GCSE Mathematics


Numbers

Definition of numbers

Sets of numbers can be described in different ways:

  • natural numbers: 1, 2, 3, 4, 5, ...
  • positive numbers: +1, +2, +3, +4, ...
  • negative numbers: -1, -2, -3, -4, ...
  • square numbers: 1, 4, 9, 16, 25, 36, ...
  • triangle numbers: 1, 3, 6, 10, 15, 21, ...

Multiples

The multiples of a number are the products of the multiplication tables. e.g. 

  • Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, ... 
  • Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, ... 

The lowest common multiple (LCM) is the lowest multiple which is common to all of the given numbers.

e.g. Common multiples of 3 and 4 are 12, 24, 36, ... The lowest common multiple is 12. 

Factors

The factors of a number are the natural numbers which divide exactly into that number (i.e. without a remainder). e.g. 

  • Factors of 8 are 1, 2, 4 and 8. 
  • Factors of 12 are 1, 2, 3, 4, 6 and 12. 

The highest common factor (HCF) is the highest factor which is common to all of the given numbers. e.g. 

  • Common factors of 8 and 12 are 1, 2 and 4. 
  • The highest common factor is 4.

Prime numbers

A prime number is a natural number with exactly two factors (i.e. 1 and itself).

The following numbers have exactly two factors so are prime numbers.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... 

e.g. 21 can be written as 3 x 7 where 3 and 7 are prime factors.

60 can be written as 2 x 2 x 3 x 5 where 2, 3 and 5 are prime factors. 

The prime factors of a number can be found by successively rewriting the number as a product of prime numbers in increasing order (i.e. 2, 3, 5, 7, 11, 13, 17, ... etc.).

e.g. 84 = 2 x 42
           = 2 x 2 x 21
           = 2 x 2 x 3 x 7

Squares

Square numbers are numbers which have been multiplied by themselves.

e.g. The square of 8 is 8 x 8 = 64 and 64 is a square number. 

Cubes

Cube numbers are numbers which have been multiplied by themselves then multiplied by themselves again.

e.g. The cube of 5 is 5 x 5 x 5 = 125 and 125 is a cube number. 

Square roots

The square root of a number such as 36 is the number which when squared equals 36 i.e. 6 (because 6 x 6 = 36).

The sign √ is used to denote the square root. √36 = 6

Cube roots

The cube root of a number such as 27 is the number which when cubed equals 27 i.e. 3 (because 3 x 3 x 3 = 27). 

The sign 3√ is used to denote the cube root. 327 = 3 

Reciprocals

The reciprocal of any number can be found by converting the number to a fraction and turning the fraction upside-down. 

The reciprocal of 2/3 is 3/2 and the reciprocal of 10 is 1/10.

Directed numbers

A directed number is one which has a + or – sign attached to it.

When adding or subtracting directed numbers, remember that signs written next to each other can be replaced by a single sign as follows: 

+ + is the same as +

+ - is the same as -

- + is the same as -

- - is the same as +

eg. (-1) + (-2) = -1-2 = -3

(+2) - (-3) = +2+3 = +5

To multiply or divide directed numbers, include the sign according to the following rules:

  •  if the signs are the same, the answer is positive 
  • if the signs are opposite, the answer is negative 
eg. (-8) x (+2) = -16
(+12) / (-4) = =3
(-2) / (-5) = +2/5
(-5)2 = +25

Positive, negative and zero indices

When multiplying a number by itself you can use the following shorthand.

7 x 7 = 72 

7 x 7 x 7 = 73

Multiplying indices

You can multiply numbers with indices as shown, by adding the powers:

74 x 76 = 710

Dividing indices

You can divide numbers with indices as shown, by subtracting the powers:

56 / 54 = 52 

Negative powers

84 / 86 = 84-6 = 8-2

8-2 = 1/82

in general, a-m = 1/am

eg 3-2 = 1/32 = 1/9

2-4 = 1/24 = 1/16

Zero powers

50 = 1 

in general, a0 = 1

Any number raised to the power of zero is equal to 1

Significant figures

Any number can be rounded off to a given number of significant figures (written s.f.) using the following rules.

  • Count along to the number of significant figures required.
  • Look at the next significant digit.
    • If it is smaller than 5, leave the ‘significant’ digits as they are.
    • If it is 5 or greater, add 1 to the last of the ‘significant’ digits.
  • Restore the number to its correct size by filling with zeros if necessary. 

e.g. Round 547.36 to 4, 3, 2, 1 significant figures.

547.36 = 547.4 (4 s.f.) 

547.36 = 547 (3 s.f.) 

547.36 = 550 (2 s.f.)

547.36 = 500 (1 s.f.) 

Decimal places

Any number can be rounded to a given number of decimal places (written d.p.) using the following rules.

  • Count along to the number of the decimal places required.
  • Look at the digit in the next decimal place.
    • If it is smaller than 5, leave the preceding digits (the digits before it) as they are.
    • If it is 5 or greater, add 1 to the preceding digit.
  • Restore the number by replacing any numbers to the left of the decimal point. 

e.g. Round 19.3461 to 4, 3, 2, 1 decimal places.

19.3461 = 19.3461 (4 d.p.) 

19.3461 = 19.346 (3 d.p.) 

19.3461 = 19.35 (2 d.p.) 

19.3461 = 19.3 (1 d.p.) 

