4 ESO Academics - Unit 01 - Real Numbers and Percentages

Unit 01 September
1. RATIONAL NUMBERS.
A Rational number is any number that can be expressed as the quotient
𝒂𝒂
𝒃𝒃
of
two integers, where the denominator b is not equal to zero. 𝐚𝐚, 𝐛𝐛 ∈ ℤ; 𝐛𝐛 ≠ 𝟎𝟎
1
2
, −
3
4
NOTE: A set is a collection of objects, these objects are called elements.
The set of all rational numbers is usually denoted ℚ (for quotient). Natural
numbers, whole numbers, integers and fractions are rational numbers.
A Decimal number is a rational number if it can be written as a fraction. Those
are decimals that either terminate or have a repeating block of digits.
Axel Cotón Gutiérrez Mathematics 4º ESO 1.1

Unit 01 September
Terminating decimals: 4.8; 3.557; 24.997897
Recurring decimals: 4.888888..; 34.345345345...
NOTE: We can write a recurring decimal in different ways:
1
3
= 0.333… = 0. 3̇ = 0. 3�
All the rational numbers can be shown in a number line. To do this we generally
divide each segment using the theorem of Thales.

Unit 01 September
MATH VOCABULARY: Rational number, Integer, Denominator, Set, Elements, Decimal
number, Terminating decimals, Recurring decimals, Number line, Thales´ theorem,
Theorem, Segment.
2. IRRATIONAL NUMBERS.
An Irrational number is a number that cannot be written as a simple fraction.
Equivalently, irrational numbers cannot be represented as terminating or recurring
decimals (the decimal part goes on forever without repeating). There are infinite
irrational numbers. Here you are some of the most interesting ones:
2.1. NUMBER π.
π (pi) is an irrational number. The value of π is
3.1415926535897932384626433832795 (and more…). There is no pattern to the
decimals, and you cannot write down a simple fraction that equals π. Remember that
π is the ratio of the circumference to the diameter of a circle. In other words, if you
measure the circumference, and then divide by the diameter of the circle you get the
number π.
MATH VOCABULARY: Irrational Numbers, Ratio, Circumference, Diameter, Circle,
Pattern.

Unit 01 September
2.2. THE SQUARE ROOT OF 2.
Pythagora’s theorem shows that the diagonal of a square with sides of one unit
of length is equal to √2:
𝐷𝐷𝐷𝐷𝐷𝐷 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 = �12 + 12 = 2
The computer says that √2 = 1.414213562373095048801688724... , but
this is not the full story! It actually goes on and on, with no pattern to the numbers.
You cannot write down a simple fraction that equals √2 .
How can we prove that √2 is an irrational number? The proof that √2 is indeed
irrational is a ‘proof by contradiction’ (if √2 WERE a rational number, then we would
get a contradiction):
PROOF:
Let’s suppose √2 were a rational number. Then we can write it as a quotient of
two integer numbers:
√2 =
𝑎𝑎
𝑏𝑏
We also suppose that is the simplest fraction, that is, a and b have no common
factors. Squaring on both sides gives:

Unit 01 September
�√2�
2
= �
𝑎𝑎
𝑏𝑏
�
2
⇒ 2 =
𝑎𝑎2
𝑏𝑏2
⇒ 2𝑏𝑏2
= 𝑎𝑎2
⇒ 𝑎𝑎2
𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛
If 𝑎𝑎 2
is an even number, then 𝑎𝑎 is also an even number. If 𝑎𝑎 were an odd
number, then 𝑎𝑎 = 2𝑛𝑛 + 1.
𝑎𝑎2
= (2𝑛𝑛 + 1)2
= (2𝑛𝑛)2
+ 2 ∙ 2𝑛𝑛 ∙ 1 + 12
= 4𝑛𝑛2
+ 4𝑛𝑛 + 1 = 2(2𝑛𝑛2
+ 𝑛𝑛) + 1
𝑎𝑎2
= 2(2𝑛𝑛2
+ 𝑛𝑛) + 1
⇒ 𝑎𝑎2
𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑏𝑏𝑏𝑏 𝑎𝑎𝑎𝑎 𝑜𝑜𝑜𝑜𝑜𝑜 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛, 𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡ℎ𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑎𝑎 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄
So we have that 𝑎𝑎 is an even number, that is, we can write 𝑎𝑎 = 2𝑝𝑝
2 =
𝑎𝑎2
𝑏𝑏2
⇒ 2 =
(2𝑝𝑝)2
𝑏𝑏2
⇒ 2 =
4𝑝𝑝2
𝑏𝑏2
⇒ 2𝑏𝑏2
= 4𝑝𝑝2
⇒ 𝑏𝑏2
= 2𝑝𝑝2
𝑏𝑏2
= 2𝑝𝑝2
⇒ 𝑏𝑏2
𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 ⇒ 𝑏𝑏 𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛
Therefore, 𝑎𝑎 and 𝑏𝑏 are both even numbers. This is a contradiction because we
started our proof saying that 𝑎𝑎 and 𝑏𝑏 have no common factors.
Then √2 is an irrational number.
Many square roots, cube roots, etc. are also irrational numbers. If 𝑝𝑝 is not a
square number, then �𝑝𝑝 is an irrational number. In general, if 𝑝𝑝 is not an exact nth
power, then �𝑝𝑝𝑛𝑛
is an irrational number.
√3, √11, √15
3
, √10
5
𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛
MATH VOCABULARY: Pythagora’s theorem, Diagonal, Square, To Prove, Proof, Proof
by contradiction, Simplest fraction, Even number, Odd number, Square root, Cube root.

