) 3% p.a.,
(ii) 6% p.a.
(b) Is the compound interest found in (a)(ii) twice the interest found in (a)(i)? Explain your
answer.
14. Judie plans to borrow $60 000 for 4 years. She can borrow the money from bank A at a simple
interest rate of 8% p.a. or from bank B at an interest rate of 7% p.a. compounded yearly. Which
bank should she choose in order to pay less interest? Explain your answer.
15. David borrows a sum of money from a bank at an interest rate of 9% p.a. compounded quarterly.
If the interest he pays after 5 years is $22 000, find the sum of money borrowed.
16.At the beginning of 2014, Mr Tam deposited a sum of money in a bank at an interest
rate of 6% p.a. compounded yearly. If he received an interest of $13 000 at the
beginning of 2017, find
(a) the sum of money deposited,
(b) the amount received at the beginning of 2019.
17. 3 years ago, Mary deposited $400 000 in a bank, and a simple interest of $96 000 is just
received. Now, she deposits the amount obtained in another bank at the same annual interest
rate, but the interest will be compounded monthly. Find the amount she can obtain after another
3 years.
18. Roy wants to deposit $20 000 in a bank for 3 years. He can deposit the money in bank A at an
interest rate of 4% p.a. compounded half-yearly, or in bank B at 3% p.a. compounded daily.
Which bank should he choose in order to earn more interest? Explain your answer.
(Assume there are 365 days in a year.)
19.Sandy plans to deposit $96 000 in a bank for 4 years. The saving schemes offered by bank H
and bank K are as follows:
Bank H: interest rate of 4% p.a. compounded monthly with a cash reward of $200 per year.
Bank K: interest rate of 4.5% p.a. compounded quarterly.
Which bank should she choose in order to earn more interest? Explain your answer.
20.Anderson is going to borrow $300 000 from a bank for 8 years. Bank A charges interest at 9% p.a.
compounded yearly. Bank B charges interest at 7% p.a. compounded monthly.
(a) Which bank should he choose in order to pay less interest? Explain your answer.
(b) Suppose the interest rate charged by bank B changes to 8.5% p.a.
(i) Which bank should he choose in order to pay less interest? Explain your answer.
(ii) When compared to the original interest rate of 7% p.a., how much more interest will
be charged by bank B now?
90
Answer
Consolidation Exercise 3B
1.
2.
3.
4.
5.
amount = $2 251, interest = $251
amount = $10 721, interest = $2 721
amount = $25 773, interest = $10 773
number of periods = 6, amount = $1 062
interest rate per period = 2%,
amount = $365 698
6. interest rate per period = 4%,
number of periods = 7, amount = $78 956
7. 52 308
8. (a) $720
(b) $776
9. $4 730
10. $38 983
11. $78 980
12. $11 802
13. (a) (i) $852.6 (ii) $1 730.4
(b) no
14. bank B
15. $39 250
16. (a) $68 057
(b) $91 076
17. $630 038 18. bank A
19. bank K
20. (a) bank B (b) (i) bank B (ii) $66 398
91
F3A: Chapter 3C
Date
Task
Lesson Worksheet
Progress
○
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○
Complete and Checked
Problems encountered
Skipped
(Full Solution)
Book Example 10
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 11
○
○
○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
3C Level 1
Maths Corner Exercise
3C Level 2
Maths Corner Exercise
3C Level 3
Maths Corner Exercise
3C Multiple Choice
E-Class Multiple Choice
Self-Test
○
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Complete and Checked
Problems encountered
Skipped
○
○
○
○
○
○
○
○
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○
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○
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
(Full Solution)
92
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 3A Lesson Worksheet 3C
(Refer to §3.3A)
3.3A Increasing at a Constant Rate
If a value P increases at a constant rate of R% per period,
its new value A after n periods is given by:
A = P(1 + R%)n
where (1 + R%) is called the growth factor.
Example 1
John weighs 50 kg this year. If his weight
increases at a constant rate of 9% per year, find
his weight after 4 years.
(Give the answer correct to the nearest 0.1 kg.)
Sol
John’s weight after 4 years
P = 50
= 50 × (1 + 9%)4 kg
= 70.6 kg, cor. to the nearest 0.1 kgR =
Instant Drill 1
A metal rod is at 78°C now. If its temperature
increases at a constant rate of 5% per hour, find
its temperature after 3 hours.
(Give the answer correct to the nearest 0.1°C.)
Sol
9
n=
1. Peter buys a gold coin at $4 000. If its
value increases steadily at a rate of 8% per
year, what will its value be after 10 years?
(Give the answer correct to the nearest
$10.)
Temperature after 3 hours
=(
)×(
)( ) °C
=
,
P = ___
R=
___
n=
2. In a city, the number of hotels was 600 in
2012. If the number of hotels in the city
increases at a steady rate of 1.5% per year,
find the number of hotels in 2017, correct
to the nearest integer.
