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 𝒂𝒂𝒃𝒃

of

two integers, where the denominator b is not equal to zero. 𝐚𝐚,𝐛𝐛 ∈ ℤ; 𝐛𝐛 ≠ 𝟎𝟎

12 ,−

34

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:

13 = 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:

𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 = �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 =𝐷𝐷𝑏𝑏

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

= �𝐷𝐷𝑏𝑏�

2⇒ 2 =

𝐷𝐷2

𝑏𝑏2 ⇒ 2𝑏𝑏2 = 𝐷𝐷2 ⇒ 𝐷𝐷2 𝐷𝐷𝑖𝑖 𝐷𝐷𝐷𝐷 𝑒𝑒𝑒𝑒𝑒𝑒𝐷𝐷 𝐷𝐷𝑛𝑛𝑛𝑛𝑏𝑏𝑒𝑒𝑛𝑛

If 𝐷𝐷 2 is an even number, then 𝐷𝐷 is also an even number. If 𝐷𝐷 were an odd

number, then 𝐷𝐷 = 2𝐷𝐷 + 1.

𝐷𝐷2 = (2𝐷𝐷 + 1)2 = (2𝐷𝐷)2 + 2 ∙ 2𝐷𝐷 ∙ 1 + 12 = 4𝐷𝐷2 + 4𝐷𝐷 + 1 = 2(2𝐷𝐷2 + 𝐷𝐷) + 1

𝐷𝐷2 = 2(2𝐷𝐷2 + 𝐷𝐷) + 1

⇒ 𝐷𝐷2 𝑤𝑤𝐷𝐷𝑛𝑛𝐷𝐷𝑤𝑤 𝑏𝑏𝑒𝑒 𝐷𝐷𝐷𝐷 𝐷𝐷𝑤𝑤𝑤𝑤 𝐷𝐷𝑛𝑛𝑛𝑛𝑏𝑏𝑒𝑒𝑛𝑛, 𝐷𝐷𝐷𝐷𝑤𝑤 𝑡𝑡ℎ𝐷𝐷𝑖𝑖 𝐷𝐷𝑖𝑖 𝐷𝐷 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒂𝒂𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄

So we have that 𝐷𝐷 is an even number, that is, we can write 𝐷𝐷 = 2𝑝𝑝

2 =𝐷𝐷2

𝑏𝑏2 ⇒ 2 =(2𝑝𝑝)2

𝑏𝑏2 ⇒ 2 =4𝑝𝑝2

𝑏𝑏2 ⇒ 2𝑏𝑏2 = 4𝑝𝑝2 ⇒ 𝑏𝑏2 = 2𝑝𝑝2

𝑏𝑏2 = 2𝑝𝑝2 ⇒ 𝑏𝑏2 𝐷𝐷𝑖𝑖 𝐷𝐷𝐷𝐷 𝑒𝑒𝑒𝑒𝑒𝑒𝐷𝐷 𝐷𝐷𝑛𝑛𝑛𝑛𝑏𝑏𝑒𝑒𝑛𝑛 ⇒ 𝑏𝑏 𝐷𝐷𝑖𝑖 𝐷𝐷𝐷𝐷 𝑒𝑒𝑒𝑒𝑒𝑒𝐷𝐷 𝐷𝐷𝑛𝑛𝑛𝑛𝑏𝑏𝑒𝑒𝑛𝑛

Therefore, 𝐷𝐷 and 𝑏𝑏 are both even numbers. This is a contradiction because we

started our proof saying that 𝐷𝐷 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, √153 , √105 𝐷𝐷𝑛𝑛𝑒𝑒 𝐷𝐷𝑛𝑛𝑛𝑛𝐷𝐷𝑡𝑡𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝐷𝐷𝑛𝑛𝑛𝑛𝑏𝑏𝑒𝑒𝑛𝑛𝑖𝑖

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:

𝟐𝟐.𝟕𝟕𝟕𝟕𝟕𝟕𝟐𝟐𝟕𝟕𝟕𝟕𝟕𝟕𝟐𝟐𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟎𝟎𝟕𝟕…

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

�𝟕𝟕

𝟕𝟕+ 𝒄𝒄�𝒄𝒄

approaches 𝒆𝒆 as 𝒄𝒄 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 𝒖𝒖𝒄𝒄𝒄𝒄𝒄𝒄𝒖𝒖 away from the origin to the right, and a negative

number – 𝒙𝒙 corresponds to a point on the line 𝒙𝒙 𝒖𝒖𝒄𝒄𝒄𝒄𝒄𝒄𝒖𝒖 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 √𝐷𝐷, 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 (𝑡𝑡𝐷𝐷 2 𝑤𝑤𝑒𝑒𝑑𝑑𝐷𝐷𝑛𝑛𝐷𝐷𝐷𝐷 𝑝𝑝𝐷𝐷𝐷𝐷𝑑𝑑𝑒𝑒𝑖𝑖)

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 (𝑡𝑡𝐷𝐷 2 𝑖𝑖𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑠𝑠𝐷𝐷𝑑𝑑𝐷𝐷𝐷𝐷𝑡𝑡 𝑠𝑠𝐷𝐷𝐷𝐷𝑛𝑛𝑛𝑛𝑒𝑒𝑖𝑖)

