Mathematics Practice Problems - Fractions, Percentages, Scientific Notation, and Expanding Brackets
Scientific Notation
A grain of sand weighs 6.45 x 10
-5
kilograms. A jar holds
4 x 10
6
grains of sand.
Calculate the weight of the sand in the jar.
(You can use a calculator for questions with this symbol)
Answer:
258kg
Scientific Notation
There are approximately 7.9 x 10
9
people on Earth.
Roughly 8% of people have blue eyes.
Estimate the number of people on Earth who have blue
eyes. Write your answer in scientific notation.
Answer:
6.32 x 10
8
Fractions
Answer:
62.8
Fractions
Fractions
Fractions
Fractions
Fractions
Percentages
Calculate 12.5% of £56.
Answer:
7
Percentages
A group of friends go out for lunch. The total of their bill
with a 12% tip comes to £76.16.
Find the cost of their bill before the tip was added.
Answer:
£68
Percentages
Lisa buys a cottage in 2018 valued at £82,000.
The value of the cottage appreciated by 3.8% per annum.
What was the value of the cottage in 2021?
Round your answer to 2 significant figures.
Answer:
£92,000
Expanding Brackets
Expand and simplify
(3x + 1)(2x – 7).
Answer:
6x
2
– 19x - 7
Expanding Brackets
Expand and simplify
(2x + 3)(x
2
– 5x + 6).
Answer:
2x
3
– 7x
2
– 3x + 18
Expanding Brackets
Expand and simplify
4(2x + 5) – 3(2x + 7).
Answer:
2x - 1
Factorising
Factorise
12p
2
+ 66pq.
Answer:
6p(2p + 11q)
Factorising
Factorise
9n
2
- 100.
Answer:
(3n + 10)(3n – 10)
Factorising
Factorise
x
2
+ 5x - 24.
Answer:
(x + 8)(x – 3)
Factorising
Factorise
3x
2
+ 14x + 8.
Answer:
(3x + 2)(x + 4)
Straight Lines
Find the equation of the line joining the points (-5,2) and
(1,4).
Straight Lines
Find the equation of the line from the diagram below:
Answer:
y = -4x - 3
Surds
Indices
Answer:
c
6
Indices
Expand and simplify
(5y
4
)
2
.
Answer:
25y
8
Indices
Arcs & Sectors
Calculate the length of the major arc AB:
A
B
6cm
O
115
0
Answer:
25.7cm
Arcs & Sectors
The length of major arc AB is 108cm. The radius of
the sector is 20cm.
Find the size of the shaded angle.
A
B
20cm
Answer:
309
0
Volume
An object is made of a cone sitting on top of a
cylinder, as shown:
8cm
20cm
14cm
Find the total volume of this object to 3 significant
figures.
Answer:
1850cm
3
Volume
An object is made of a cylinder with a
hemispherical
indent as shown:
9cm
8cm
10cm
Find the total volume of this object to 3 significant
figures.
Answer:
573cm
3
Completing the Square
Write x
2
+ 14x + 50 in the form (x + p)
2
+ q.
Answer:
(x + 7)
2
+ 1
Pythagoras’ Theorem
In the diagram below, radius OB = 9cm and length
AB = 14cm.
O
A
B
h
Calculate the length of h to 1 decimal place.
Answer:
14.7cm
Pythagoras’ Theorem
Is the triangle above right-angled? Justify your
answer.
8cm
17cm
15cm
Answer:
Yes since 17
2
= 15
2
+ 8
2
Pythagoras’ Theorem
Calculate the length of the space diagonal AB.
7cm
20cm
12cm
A
B
Answer:
24.4cm
Angle Properties
Calculate the size of the shaded angle ABC.
A
B
O
C
D
52
o
Answer:
84
0
Angle Properties
Calculate the size of the shaded angle COD.
A
B
O
C
D
Answer:
142
0
Angle Properties
ABCDE is a regular pentagon.
Find the size of the shaded angle EFA.
A
B
C
D
E
F
Answer:
36
0
Algebraic Fractions
Algebraic Fractions
Algebraic Fractions
Equations and Inequalities
Solve the inequality 15 - 3(2x - 4) < 25.
Equations and Inequalities
Changing the Subject of a Formula
Simultaneous Equations
Solve the system of simultaneous equations:
2x + 4y = -2
5x – 3y = 21
Answer:
x = 3, y = -2
Simultaneous Equations
The cost of 10 textbooks and 8 boxes of pens is £105.
The cost of 4 textbooks and 10 boxes of pens is £59.
Create an equation each for the information above and
then find the cost of one textbook and of one box of
pens.
Answer:
one textbook costs £8.50, one box of pens
costs £2.50
Further Trigonometry
Find the size of length PQ.
24cm
P
Q
R
75
0
48
0
Answer:
18.5cm
Further Trigonometry
Find the size of angle PQR.