Number of periods
=(
)–(
)
=
Number of hotels in 2017
=
○→ Ex 3C 1–6
93
3. The height of a tree increases at a constant
rate of 20% per year. If the present height
of the tree is 16 m, find its height 2 years
ago.
(Give the answer correct to the nearest
0.01 m.)
Let P m be its height 2 years ago.
(
)=P×(
)( )
4. After a promotion campaign, the number
of blood donors increases at a constant rate
of 4% per day.
(a) Find the growth factor.
(b) If there are 1 200 blood donors today,
how many blood donors were there
one week ago?
(Give the answer correct to the
nearest integer.)
=
○→ Ex 3C 10–13
5. The profit of a company is $100 000 this
year. It is estimated that the profit of the
company will increase by 16% every
2 years. Estimate the profit after 10 years.
(Give the answer correct to the
2 significant figures.)
6. In an experiment, the number of bacteria
increases by 3% every half an hour. If
there are 5 000 bacteria now, how many
bacteria will there be after 12 hours?
(Give the answer correct to the nearest
integer.)
Taking _____ years as a period,
number of periods in 10 years
10
=
( )
=
Estimated profit after 10 years
=
○→ Ex 3C 7, 8
94
‘Explain Your Answer’ Question
7. The population of a town is 75 000 this year. If its population grows at a steady rate of 2.3%
per year, will the population of the town exceed 90 000 after 8 years? Explain your answer.
 Level Up Questions
8. Due to technological improvement, the rice production of a farm increases at a constant rate
of 10% every 5 years. Suppose its rice production is 65 tonnes this year. Find the rice
production of the farm 20 years ago, correct to the nearest 0.1 tonnes.
1 tonne = 1 000 kg
9. In 2010, the monthly salary of Fred was $20 000. His monthly salary increased at a constant
rate of 3% per year from 2010 to 2016.
(a) Find the growth factor.
(b) Find the increase in his monthly salary from 2010 to 2016.
(Give the answer correct to 3 significant figures.)
95
New Century Mathematics (2nd Edition) 3A
3
Percentages (II)

Consolidation Exercise
3C
Level 1
Complete the following table. [Nos. 1–3]
Growth rate
Original value
Growth factor
per year
300
50%
1.5
1.
6 000
40%
2.
$20 000
30%
3.
Time
New value
1 year
3 years
5 years
4. The water consumption of a village is 30 units in a certain week, and it increases steadily at a
rate of 6% per week. Find the water consumption after 4 weeks, correct to 3 significant figures.
5. The population of a city was 540 000 in 2015 and it increases steadily at a rate of 3.8% per year.
Find the population in 2021. (Give the answer correct to the nearest integer.)
6. The present value of an oil painting is $256 000. If its value increases by 30% every 4 years,
what will its value be after 12 years?
7. A computer virus spreads through a certain network. The number of infected computers
increases at a steady rate of 300% every 10 minutes. If 2 computers are infected initially, find
the number of infected computers after 1 hour.
8. The average temperature of a town in January is 20°C. It increases steadily at a rate of 4% every
month until August. Will the average temperature in June be higher than 25°C? Explain your
answer.
Complete the following table. [Nos. 9–10]
Growth rate
Original value
Growth factor
per year
10%
1.1
9.
60%
10.
96
Time
New value
1 year
3 years
1 210
49 152
Level 2
11. The weight of a dog was 115 g at the beginning of 2013. Then, its weight increased steadily at a
rate of 2% per year from 2013 to 2016.
(a) Find the weight of the dog at the end of 2016.
(b) If the weight of the dog increases by 4% every year from 2017 onwards, find its weight at
the end of 2018.
(Give the answers correct to the nearest g.)
12. A tree was 210 cm tall 4 years ago. Then, its height increases steadily at a rate of 3% per year.
(a) Find its height at present.
(b) Find the percentage increase in its height over these 4 years.
(Give the answers correct to 3 significant figures.)
13. The average stock price of a company increased by 1% every month over the past 9 months. It
is known that the average stock price is $78 this month.
(a) Find the growth factor.
(b) Find the increase in the average stock price over the past 9 months.
(Give the answer correct to 3 significant figures.)
14. Over the past 3 years, the monthly income of Joe increased at a constant rate of 25% per year. It
is known that the monthly income of Joe at present is $50 000.
(a) What was the monthly income of Joe 3 years ago?
(b) Find the increase in the monthly income of Joe over the past 3 years.
15. In 2015, the number of fish in a pond was 260. In 2016, the number of fish was 273.
(a) Find the growth factor.
(b) Suppose the growth factor remains unchanged. Find the number of fish in the pond in
2020.
(Give the answer correct to the nearest integer.)
16. From 2014 to 2016, company A’s profit increased by 3% per year and company B’s profit
increased by 4% per year. It is given that both companies made the same profit in 2016.
(a) Which company’s profit was higher in 2014? Explain your answer.
(b) If the difference of profits of companies A and B in 2014 was $40 000, find the profit of
each company in 2016.