Unit 01 September
2.3. THE GOLDEN NUMBER.
In a regular pentagon the ratio between a diagonal and a side is the irrational
number:
Φ =
√5 + 1
2
This number is called 𝚽𝚽, the Golden number (or the Golden ratio).
If you divide a line into two parts so that: “The whole length divided by the
longest part is equal to the longest part divided by the smallest part, then you will have
the golden number”.
In this case we say that a and 𝒃𝒃 are in the Golden ratio. The golden ratio
appears many times in geometry, art, architecture and other areas.

Unit 01 September
A rectangle, in which the ratio of the longest side to the shortest is the golden
ratio, is called Golden rectangle. A golden rectangle can be cut into a square and a
smaller rectangle that is also a golden rectangle.
Some artists and architects believe the golden ratio makes the most pleasing
and beautiful shape. Many buildings ant artworks have the Golden Ratio in them, such
as the Parthenon in Greece, but it is not really known if it was designed that way.
MATH VOCABULARY: Pentagon, Side, Golden ratio, Golden number, Length, Geometry,
Rectangle, Shape.

Unit 01 September
2.4. THE NUMBER e.
The number e is a famous irrational number, and one of the most important
numbers in mathematics. It is often called Euler’s number, after Leonhard Euler. The
first few digits are:
𝟐𝟐. 𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕…
There are many ways of calculating the value of 𝒆𝒆, but none of them ever give
an exact answer. Nevertheless, it is known to over 1 trillion digits of accuracy. For
example, the value of
�
𝟏𝟏
𝟏𝟏 + 𝒏𝒏
�
𝒏𝒏
approaches 𝒆𝒆 as 𝒏𝒏 gets bigger and bigger.
MATH VOCABULARY: Euler’s number, Accuracy, To Approach.
3. REAL NUMBERS.
The set of the rational numbers and the irrational numbers is called the set of
the Real numbers. It is usually denoted ℝ.

Unit 01 September
Natural numbers are a subset of integers. Integers are a subset of rational
numbers. Rational numbers are a subset of real numbers.
MATH VOCABULARY: Real numbers, Subset.
3.1. THE REAL NUMBER LINE.
The real number system can be visualized as a horizontal line that extends from
a special point called the origin in both directions towards infinity. Also associated with
the line is a unit of length. The origin corresponds to the number 𝟎𝟎. A positive number
𝒙𝒙 corresponds to a point 𝒙𝒙 𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖 away from the origin to the right, and a negative
number – 𝒙𝒙 corresponds to a point on the line 𝒙𝒙 𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖 away from the origin to the
left.

Unit 01 September
This horizontal line is called the Real number line. Any point on the number line
is a real number and vice versa, any real number is a point on the number line. To plot
irrational numbers like √𝑛𝑛, we use the Pythagora’s theorem. For irrational numbers
like 𝜋𝜋, 𝑒𝑒, … we plot them approximately.
MATH VOCABULARY: Horizontal, To Extend, Point, Origin, Real number line.
4. APPROXIMATION AND ROUNDING.
Rounding a number is another way of writing a number approximately. We
often don’t need to write all the figures in a number, as an approximate one will do.
The population of Villanueva de la Serena is 28,789. Since populations change
frequently, we use a rounded number instead of the exact number. It is better to
round up and say 29,000.
To round a number to a given place:
• Find the place you are rounding to.
• Look at the digit to its right.