53cm
P
Q
R
44cm
93
0
Answer:
56
0
Further Trigonometry
Find the size of length PQ.
7cm
P
Q
R
8.5cm
87
0
Answer:
10.7cm
Further Trigonometry
Find the size of angle RPQ.
22cm
P
Q
R
12cm
16cm
Answer:
45
0
Further Trigonometry
Find the area of triangle PQR.
18cm
P
Q
R
14cm
99
o
Answer:
124.4cm
2
Further Trigonometry
Find the size of length SR.
17
0
P
Q
R
6cm
21
0
S
Answer:
5.8cm
Statistics
Find the mean and standard deviation of this data set:
27
25
32
28
Answer:
mean = 28, s.d. = 2.9
Statistics
A group of dancers are scored on two rounds of dancing.
After the first round, the mean score was 6.7 and the
standard deviation of the scores was 3.5.
After the second round, the mean score was 7.3 and the
standard deviation was 1.8.
Make two valid comments comparing the scores of the
dancers after both rounds.
Answer:
eg
“On average the dancers scored higher in
round 2.”
eg
“The dancers’ scores were more consistent
in round 2.”
Trigonometric Graphs
This graph has an equation of the form y = acosbx.
State the values of a and b.
Answer:
a = 3, b = 6
Trigonometric Graphs
State the period of the graph with equation y = -2sin4x.
Answer:
90
0
Trigonometric Graphs
List the following values in order of size, starting with the
smallest:
sin(120)
sin(240)
sin(360)
Justify your answer.
Answer:
sin(240) < 0; sin(360) = 0; sin(120) > 0
Trigonometric Equations
Solve the equation 6cos(x) + 3 = 8, for 0 < x < 360.
Answer:
x = 33.6
0
or 326.4
0
Trigonometric Equations
Solve the equation 4tan(x) + 7 = 1, for 0 < x < 360.
Answer:
x = 123.7
0
or 303.7
0
Trigonometric Identities
Answer:
sin(x)
Trigonometric Identities
Answer:
sin(x)
Quadratic Graphs
A quadratic graph has the equation y = (x - 3)
2
+ 5.
a)
Determine the nature and coordinates of the turning
point.
b)
Determine the coordinates of the y-intercept.
Answer:
a)
minimum TP; (3,5)
b)
(0,14)
Quadratic Graphs
A quadratic graph has the equation y = x
2
- 6x - 16.
a)
Find the roots of the parabola.
b)
Determine the coordinates of the y-intercept.
c)
Find the coordinates of the turning point.
Answer:
a)
x = 8 or -2
b)
(0,-16)
c)
(3,-25)
Quadratic Graphs
Sketch the graph of the parabola with equation
y = (x - 3)(x + 5).
Clearly indicate the turning point and the points where
the parabola crosses the y-axis and the x-axis.
Answer:
Quadratic Equations
Solve x
2
- 6x + 5 = 0.
Answer:
x = 1 or 5
Quadratic Equations
Solve 2x
2
+ 4x - 5 = 0.
Round your answers to 1 decimal place.
Answer:
x = -2.9 or 0.9
Quadratic Equations
The rectangle below has a length of (x + 5) metres
and a breadth of (2x - 3) metres.
(x + 5) metres
(2x - 3) metres
a)
Show that the area of the rectangle is given by
A = 2x
2
+ 7x - 15
b)
The area of the rectangle is 24m
2
.
Calculate, algebraically, the length and breadth of the
rectangle.
Answer:
length = 8m,
breadth = 3m
Functions
Function
f
is defined as f(x) = 3x
2
+ 7x.
Evaluate f(-5).
Answer:
40
Functions
Function
f
is defined as f(x) = 8x - 11.
Find the value of a if f(a) = 37.
Answer:
a = 6
Practice various mathematics topics such as fractions, percentages, scientific notation, and expanding brackets with sample problems and solutions.