(Give the answer correct to the nearest dollar.)
97
Answer
Consolidation Exercise 3C
1. 450
2. growth factor = 1.4, new value = 16 464
3. growth factor = 1.3, new value = $74 258.6
4. 37.9 units 5. 675 426
6. $562 432
7. 8 192
8. no
9. 1 100
10. original value = 12 000, growth factor = 1.6
11. (a) 124 g
(b) 135 g
12. (a) 236 cm
(b) 12.6%
13. (a) 1.01
(b) $6.68
14. (a) $25 600
(b) $24 400
15. (a) 1.05
(b) 332
16. (a) company A (b) $2 217 332
98
F3A: Chapter 3D
Date
Task
Lesson Worksheet
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Book Example 12
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Book Example 13
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Book Example 14
○
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Consolidation Exercise
Maths Corner Exercise
3D Level 1
Maths Corner Exercise
3D Level 2
Maths Corner Exercise
3D Level 3
Maths Corner Exercise
3D Multiple Choice
E-Class Multiple Choice
Self-Test
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99
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 3A Lesson Worksheet 3D
(Refer to §3.3B)
3.3B Decreasing at a Constant Rate
If a value P decreases at a constant rate of R% per period,
its new value A after n periods is given by:
A = P(1 – R%)n
where (1 – R%) is called the decay factor.
Example 1
A pool contains 300 m3 of water originally. If
the volume of water decreases at a constant
rate of 10% per hour, find the volume of water
in the pool after 6 hours.
(Give the answer correct to the nearest 0.1 m3.)
Sol Volume of water in the pool after 6
hours
= 300 × (1 – 10%)6 m3
P = 300
= 159.4 m3, cor. to the
R=
nearest 0.1 m3
Instant Drill 1
The monthly charge of a mobile data plan is
$180. If the monthly charge decreases at a
constant rate of 5% per year, find the monthly
charge after 3 years.
(Give the answer correct to the nearest
dollar.)
P = _____
R=
Sol Monthly charge after 3 years
(
)
_____
= $(
)×(
)
n=
=
,
1. The number of tigers in a country is 1 400
at present. It is known that the number of
tigers decreases by 4% per year.
(a) Find the decay factor.
(b) Find the number of tigers after 5
years.
(Give the answer correct to the
nearest integer.)
2. In a shop, the profit from selling a model
of calculator was $50 000 in 2013. The
profit from selling this model decreases at
a rate of 15% per year. Find the profit from
selling this model in 2017.
(Give the answer correct to the nearest
$100.)
10
n=6
Number of periods
=(
)–(
)
=
Profit from selling this model in 2017
=
○→ Ex 3D 1, 2, 5–7
100
3. Suppose the weight of a block of dry ice
decreases steadily at a rate of 2% per
minute. If its present weight is 200 g, find
its weight 5 minutes ago.
(Give the answer correct to the nearest
0.01 g.)
Let P g be its weight 5 minutes ago.
(
)=P×(
)( )
4. The number of newborn babies in a city
decreases steadily at 4% per year. If the
number of newborn babies in the city is
18 000 this year, find the number of
newborn babies 9 years ago.
(Give the answer correct to the nearest
thousand.)
=
○→ Ex 3D 3, 4
(a) Depreciation is the decrease in value of a product after it
has been used for a period of time, where
depreciation = original value – new value
depreciation
depreciation rate =
× 100%
original value
(b) For a product with the original value $P and depreciation rate
R% per period, its new value $A after n periods is given by:
A = P(1 – R%)n
Example 2
The original price of a watch is $4 000. If its
value depreciates by 7% per year, what will the
value of the watch be after 2 years?
Sol Value of the watch after 2 years
= $4 000 × (1 – 7%)2
= $3 459.6
Depreciation is a
kind of decrease
at a constant rate.
Instant Drill 2
The original price of a machine is $30 000. If
its value depreciates by 10% per year, what
will the value of the machine be after 3 years?
Sol Value of the machine after
years
= $(
)×(
)( )
=
101
5. The present value of a sofa is $5 600. If its
depreciation rate is 11% per year, find its
value after 5 years, correct to the nearest
dollar.
6. The value of a smartphone depreciates by
20% every year. If its present value is
$3 072, find its value 3 years ago.
○→ Ex 3D 10
7. The original price of a piano is $47 000. If
its value depreciates at 18% every 2 years,
find the value of the piano after 6 years,
correct to the nearest $100.
8. The value of a camera depreciates by 5%
every 6 months. Its value was $7 500 in
2015. Find the depreciation of the camera
from 2015 to 2017, correct to the nearest
$10.
○
→ Ex 3D 11
depreciation
= original value –
new value
○→ Ex 3D 8, 9
‘Explain Your Answer’ Question
9. Steven bought a printer 2 years ago. The depreciation rate is 25% per year and its present
value is $810. Steven claims that he spent less than $1 500 for buying the printer. Do you
agree? Explain your answer.