0.00405 ≈ 0.0041 (𝑡𝑡𝐷𝐷 2 𝑖𝑖𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑠𝑠𝐷𝐷𝑑𝑑𝐷𝐷𝐷𝐷𝑡𝑡 𝑠𝑠𝐷𝐷𝐷𝐷𝑛𝑛𝑛𝑛𝑒𝑒𝑖𝑖) 6.339 ≈ 6.34 (𝑡𝑡𝐷𝐷 3 𝑖𝑖𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑠𝑠𝐷𝐷𝑑𝑑𝐷𝐷𝐷𝐷𝑡𝑡 𝑠𝑠𝐷𝐷𝐷𝐷𝑛𝑛𝑛𝑛𝑒𝑒𝑖𝑖)

0.000000338754 ≈ 0.000000339 (𝑡𝑡𝐷𝐷 3 𝑖𝑖𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑠𝑠𝐷𝐷𝑑𝑑𝐷𝐷𝐷𝐷𝑡𝑡 𝑠𝑠𝐷𝐷𝐷𝐷𝑛𝑛𝑛𝑛𝑒𝑒𝑖𝑖)

You can estimate the answer to a calculation by rounding the numbers:

Estimate the answer to

6.23 ∙ 9.8918.7

You can round each of the numbers to 1 significant figure:

6 ∙ 1020 = 3 ⇒

6.23 ∙ 9.8918.7 ≈ 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 𝒗𝒗𝒂𝒂𝒂𝒂𝒂𝒂𝒄𝒄𝒄𝒄𝒙𝒙, the Absolute error is:

𝑨𝑨𝒃𝒃𝒖𝒖𝒄𝒄𝑨𝑨𝒖𝒖𝒄𝒄𝒆𝒆 𝒆𝒆𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 = ∆𝒙𝒙 = 𝝐𝝐 = �𝒗𝒗 − 𝒗𝒗𝒂𝒂𝒂𝒂𝒂𝒂𝒄𝒄𝒄𝒄𝒙𝒙�

Relative error gives an indication of how good a measurement is relative to the

size of the thing being measured. The Relative error is:

𝑹𝑹𝒆𝒆𝑨𝑨𝒂𝒂𝒄𝒄𝒄𝒄𝒗𝒗𝒆𝒆 𝒆𝒆𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 = 𝜼𝜼 =𝝐𝝐

|𝒗𝒗| =�𝒗𝒗 − 𝒗𝒗𝒂𝒂𝒂𝒂𝒂𝒂𝒄𝒄𝒄𝒄𝒙𝒙�

|𝒗𝒗| = �𝟕𝟕 −𝒗𝒗𝒂𝒂𝒂𝒂𝒂𝒂𝒄𝒄𝒄𝒄𝒙𝒙

𝒗𝒗 �

This error can be converter in a percentage by multiplying by 100

called Percent Error:

𝜹𝜹 = 𝜼𝜼 ∙ 𝟕𝟕𝟎𝟎𝟎𝟎

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| =0.150 = 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:

0.012 = 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 𝐷𝐷𝑛𝑛 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% 𝒄𝒄𝒐𝒐 𝑸𝑸 = 𝒂𝒂𝟕𝟕𝟎𝟎𝟎𝟎 ∙ 𝑸𝑸

Example: 9% of 24 m.

9% 𝐷𝐷𝑠𝑠 24 =9

100 ∙ 100 = 2.16 𝑛𝑛

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:

𝑷𝑷𝒆𝒆𝒄𝒄𝒄𝒄𝒆𝒆𝒄𝒄𝒄𝒄𝒂𝒂𝑷𝑷𝒆𝒆 𝑰𝑰𝒄𝒄𝒄𝒄𝒄𝒄𝒆𝒆𝒂𝒂𝒖𝒖𝒆𝒆 𝒂𝒂% 𝒄𝒄𝒗𝒗𝒆𝒆𝒄𝒄 𝑸𝑸 = 𝑸𝑸 ∙ �𝟕𝟕𝟎𝟎𝟎𝟎+ 𝒂𝒂𝟕𝟕𝟎𝟎𝟎𝟎 �

𝑷𝑷𝒆𝒆𝒄𝒄𝒄𝒄𝒆𝒆𝒄𝒄𝒄𝒄𝒂𝒂𝑷𝑷𝒆𝒆 𝑫𝑫𝒆𝒆𝒄𝒄𝒄𝒄𝒆𝒆𝒂𝒂𝒖𝒖𝒆𝒆 𝒂𝒂% 𝒄𝒄𝒗𝒗𝒆𝒆𝒄𝒄 𝑸𝑸 = 𝑸𝑸 ∙ �𝟕𝟕𝟎𝟎𝟎𝟎 − 𝒂𝒂𝟕𝟕𝟎𝟎𝟎𝟎 �

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?