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Presentation Transcript
Scientific Notation (You can use a calculator for questions with this symbol) A grain of sand weighs 6.45 x 10-5 kilograms. A jar holds 4 x 106 grains of sand. Calculate the weight of the sand in the jar. Answer: 258kg
Scientific Notation There are approximately 7.9 x 109 people on Earth. Roughly 8% of people have blue eyes. Estimate the number of people on Earth who have blue eyes. Write your answer in scientific notation. Answer: 6.32 x 108
Fractions Calculate 240 360 3.14 30. Answer: 62.8
Fractions Calculate 31 2+ 22 9 . 103 18 or 5 13 Answer: 18
Fractions Calculate 51 4 24 5 . 49 20 or 2 9 Answer: 20
Fractions Calculate 21 3 14 5 . 21 5 Answer:
Fractions Calculate 91 2 33 8 . 76 27 Answer:
Fractions Calculate2 5(13 5+ 21 2). 41 25 Answer:
Percentages Calculate 12.5% of 56. Answer: 7
Percentages A group of friends go out for lunch. The total of their bill with a 12% tip comes to 76.16. Find the cost of their bill before the tip was added. Answer: 68
Percentages Lisa buys a cottage in 2018 valued at 82,000. The value of the cottage appreciated by 3.8% per annum. What was the value of the cottage in 2021? Round your answer to 2 significant figures. Answer: 92,000
Expanding Brackets Expand and simplify (3x + 1)(2x 7). Answer: 6x2 19x - 7
Expanding Brackets Expand and simplify (2x + 3)(x2 5x + 6). Answer: 2x3 7x2 3x + 18
Expanding Brackets Expand and simplify 4(2x + 5) 3(2x + 7). Answer: 2x - 1
Factorising Factorise 12p2 + 66pq. Answer: 6p(2p + 11q)
Factorising Factorise 9n2 - 100. Answer: (3n + 10)(3n 10)
Factorising Factorise x2 + 5x - 24. Answer: (x + 8)(x 3)
Factorising Factorise 3x2 + 14x + 8. Answer: (3x + 2)(x + 4)
Straight Lines Find the equation of the line joining the points (-5,2) and (1,4). Answer: y = 1 3x + 11 3
Straight Lines Find the equation of the line from the diagram below: Answer: y = -4x - 3
Surds Simplify 75 + 2 3 27. Answer: 4 3
Indices ?7 ?2 ?3 . Simplify Answer: c6
Indices Expand and simplify (5y4)2. Answer: 25y8
Indices Simplify, writing your answer with a positive index: ? 5 1 2 2 ? 1 ?2 Answer:
Arcs & Sectors Calculate the length of the major arc AB: A 1150 B O 6cm Answer: 25.7cm
Arcs & Sectors The length of major arc AB is 108cm. The radius of the sector is 20cm. Find the size of the shaded angle. 20cm A B Answer: 3090
Volume An object is made of a cone sitting on top of a cylinder, as shown: 20cm 8cm Answer: 1850cm3 14cm Find the total volume of this object to 3 significant figures.
Volume An object is made of a cylinder with a hemispherical indent as shown: 8cm 9cm Answer: 573cm3 10cm Find the total volume of this object to 3 significant figures.
Completing the Square Write x2 + 14x + 50 in the form (x + p)2 + q. Answer: (x + 7)2 + 1
Pythagoras Theorem In the diagram below, radius OB = 9cm and length AB = 14cm. h O B A Calculate the length of h to 1 decimal place. Answer: 14.7cm
Pythagoras Theorem 8cm 15cm 17cm Is the triangle above right-angled? Justify your answer. Answer: Yes since 172 = 152 + 82
Pythagoras Theorem A 12cm 7cm B 20cm Calculate the length of the space diagonal AB. Answer: 24.4cm
Angle Properties A B 88 O D 52o C Calculate the size of the shaded angle ABC. Answer: 840
Angle Properties C A 41 B 46 O D Calculate the size of the shaded angle COD. Answer: 1420
Angle Properties D E C F B A ABCDE is a regular pentagon. Find the size of the shaded angle EFA. Answer: 360
Algebraic Fractions 14 9?2. 8 Simplify 3? 12? 7 Answer:
Algebraic Fractions 3? 2?+7 2?+7 3? 1. Simplify 3? 3? 1 Answer:
Algebraic Fractions ?+5as a single fraction in its simplest form. 5 3 Write ?+2 2?+19 ?+2 ?+5 Answer:
Equations and Inequalities Solve the inequality 15 - 3(2x - 4) < 25. Answer: ? >1 3
Equations and Inequalities 4. Solve the equation 2? 3 2 =? 4 Answer: x = 12 5
Changing the Subject of a Formula Change the subject of this formula to n: ? =3?2 ? ?? 3 Answer: n =
Simultaneous Equations Solve the system of simultaneous equations: 2x + 4y = -2 5x 3y = 21 Answer: x = 3, y = -2
Simultaneous Equations The cost of 10 textbooks and 8 boxes of pens is 105. The cost of 4 textbooks and 10 boxes of pens is 59. Create an equation each for the information above and then find the cost of one textbook and of one box of pens. Answer: one textbook costs 8.50, one box of pens costs 2.50
Further Trigonometry Find the size of length PQ. P 24cm 750 480 Q R Answer: 18.5cm
Further Trigonometry Find the size of angle PQR. P 44cm 930 R 53cm Q Answer: 560
Further Trigonometry Find the size of length PQ. Q P 7cm 8.5cm 870 R Answer: 10.7cm
Further Trigonometry Find the size of angle RPQ. P 22cm 12cm Q 16cm R Answer: 450
Further Trigonometry Find the area of triangle PQR. P 14cm 99o 18cm R Q Answer: 124.4cm2
Further Trigonometry Find the size of length SR. P 6cm S 170 210 Q R Answer: 5.8cm
Statistics Find the mean and standard deviation of this data set: 27 25 32 28 Answer: mean = 28, s.d. = 2.9