 Level Up Questions
10. The expenditure of a family in March is $35 200. The family decreases their expenditure at a
rate of 5% every 2 months since March. Find the expenditure of the family in
(a) July,
(b) September
in the same year.
102
11. Four years ago, the number of traffic accidents in a district was 400. Suppose the number of
traffic accidents decreases steadily at a rate of 3.7% per year.
(a) Find the decay factor.
(b) Find the percentage decrease in the number of traffic accidents over the past four years.
(Give the answer correct to 2 significant figures.)
Find the number of
traffic accidents this
year first.
103
New Century Mathematics (2nd Edition) 3A
3
Percentages (II)

Consolidation Exercise
3D
Level 1
Complete the following table. [Nos. 1–3]
Rate of decrease
Original value
Decay factor
per year
800
30%
1.
$7 500
0.75
2.
20%
3.
Time
New value
4 years
2 years
3 years
1 600 cm3
4. A tank contains 20 L of water. Owing to an accident, the volume of water in the tank decreases
at a rate of 2% per minute. Find the volume of water in the tank after 4 minutes.
(Give the answer correct to the nearest L.)
5. In February, Ron joins a slimming programme and his weight decreases by 3% per month. If
his weight in February is 120 kg, find his weight in July, correct to the nearest kg.
6. The original price of a mobile phone is $6 000. If its value depreciates by 4% every 3 months,
find the value of the mobile phone after 1 year.
(Give the answer correct to the nearest dollar.)
7. The value of a book decreases by 20% every year. If its present value is $400, find its value 2
years ago.
8. The value of a bike depreciates by 10% every 3 years. Its value was $8 748 in 2016. Find its
value in 2007.
Level 2
9. The distance between a car and a building is 80 km. The car is now moving towards the
building so that the distance between them decreases by 6% every 10 minutes. Find the distance
between them after
(a) 20 minutes,
(b) 1 hour.
(Give the answers correct to 3 significant figures.)
104
10.Thomas bought a car 5 years ago. Its value depreciates by 7% every year. Its present value is
$142 000.
(a) Find the original price of the car, correct to the nearest dollar.
(b) When the depreciation of the car is greater than $60 000, Thomas will buy a new car. Will
he buy a new car now? Explain your answer.
11.The value of a computer was $4 000 in 2015. Its value then decreases by 13% every year.
(a) Find the value of the computer in 2017.
(b) What is the percentage decrease in the value of the computer from 2015 to 2017?
12.Kate bought a piano for $64 000 in 2015.
(a) If the value of the piano decreases by 6% every 6 months, find its value in 2017.
(b) If the value of the piano decreases at a rate of 12% every year, will its value in 2017 be the
same as that found in (a)? Explain your answer. If not, what is the difference?
(Give the answers correct to the nearest dollar if necessary.)
13.The population of a city was 350 000 in 1961. It decreased to 346 150 in 1962.
(a) Find the percentage change in the population over that year.
(b) If the population continues to change at the same rate as in (a) per year, find the population
in
(i) 1980,
(ii) 2000.
(Give the answers correct to the nearest integer.)
14.The value of an ebook reader depreciates by x% every 8 months. It is known that the present
value of the reader is $1 100 and its value will be $770 after 8 months.
(a) Find the value of x.
(b) Find the depreciation of the ebook reader after 2 years as compared to the present value.
105
Answer
Consolidation Exercise 3D
1. decay factor = 0.7, new value = 192.08
2. rate of decrease per year = 25%,
new value = $4 218.75
3. original value = 3 125 cm3, decay factor = 0.8
4. 18 L
5. 103 kg
6. $5 096
7. $625
8. $12 000
9. (a) 70.7 km
(b) 55.2 km
10. (a) $204 114
(b) yes
11. (a) $3 027.6
(b) 24.31%
12. (a) $49 968
(b) no, $406
13. (a) −1.1% (b) (i) 283 660 (ii) 227 365
14. (a) 30
(b) $722.7
106
F3A: Chapter 3E
Date
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Lesson Worksheet
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Book Example 15
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Book Example 16
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Book Example 17
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Book Example 18
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Book Example 19
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(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
3E Level 1
Maths Corner Exercise
3E Level 2
Maths Corner Exercise
3E Level 3
Maths Corner Exercise
3E Multiple Choice
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(Full Solution)
107
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
E-Class Multiple Choice
Self-Test
○
○
○
Complete and Checked
Problems encountered
Skipped
108
Mark:
_________
Book 3A Lesson Worksheet 3E
(Refer to §3.4)
3.4A Successive Percentage Changes
If a value N increases by x% and then decreases by y%, then
new value = N(1 + x%)(1 – y%)
Example 1
75 increases by 20% and then decreases by
30%. Find the new value.
Instant Drill 1
120 increases by 50% and then decreases by
15%. Find the new value.
increase by 50% :
(_________)
Sol
New value
by 20% : (1 + 20%)
= 75 × (1 + 20%) ×increase
(1 – 30%)
decrease by 30% : (1 – 30%)
= 75 × 1.2 × 0.7
= 63
1. $300 decreases by 10% and then increases
by 40%. Find the new value.