𝑆𝑆𝐷𝐷𝐷𝐷𝑒𝑒 𝑝𝑝𝑛𝑛𝐷𝐷𝑑𝑑𝑒𝑒 = £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:

𝑴𝑴𝒖𝒖𝑨𝑨𝒄𝒄𝒄𝒄𝒂𝒂𝑨𝑨𝒆𝒆 𝒂𝒂𝒆𝒆𝒄𝒄𝒄𝒄𝒆𝒆𝒄𝒄𝒄𝒄𝒂𝒂𝑷𝑷𝒆𝒆𝒖𝒖 𝒄𝒄𝒗𝒗𝒆𝒆𝒄𝒄 𝑸𝑸 = 𝑸𝑸 ∙ �𝟕𝟕𝟎𝟎𝟎𝟎± 𝒂𝒂𝟕𝟕𝟕𝟕𝟎𝟎𝟎𝟎 � ∙ �

𝟕𝟕𝟎𝟎𝟎𝟎 ± 𝒂𝒂𝟐𝟐𝟕𝟕𝟎𝟎𝟎𝟎 �…

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 = 𝑰𝑰𝒄𝒄𝒄𝒄𝒆𝒆𝒄𝒄𝒆𝒆𝒖𝒖𝒄𝒄 𝒆𝒆𝒂𝒂𝒄𝒄𝒄𝒄𝒆𝒆𝒄𝒄; 𝑷𝑷 = 𝑷𝑷𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒂𝒂𝒂𝒂𝑨𝑨; 𝑹𝑹 = 𝑰𝑰𝒄𝒄𝒄𝒄𝒆𝒆𝒄𝒄𝒆𝒆𝒖𝒖𝒄𝒄 𝒄𝒄𝒂𝒂𝒄𝒄𝒆𝒆; 𝑻𝑻 = 𝑻𝑻𝒄𝒄𝑻𝑻𝒆𝒆

Usually:

𝑹𝑹 =𝒄𝒄𝟕𝟕𝟎𝟎𝟎𝟎 ,𝒘𝒘𝒘𝒘𝒆𝒆𝒄𝒄𝒆𝒆 𝒄𝒄 𝒄𝒄𝒖𝒖 𝒄𝒄𝒘𝒘𝒆𝒆 𝒂𝒂𝒆𝒆𝒄𝒄𝒄𝒄𝒆𝒆𝒄𝒄𝒄𝒄𝒂𝒂𝑷𝑷𝒆𝒆 𝒆𝒆𝒂𝒂𝒄𝒄𝒄𝒄𝒆𝒆𝒄𝒄 𝒂𝒂𝒆𝒆𝒄𝒄 𝒚𝒚𝒆𝒆𝒂𝒂𝒄𝒄

Calculate the interest when £1,000 is invested for 4 years at a 5% simple

interest (I).

𝐼𝐼 = 1,000 ∙5

100 ∙ 4 = £200

𝑇𝑇𝐷𝐷𝑡𝑡𝐷𝐷𝐷𝐷 = 𝑃𝑃𝑛𝑛𝐷𝐷𝐷𝐷𝑑𝑑𝐷𝐷𝑝𝑝𝐷𝐷𝐷𝐷 + 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 ∙ �𝟕𝟕 +𝑹𝑹𝒄𝒄�

𝒄𝒄∙𝒄𝒄

𝑨𝑨 = 𝑨𝑨𝑻𝑻𝒄𝒄𝒖𝒖𝒄𝒄𝒄𝒄;

𝑷𝑷 = 𝑷𝑷𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒂𝒂𝒂𝒂𝑨𝑨;

𝑹𝑹 = 𝒂𝒂𝒄𝒄𝒄𝒄𝒖𝒖𝒂𝒂𝑨𝑨 𝒄𝒄𝒄𝒄𝑻𝑻𝒄𝒄𝒄𝒄𝒂𝒂𝑨𝑨 𝒄𝒄𝒄𝒄𝒄𝒄𝒆𝒆𝒄𝒄𝒆𝒆𝒖𝒖𝒄𝒄 𝒄𝒄𝒂𝒂𝒄𝒄𝒆𝒆;

𝒄𝒄 = 𝒄𝒄𝒖𝒖𝑻𝑻𝒃𝒃𝒆𝒆𝒄𝒄𝒄𝒄𝒐𝒐 𝒄𝒄𝒄𝒄𝑻𝑻𝒆𝒆𝒖𝒖 𝒄𝒄𝒘𝒘𝒆𝒆 𝒄𝒄𝒄𝒄𝒄𝒄𝒆𝒆𝒄𝒄𝒆𝒆𝒖𝒖𝒄𝒄 𝒄𝒄𝒖𝒖 𝒄𝒄𝒄𝒄𝑻𝑻𝒂𝒂𝒄𝒄𝒖𝒖𝒄𝒄𝒄𝒄𝒆𝒆𝒄𝒄 𝒂𝒂𝒆𝒆𝒄𝒄 𝒚𝒚𝒆𝒆𝒂𝒂𝒄𝒄;

𝒄𝒄 = 𝒄𝒄𝒖𝒖𝑻𝑻𝒃𝒃𝒆𝒆𝒄𝒄 𝒄𝒄𝒐𝒐 𝒚𝒚𝒆𝒆𝒂𝒂𝒄𝒄𝒖𝒖

£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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