Unit 01 September
• If the digit is less than 5, round down.
• If the digit is 5 or greater, round up.
Round 83,524 to the nearest ten:
The digit to the right is 4. 4 < 5. Round down to 83,520
Round 83,524 to the nearest hundred:
The digit to the right is 2. 2 < 5. Round down to 83,500
Round 83,524 to the nearest thousand:
The digit to the right is 5. 5 = 5. Round up to 84,000
To round 718.394 to 2 decimal places, look at the thousandths digit.
The thousandths digit is 4, so round down to 718.39.
718.394 ≈ 718.39 (𝑡𝑡𝑡𝑡 2 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝)
Numbers can be rounded:
• To decimal places 4.16 = 4.2 to 1 decimal place
• To the nearest unit, 10, 100, 1000, …
Remember that a method of giving an approximate answer to a problem is to
round off using significant figures. The first non-zero digit in a number is called the

Unit 01 September
first significant figure –it has the highest value in the number. When rounding to
significant figures, count from the first non-zero digit.
54.76 ≈ 55 (𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓)
0.00405 ≈ 0.0041 (𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓)
You can estimate the answer to a calculation by rounding the numbers:
Estimate the answer to
6.23 ∙ 9.89
18.7
You can round each of the numbers to 1 significant figure:
6 ∙ 10
20
= 3 ⇒
6.23 ∙ 9.89
18.7
≈ 3
dp and sf are abbreviations for ‘decimal places’ and ‘significant figures’. When
a measurement is written, it is always written to a given degree of accuracy. The real
measurement can be anywhere within ± half a unit.
A man walks 23 km (to the nearest km). Write the maximum and minimum
distance he could have walked.
Because the real measurement has been rounded, it can lie anywhere between
22.5 km (minimum) and 23.5 (maximum).
Another way of approximating a number is called Truncating a number, is a
method of approximating a decimal number by dropping all decimal places past a
certain point without rounding.

Unit 01 September
3.14159265... can be truncated to 3.1415
MATH VOCABULARY: Rounding, Approximation, Population, Frequently, Exact number,
Round up, Round down, Thousandths, First significant figure, Estimate, Truncate.
5. APPROXIMATION ERRORS.
Absolute and Relative error are two types of error. The differences are
important.
Absolute error is the amount of physical error in a measurement, period. Given
some value 𝒗𝒗 and its approximation 𝒗𝒗𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂, the Absolute error is:
𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨 𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆 = ∆𝒙𝒙 = 𝝐𝝐 = �𝒗𝒗 − 𝒗𝒗𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂�
Relative error gives an indication of how good a measurement is relative to the
size of the thing being measured. The Relative error is:
𝑹𝑹𝑹𝑹𝑹𝑹𝑹𝑹𝑹𝑹𝑹𝑹𝑹𝑹𝑹𝑹 𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆 = 𝜼𝜼 =
𝝐𝝐
| 𝒗𝒗|
=
�𝒗𝒗 − 𝒗𝒗𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂�
| 𝒗𝒗|
= �𝟏𝟏 −
𝒗𝒗𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂
𝒗𝒗
�
This error can be converter in a percentage by multiplying by 100
called Percent Error:
𝜹𝜹 = 𝜼𝜼 ∙ 𝟏𝟏𝟏𝟏𝟏𝟏
In words, the Absolute error is the magnitude of the difference between the
exact value and the approximation. The Relative error is the absolute error divided by
the magnitude of the exact value. The Percent error is the relative error expressed in
terms of per 100.
If the exact value is 50 and its approximation 49.9, then the absolute error is:
ϵ = |50 − 49.9| = |0.1| = 0.1

Unit 01 September
and the relative error is:
η =
ϵ
|v|
=
0.1
50
= 0.002
so the percent error is:
𝛿𝛿 = 𝜂𝜂 ∙ 100 = 0.002 ∙ 100 = 0.2%
Sometimes a maximum assumable error is given; this error is called
“Error bound”. This error can be absolute or relative.
Absolute error bound is 0.005 cm
Relative error bound is 5 %
An upper bound for the absolute error is half a unit of the last significant figure
in the approximate value.
3.14 is the approximation of the number 𝜋𝜋 to 3 significant figures, that is, to the
hundredths. An upper bound for the absolute error of this approximation is: half one
hundredth, that is:
0.01
2
= 0.005 ⇒ | 𝜋𝜋 − 3.14| < 0.005
MATH VOCABULARY: Error, Absolute error, Relative error, Percent error, Error bound.
6. INTERVALS.
An Interval is a set formed by the real numbers between, and sometimes
including, two numbers. They can also be non-ending intervals as we are going to see.
It can also be thought as a segment of the real number line. An endpoint of an interval
is either of the two points that mark end on the line segment. An interval can include
either endpoint, both endpoints, or neither endpoint.