Sol
New value decrease by 15% :
= 120 × (
)×(
=
)
2. 5 kg increases by 30% and then increases
by a further 70%. Find the new value.
○→ Ex 3E 1
3. In a tutorial school, there were 1 250
students in 2013. The number of students
increased by 10% in 2014 and then
decreased by 4% in 2015. Find the number
of students in 2015.
4. On Monday, Helen practised on violin for
80 minutes. Her practising time decreased
by 20% on Tuesday, and then decreased by
a further 25% on Wednesday. Find her
practising time on Wednesday.
○→ Ex 3E 3, 4
109
5. A shop sold 200 TVs in May. The number
of TVs sold decreased by 50% in June and
then increased by 80% in July.
(a) Find the number of TVs sold in July.
(b) What was the increase or decrease in
the number of TVs sold from May to
July?
6. The original price of a suit is $5 600. The
price of the suit first increases by 20% and
then decreases by 10%.
(a) Find the final price of the suit.
(b) Find the increase or decrease in the
price of the suit as compared to the
original price.
○→ Ex 3E 5, 6
7. The production cost of a tennis racket is $P. If the production cost is first increased by 16%
and then decreased by 25%, the new production cost will be $609. Find the value of P.
P(
)(
)=(
=
)
○→ Ex 3E 7–9
110
3.4B Percentage Changes of Different Components
To find the overall percentage change in a quantity with different components:
Step1
1: Find the original value of the quantity.
Step2
2: Find the new value of each component and hence the new value of the
quantity.
change
Step3
3: Use the formula ‘percentage change =
× 100%’ to find
original value
the overall percentage change.
Example 2
Last year, there were 10 boys and 30 girls in a
choir. This year, the number of boys increases
by 20% and the number of girls decreases by
40%. Find the percentage change in the total
number of children in the choir.
Sol Original total number of children Step 1
= 10 + 30
= 40
Step 2
New number of boys
= 10 × (1 + 20%)
= 12
New number of girls
= 30 × (1 – 40%)
= 18
New total number of children
= 12 + 18
= 30
∴ Percentage change in the
Step 3
total number of children
30 − 40
=
× 100%
40
= –25%
Instant Drill 2
There are 40 large tables and 20 small tables in
a hall. Now, the number of large tables
increases by 25% while the number of small
tables decreases by 5%. Find the percentage
change in the total number of tables in the hall.
Sol Original total number of tables Step 1
=(
)+(
)
=
Step 2
New number of large tables
=(
)×(
)
=
New number of small tables
)
=(
)×(
=
New total number of tables
=(
)+(
)
=
∴ Percentage change in the
Step 3
total number of tables
(
)−(
)
=
× 100%
(
)
=
111
8. Last month, the income and the expenditure of a flower shop were $24 000 and $18 000
respectively. This month, the income decreases by 10% while the expenditure increases by
10%. Find the percentage change in the profit of the shop.
Profit of the shop last month = $[(
=
Income of the shop this month =
)–(
)]
Expenditure of the shop this month =
Profit of the shop this month =
∴ Percentage change in the profit of the shop =
○→ Ex 3E 10, 11
9. In the figure, the length and the width of a rectangle are 25 cm and
cm respectively.
(a) Find the area of the rectangle.
(b) If the length of the rectangle increases by 80% while the width
decreases by 40%, find the percentage change in the area of
the rectangle.
18
25 cm
18 cm
○→ Ex 3E 14, 15
112
10. A professional examination consists of papers I, II and III. In the first attempt, Calvin scored 50,
70 and 80 in papers I, II and III respectively.
(a) Find the total score of Calvin in the first attempt.
(b) In the second attempt, Calvin’s scores in papers I and II both increase by 20% and his
score in paper III remains unchanged. Find the percentage change in his total score as
compared to the first attempt.
Paper I
Paper II
Paper III
1st
atte
mpt
2nd
atte
50
70
80
○→ Ex 3E 12, 13
‘Explain Your Answer’ Question
11. A bowl contains 300 mL of water originally. The volume of water in the bowl increases by
30% and then decreases by 40%. Sandy claims that the overall percentage change of the
volume of water in the bowl is –10%. Do you agree? Explain your answer.
New volume of water = 300 × (
=
)×(
) mL
Overall percentage change =
∵ _______ ( = / ≠ ) –10%
∴ The claim is (agreed / disagreed).
 Level Up Questions
12. A value decreases by 30% and then increases by 50%. Find the overall percentage change.
Let P be the original value.
New value =
∴ Overall percentage change
113
=
13. After the fare of a bus route increases from $6 to $6.3, the number of passengers reduces by
14%. Find the percentage change in the revenue obtained from the bus route.
Let x be the original number of passengers.
Original revenue = $6x
New number of passengers = x × (
=
New revenue = $(
)×(
)
= $(
)
∴ The required percentage change
)
=
114
New Century Mathematics (2nd Edition) 3A
3
Percentages (II)
Consolidation Exercise
3E

Level 1
1. Find the new value in each of the following situations.
(a) 200 increases by 30% and then increases by a further 40%.
(b) 8 m decreases by 50% and then decreases by a further 10%.
(c) $1 600 decreases by 25% and then increases by 60%.
2. Find the original value in each of the following situations.
(a) After increasing by 6% and then increasing by a further 7%, the new value is 90 736.
(b) After increasing by 30% and then decreasing by 20%, the new value is 208 g.
(c) After decreasing by 40% and then increasing by 60%, the new value is 76.8 cm3.
3. In January, the number of visitors of a theme park is 50 000. The number decreases by 24%
from January to February, and then increases by 7% from February to March. Find the number
of visitors in March.
4. The income of a taxi driver on Friday is 40% more than that on Saturday. The income on
Saturday is 8% less than that on Sunday. If the income on Sunday is $2 000, find the income on
Friday.
5. Ron’s first examination result is 60 marks. His second examination result is 20% higher than his
first one, while his third examination result is 25% lower than his second one.
(a) What is Ron’s third examination result?
(b) What is the increase or decrease in Ron’s third examination result as compared with his
first examination result?
6. A hawker sells 200 apples on Thursday. The number of apples sold increases by 40% from
Thursday to Friday and then decreases by 30% from Friday to Saturday. Is the number of
apples sold on Saturday less than that on Thursday? Explain your answer.
7. The cost of a product decreases by 20% and then increases by 25%. The final cost of the
product is $77. Find the original cost of the product.
115
8. Last year, there were 800 male students and 900 female students in a university. This year, the
number of male students increases by 30% and the number of female students decreases by
15%.
(a) (i) Find the number of male students this year.
(ii) Find the number of female students this year.
(b) Find the percentage change in the total number of students, correct to 3 significant figures.
9. Last year, 6 000 candidates took an accountant examination and 1 500 of them failed the
examination. This year, the number of candidates who pass the examination increases by 18%,
and the number of candidates who fail the examination decreases by 16%. Find the percentage
change in the total number of candidates.
10. Last week, the number of books sold in bookstores P, Q and R were 150, 240 and 320
respectively. As compared to last week, the number of books sold this week in bookstore P
decreases by 2%, that in bookstore Q increases by 5% and that in bookstore R remains
unchanged. Find the percentage change in the total number of books sold in the three bookstores,
correct to 3 significant figures.
11.The length of a rectangle is 10 cm and the width is 7 cm.
(a) Find the area of the rectangle.
(b) If the length of the rectangle decreases by 10% and the width increases by 10%, find
(i) the new area,
(ii) the percentage change in the area.
12. Last month, there were 500 workers in a factory and the wage of each worker was $12 000. This
month, the number of workers decreases by 12% and the wage of each worker increases by 6%.
Find the percentage change in the sum of wages of all workers over these two months.
Level 2
13.Find the percentage change in each of the following situations.
(a) A value increases by 80% and then decreases by 25%.
(b) A value decreases by 17% and then increases by 17%.
14.The profit of a company increased by 40% from 2014 to 2015, and then decreased by 35% from
2015 to 2016. What was the percentage change in the profit of the company from 2014 to 2016?
15. In a shop, the price of a product is decreased by 20% and then increased by 30%. Find the
percentage change in the price of the product.
116
16. The tax revenue of a government increased by 20%, 10%, 35% respectively in the first three
quarters of 2016, and then decreased by 45% in the fourth quarter of the year. What was the
percentage change in the quarterly tax revenue over the whole year 2016?
17. If a number is decreased by 90% and then increased by x%, the overall percentage change is
−83%. Find x.
18.The operating cost ($F) of a yoga club can be calculated by the following formula:
F = 150 000 + 20 000N + R,
where N is the number of yoga instructors and $R is the monthly rent of the yoga club.
(a) Last month, there were 20 yoga instructors and the monthly rent was $300 000. Find the
operating cost.
(b) This month, the number of yoga instructors decreases by 25% and the monthly rent increases
by 15%. Does the operating cost decrease by 10% as compared to last month? Explain your
answer.
19.The numbers of tourists of three cities A, B and C in 2015 are listed below:
A
B
C
City
2 500
900
1 600
Number of tourists
In 2016, the numbers of tourists of cities B and C increased by 40% and 20% respectively. If the
total number of tourists of the three cities increased by 4%, find the percentage change in the
number of tourists of city A.
20.If the base of a parallelogram increases by 18% and its height decreases by 30%, find the
percentage change in its area.
21. Last week, the numbers of male customers and the female customers in a shopping mall were in
the ratio 1 : 3. This week, the number of male customers increases by 16% while the number of
female customers decreases by 12%. Is there an increase in the total number of customers as
compared to last week? Explain your answer.