Unit 01 September
There are different notations for intervals:
Let 𝒂𝒂 and 𝒃𝒃 be real numbers such that 𝒂𝒂 < 𝒃𝒃:
• The Open interval (𝒂𝒂, 𝒃𝒃) is the set of real numbers between 𝒂𝒂 and 𝒃𝒃, excluding
𝒂𝒂 and 𝒃𝒃.
• The Closed interval [ 𝒂𝒂, 𝒃𝒃] is the set of real numbers between 𝒂𝒂 and 𝒃𝒃, including
𝒂𝒂 and 𝒃𝒃.
• The Left half-open interval (𝒂𝒂, 𝒃𝒃] is the set of real numbers between 𝒂𝒂 and 𝒃𝒃,
including 𝒃𝒃 but not a.

Unit 01 September
• The Right half-open interval [ 𝒂𝒂, 𝒃𝒃) is the set of real numbers between 𝒂𝒂 and 𝒃𝒃,
including 𝒂𝒂 but not 𝒃𝒃.
• The Infinite intervals are those that do not have an endpoint in either the
positive or negative direction, or both. The interval extends forever in that
direction.
Examples:

Unit 01 September
MATH VOCABULARY: Interval, Endpoint, Square bracket, Round bracket, Notation,
Open interval, Closed interval, Left half-open interval, Right half-open interval, Infinite
interval.
6.1. UNION AND INTERSECTION.
We can to join two sets using "Union" (and the symbol ∪). There is also
"Intersection" which means "has to be in both". Think "where do they overlap?". The
Intersection symbol is "∩".
Example: 𝑥𝑥 ≤ 2 𝑜𝑜𝑜𝑜 𝑥𝑥 > 3. On the number line it looks like this:
And interval notation looks like this:
(−∞, 2] 𝑈𝑈 (3,+∞)
Example: (−∞, 6] ∩ (1, ∞). The first interval goes up to (and including) 6. The second
interval goes from (but not including) 1 onwards.
The Intersection (or overlap) of those two sets goes from 1 to 6 (not including
1, including 6):
MATH VOCABULARY: Union, Intersection.

Unit 01 September
7. PERCENTAGES.
7.1. FINDING A PERCENTAGE OF A QUANTITY.
You often need to calculate a Percentage of a quantity 𝑸𝑸:
𝒂𝒂% 𝒐𝒐𝒐𝒐 𝑸𝑸 =
𝒂𝒂
𝟏𝟏𝟏𝟏𝟏𝟏
∙ 𝑸𝑸
Example: 9% of 24 m.
9% 𝑜𝑜𝑜𝑜 24 =
9
100
∙ 100 = 2.16 𝑚𝑚
7.2. PERCENTAGE INCREASE AND DECREASE.
Percentages are used in real life to show how much an amount has increased
or decreased.
• To calculate a Percentage increase, work out the increase and add it to the
original amount.
• To calculate a Percentage decrease, work out the decrease and subtract it from
the amount.
Alan is paid £940 a month. His employer increases his wage by 3%. Calculate
the new wage Alan is paid each month.
Increase in wage = 3% of £940 = 0.03 × £940 = £28.20
Alan’s new wage = £940 + £28.20 = £968.20
A new car costs £19 490. After one year the car depreciates in value by 8.7%.
What is the new value of the car?
Depreciation = 8.7% of £19 490 = 0.087 × £19490 = £1695.63
New value of car = £19490 − £1695.63 = £17 794.37