117
Answer
Consolidation Exercise 3E
1. (a) 364
(b) 3.6 m
(c) $1 920
2. (a) 80 000 (b) 200 g
(c) 80 cm3
3. 40 660
4. $2 576
5. (a) 54 marks
(b) a decrease of 6 marks
6. yes
7. $77
8. (a) (i) 1 040 (ii) 765
(b) +6.18%
9. +9.5%
10. +1.27%
2
11. (a) 70 cm (b) (i) 69.3 cm2 (ii) −1%
12. −6.72%
13. (a) +35%
(b) −2.89%
14. −9%
15. +4%
16. −1.99%
17. 70
18. (a) $850 000
(b) no
19. −19.2%
20. −17.4%
21. no
118
F3A: Chapter 3F
Date
Task
Lesson Worksheet
Progress
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Complete and Checked
Problems encountered
Skipped
(Full Solution)
Book Example 20
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Complete
Problems encountered
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(Video Teaching)
Book Example 21
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Problems encountered
Skipped
(Video Teaching)
Book Example 22
○
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Complete
Problems encountered
Skipped
(Video Teaching)
Book Example 23
○
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○
Complete
Problems encountered
Skipped
(Video Teaching)
Consolidation Exercise
Maths Corner Exercise
3F Level 1
Maths Corner Exercise
3F Level 2
Maths Corner Exercise
3F Level 3
Maths Corner Exercise
3F Multiple Choice
E-Class Multiple Choice
Self-Test
○
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Complete and Checked
Problems encountered
Skipped
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Complete and Checked
Problems encountered
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Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
Complete and Checked
Problems encountered
Skipped
(Full Solution)
119
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Teacher’s
Signature
___________
(
)
Mark:
_________
Book 3A Lesson Worksheet 3F
(Refer to §3.5)
3.5A Rates
Rates for the year = rateable value × rates percentage charge
rates for the year
Rates for a quarter =
4
(In this worksheet, the rates percentage charge is set as 5%.)
Example 1
Instant Drill 1
The rateable value of a flat is $300 000. Find
The rateable value of a flat is $85 000. Find the
the rates payable in a year.
rates payable in a year.
Sol Rates payable in a year
Sol Rates payable in a year
= $300 000 × 5%
=(
)×(
)
= $15 000
=
○→ Ex 3F 1, 2
1. The rateable value of a building is
$1 600 000. Find the rates for a quarter of
a year.
2. The rateable value of a piece of land is
$2 304 000. Find the rates for a quarter of
a year.
Rates for a quarter of a year
=
Rateable value
↓ × 5%
Rates for a year
↓÷4
Rates for a quarter
○→ Ex 3F 3, 4
3. Mr Luk pays $9 000 for the annual rates
on his flat. Find the rateable value of the
flat.
4. The owner of a building pays $57 100 of
rates quarterly. Find the rateable value of
the building.
Rates for a quarter
rates for the year
=
4
Let $P be the rateable value of the flat.
P×(
)=(
)
=
○→ Ex 3F 5–7
3.5B Salaries Tax
To calculate the salaries tax payable:
Step1
1: Split up the net chargeable income
into several parts, which are called
‘tax bands’. (see Table 1)
Step2
2: Calculate the tax for each part by
120
Net chargeable income
Tax rates
On the first $40 000
On the next $40 000
On the next $40 000
Remainder
2%
7%
12%
17%
Table 1
multiplying the corresponding tax rates.
Step3
3: Salaries tax payable
= sum of the taxes in Step2
2
In this worksheet, refer to
Table 1 when calculating
salaries tax.
Note: The government may adjust the tax bands and the tax rates.
Example 2
The net chargeable income of May is $125 000.
Find her salaries tax payable.
Sol Step1
1:
Net chargeable income
= $125 000
= $(40 000 + 40 000 + 40 000 + 5 000)
Step2
2:
Net chargeable
income
On the first
$40 000
On the next
$40 000
On the next
$40 000
Remainder
$5 000
7%
12%
17%
Step2
2:
Net chargeable
income
On the first
$40 000
On the next
$40 000
On the next
_________
Remainder
_________
Tax
Rate
2%
Instant Drill 2
The net chargeable income of Alex is $130 000.
Find his salaries tax payable.
Sol Step1
1:
Net chargeable income
= $130 000
= $(40 000 + 40 000 + ______________ )
$40 000 × 2%
= $800
$40 000 × 7%
= $2 800
$40 000 × 12%
= $4 800
$5 000 × 17%
= $850
Rate
2%
7%
Tax
$40 000 × 2%
= $800
$40 000 × 7%
= _________
____
____
Step3
3:
His salaries tax payable
= $(800 + ________________________ )
=
Step3
3:
Her salaries tax payable
= $(800 + 2 800 + 4 800 + 850)
= $9 250
121
5. The net chargeable income of Ann is $150 000. Find her salaries tax payable.
Net chargeable income =
Net chargeable income
On the first $40 000
Rate
2%
Tax
$40 000 × ______ = _________
Her salaries tax payable =
○→ Ex 3F 8–10
Net chargeable income = annual income – allowances
Example 3
The annual income of Kary is $220 000. If she
has a total allowance of $120 000, find
(a) her net chargeable income,
(b) her salaries tax payable.
Sol (a) Net chargeable income
= $(220 000 – 120 000)
= $100 000
(b) Net chargeable income
= $100 000
= $(40 000 + 40 000 + 20 000)
Net chargeable
Rate
Tax
income
On the first
$40 000 × 2%
2%
$40 000
= $800
On the next
$40 000 × 7%
7%
$40 000
= $2 800
Remainder
$20 000 × 12%
12%
$20 000
= $2 400
Her salaries tax payable
= $(800 + 2 800 + 2 400)
= $6 000
Instant Drill 3
The annual income of Tim is $240 000. If he
has a total allowance of $150 000, find
(a) his net chargeable income,
(b) his salaries tax payable.
Sol (a) Net chargeable income
= $[(
)–(
)]
=$
(b) Net chargeable income
= $(
)
= $(40 000 + 40 000 + ________ )
Net chargeable
Rate
Tax
income
On the first
$40 000 × 2%
2%
$40 000
= _________
His salaries tax payable
=
6. The annual income of Charles is $408 000. If he has a total allowance of $132 000, find his
salaries tax payable.
Net chargeable income =
Net chargeable income
On the first $40 000
Rate
2%
Tax
$40 000 × ______ = _________
122
His salaries tax payable =
○→ Ex 3F 12, 13
 Level Up Question
7. The average monthly income of Vanessa is $13 000 and her total allowance is $120 000. Find
her salaries tax payable.
123
New Century Mathematics (2nd Edition) 3A
3
Percentages (II)
Consolidation Exercise
3F

[In this exercise, when calculating salaries tax, refer to the tax rates as shown in Table 1 on P.3.44
of the textbook.]
Level 1
1. The rateable value of a flat is $360 000. Find the rates payable in a year.
2. The rateable value of a shopping centre is $3 000 000. Find the rates payable in a quarter of a
year.
3. The rateable value of a piece of land is $76 900 000. Find the rates payable in a quarter
of a year.
4. Mr Kan pays $18 000 for the annual rates on his property. Find the rateable value of the
property.
5. David pays $7 200 for the quarterly rates on his flat. Find the rateable value of the flat.
6. The owner of an apartment pays $22 000 of rates quarterly. What is the rateable value of the
apartment?
Find the salaries tax payable for each of the persons below. [Nos. 7–9]
7. A taxi driver has a net chargeable income of $32 000.
8. Carrie is a dancer with a net chargeable income of $79 000.
9. Jon is an artist with a net chargeable income of $101 000.
10.The net chargeable incomes of Ramsey and Paul are $8 000 and $24 000 respectively. Is the
salaries tax paid by Paul 3 times that paid by Ramsey? Explain your answer.
11.The annual income of Ellen is $186 000. If she has a total allowance of $110 000, find her
salaries tax payable.
Level 2
12.Priscilla paid $6 500 for the rates last year. At the beginning of this year, she moves to a new
flat and the rates payable decreases by 10% as compared to last year. What is the rateable value
of the new flat?
124
13.Sally paid $9 400 for the rates last year. At the beginning of this year, she moves to a new flat
and its rateable value is $142 800. What is the percentage change in the rates payable this year
as compared to last year?
(Give the answer correct to the nearest 1%.)
14.The average monthly income of Gigi is $25 000. If she has a total allowance of $120 000, find
her salaries tax payable.
15.(a) Find the salaries tax payable in each of the following situations:
(i) Net chargeable income = $40 000
(ii) Net chargeable income = $80 000
(iii) Net chargeable income = $120 000
(b) Joe has to pay the salaries tax of $10 100. Using the results of (a), calculate his net
chargeable income.
16.Benny’s salaries tax payable is $13 500. Find his net chargeable income.
17.Glen has a total allowance of $130 000. He has to pay $1 500 in salaries tax.
(a) Find his net chargeable income.
(b) Find his average monthly income.
18.(a) The net chargeable income of Gary is $123 000. Find his salaries tax payable.
(b) The net chargeable income of Raymond is one third of that of Gary. Is the salaries tax
payable of Raymond one third of that of Gary? Explain your answer. If not, find the
difference in their salaries tax payable.
125
Consolidation Exercise 3F
1.
4.
7.
10.
13.
15.
$18 000
2. $37 500
3. $961 250
$360 000 5. $576 000 6. $1 760 000
$640
8. $3 530
9. $6 120
yes
11. $3 320
12. $117 000
−24%
14. $18 600
(a) (i) $800 (ii) $3 600 (iii) $8 400
(b) $130 000
16. $150 000
17. (a) $50 000
(b) $15 000
18. (a) $8 910
(b) no, $8 040
126