Unit 01 September
You can also calculate a percentage increase or decrease in a single calculation:
𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰 𝒂𝒂% 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐 𝑸𝑸 = 𝑸𝑸 ∙ �
𝟏𝟏𝟏𝟏𝟏𝟏 + 𝒂𝒂
𝟏𝟏𝟏𝟏𝟏𝟏
�
𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫 𝒂𝒂% 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐 𝑸𝑸 = 𝑸𝑸 ∙ �
𝟏𝟏𝟏𝟏𝟏𝟏 − 𝒂𝒂
𝟏𝟏𝟏𝟏𝟏𝟏
�
In a sale all prices are reduced by 16%. A pair of trousers normally costs £82.
What is the sale price of the pair of trousers?
𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = £82 ∙ �
100 − 16
100
� = £82 ∙ 0.84 = £68.88
MATH VOCABULARY: Percentage increase, Percentage decrease.
7.3. MULTIPLE PERCENTAGES.
Sometimes we need to calculate Multiple percentages simultaneously. We
have to apply the above formulas several times:
𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴 𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐 𝑸𝑸 = 𝑸𝑸 ∙ �
𝟏𝟏𝟏𝟏𝟏𝟏 ± 𝒂𝒂𝟏𝟏
𝟏𝟏𝟏𝟏𝟏𝟏
� ∙ �
𝟏𝟏𝟏𝟏𝟏𝟏 ± 𝒂𝒂𝟐𝟐
𝟏𝟏𝟏𝟏𝟏𝟏
�…
During Christmas, a phone shop prices up 21%. On January, during sales, prices
fell 19%. Before Christmas a phone cost 645€. How much cost in January?
January cost = 645 ∙ �
100 + 21
100
� ∙ �
100 − 19
100
� = 632.16 €
MATH VOCABULARY: Multiple percentages, Formulae.

Unit 01 September
8. SIMPLE INTEREST.
You earn Interest when you invest in a savings account at a bank. However, you
pay Interest if you borrow money for a mortgage. The original sum you invest is called
the principal.
Simple interest is money you can earn by initially investing some money (the
principal). A percentage (the interest) of the principal is added to the principal, making
your initial investment grow!
To calculate simple interest, use the interest rate to work out the amount
earned. If simple interest is paid for several years, the amount paid each time stays the
same, because the interest is paid elsewhere and the principal stays the same.
𝑰𝑰 = 𝑷𝑷 ∙ 𝑹𝑹 ∙ 𝑻𝑻
𝑰𝑰 = 𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰 𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆; 𝑷𝑷 = 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝒑𝒑𝒂𝒂𝒂𝒂; 𝑹𝑹 = 𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰 𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓; 𝑻𝑻 = 𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻
Usually:
𝑹𝑹 =
𝒓𝒓
𝟏𝟏𝟏𝟏𝟏𝟏
, 𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘 𝒓𝒓 𝒊𝒊𝒊𝒊 𝒕𝒕𝒕𝒕𝒕𝒕 𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑 𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆 𝒑𝒑𝒑𝒑𝒑𝒑 𝒚𝒚𝒚𝒚𝒚𝒚𝒚𝒚
Calculate the interest when £1,000 is invested for 4 years at a 5% simple
interest (I).
𝐼𝐼 = 1,000 ∙
5
100
∙ 4 = £200
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 + 𝐼𝐼 = £1,000 + £200 = £1200
MATH VOCABULARY: Interest, Saving account, to Invest, Mortgage, Principal, Simple
interest.

Unit 01 September
9. COMPOUND INTEREST.
The addition of interest to the principal sum of a loan or deposit is called
compounding. Compound interest is interest on interest. It is the result of reinvesting
interest, rather than paying it out, so that interest in the next period is then earned on
the principal sum plus previously-accumulated interest. Compound interest is standard
in finance and economics.
To calculate compound interest, work out the interest in the same way, but
add the interest earned to the principal. If compound interest is paid for several years,
the amount of interest earned each year increases, because the principal increases.
𝑨𝑨 = 𝑷𝑷 ∙ �𝟏𝟏 +
𝑹𝑹
𝒏𝒏
�
𝒏𝒏∙𝒕𝒕
𝑨𝑨 = 𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨;
𝑷𝑷 = 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷;
𝑹𝑹 = 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂 𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓;
𝒏𝒏 = 𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒕𝒕𝒕𝒕𝒕𝒕 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝒊𝒊𝒊𝒊 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 𝒑𝒑𝒑𝒑𝒑𝒑 𝒚𝒚𝒚𝒚𝒚𝒚𝒚𝒚;
𝒕𝒕 = 𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏 𝒐𝒐𝒐𝒐 𝒚𝒚𝒚𝒚𝒚𝒚𝒚𝒚𝒚𝒚
£2,000 is invested at 6.5% compound interest. Find the principal after 15 years.
A = 2,000 ∙ �1 +
0,065
1
�
1∙15
= 2,000 ∙ (1,065)15
= £5,143.68
If you have a bank account whose principal is $1,000, and your bank
compounds the interest twice a year at an interest rate of 5%, how much money do
you have in your account at the year's end?
A = 1,000 ∙ �1 +
0,05
2
�
2∙1
= 1,000 ∙ (1,025)2
= $1,050.63
MATH VOCABULARY: Compound interest, to Reinvest.

4 ESO Academics - Unit 01 - Real Numbers and Percentages

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4 ESO Academics - Unit 01 - Real Numbers and Percentages