Multiplying decimals

To multiply two decimals without using a calculator:

  • ignore the decimal points and multiply the numbers
  • add the number of digits after the decimal point in the numbers in the question
  •  position the decimal point so that the number of digits after the decimal point in the answer is the same as the total number of decimal places in the question. 

e.g. Calculate 1.67 5.3
167 x
 53 = 8851 (Ignoring the decimal points and multiplying the numbers.) 

The number of digits after the decimal point in the numbers = 2 + 1 = 3.

1.67 x 5.3 = 8.851 (Replacing the decimal point so that the number of digits after the decimal point in the answer is 3.)

It is helpful to check that the answer is approximately correct i.e. 1.67 x 5.3 is approximately 2 x 5 = 10 so the answer of 8.851 looks correct. 

Dividing decimals

You can use the idea of equivalent fractions to divide decimals.

e.g. Work out 0.00308 ÷ 0.00014 

0.00308 ÷ 0.00014 = 0.00308/0.00014 = 308/14 (Multiplying top and bottom by 100 000 to obtain an equivalent fraction.)

Now divide 308 ÷ 14

So 0.00308 ÷ 0.00014 = 22 

Estimation and approximation

It is useful to check your work by approximating your answers to make sure that they are reasonable. Estimation and approximation questions are popular questions on the examination syllabus. You will usually be required to give an estimation by rounding numbers to 1 (or 2) significant figures. 

eg. Estimate the value of (6.98 x (10.16)2)/(9.992 x √50)

Rounding the figures to 1 significant figure and approximating √50 as 7:

(6.98 x (10.16)2)/(9.992 x √50) --> (7 x 102)/(10 x 7) = 700/70 = 10

A calculator gives an answer of 10.197 777 so that the answer is quite a good approximation. 

Imperial/Metric units

In number work, it is common to be asked to convert between imperial and metric units. In particular, the following conversions may be tested in the examination. 


Fractions

The top part of a fraction is called the numerator and the bottom part is called the denominator

Equivalent fractions

Equivalent fractions are fractions which are equal in value to each other. The following fractions are all equivalent to 1/2

1/2 = 2/4 = 3/6 = 5/10 = ...

Equivalent fractions can be found by multiplying or dividing the numerator and denominator by the same number. 

One number as a fraction of another

To find one number as a fraction of another, write the numbers in the form of a fraction.

e.g. Write 4mm as a fraction of 8cm.

First ensure that the units are the same. Remember 8cm = 80mm

4mm as a fraction of 80mm = 4/80 = 1/20

so 4mm is 1/20 of 8cm


Addition and subtraction

Before adding (or subtracting) fractions, ensure that they have the same denominator. 

eg 7/8 - 1/5 = 35/40 - 8/40 (writing both fractions with a denominator of 40)

                   = 27/40

To find the common denominator of two numbers, find their lowest common multiple or LCM.

The LCM of 8 and 5 is 40 

Multiplication of fractions

To multiply fractions, multiply the numerators and multiply the denominators. 

eg. 4/7 x 2/11 = (4 x 2)/(7 x 11) = 8/77

eg. 1/1/5 x 6/2/3 = 6/5 x 20/3 = (6 x 20)/(5 x 3) = 120/15 = 8 (converting to top heavy fractions. Multiplying the numerators and multiplying the denominators)

Division of fractions

To divide one fraction by another, multiply the first fraction by the reciprocal of the second fraction. 

eg. 3/7 ÷ 1/7 = 3/7 x 7/1 = 3

eg. 4/4/5 ÷ 1/1/15 = 24/5 ÷ 16/15 = 24/5 x 15/16 = (3 x 3)/(1 x 2) = 9/2 = 4/1/2

Fractions to decimals

A fraction can be changed to a decimal by carrying out the division.

eg. change 3/8 to a decimal

3/8 = 3 ÷ 8 = 0.375

eg. change 4/15 to a decimal.

4/15 = 4 ÷ 15 = 0.266666...

The decimal 0.266 666 6... carries on infinitely and is called a recurring decimal. You can write the recurring decimal 0.266 666 6... as 0.26.. The dot over the 6 means that the number carries on infinitely. If a group of numbers carries on infinitely, 2 dots can be used to show the repeating numbers.


Decimals to fractions

A decimal can be changed to a fraction by considering place value.

e.g. Change 0.58 to a fraction.

0.58 = 58/100 = 29/50 

Percentages

Percentages are fractions with a denominator of 100.

1% means 1 out of 100 or 1/100

25% means 25 out of 100 or 25/100 = 1/4 (in lowest term)

Percentages to fractions

To change a percentage to a fraction, divide by 100.

eg. Change 65% to a fraction

65% = 65/100 = 13/20

eg. Change 33/1/2% to a fraction.

33/1/2%  = (33/1/2)/100 = 67/100

Fractions to percentages

To change a fraction to a percentage, multiply by 100.

e.g. Change 1/4  to a percentage. 

1/4 = 1/4 x 100% = 25%

Percentages to decimals

To change a percentage to a decimal, divide by 100. 

e.g. Change 65% to a decimal. 

65% = 65 ÷ 100 = 0.65

Decimals to percentages

To change a decimal to a percentage, multiply by 100.

e.g. Change 0.005 to a percentage.

0.005 = 0.005 x 100% = 0.5%

To compare and order percentages, fractions and decimals, convert them all to percentages.

Percentage change

To work out the percentage change, work out the change and use the formula:

where change might be increase, decrease, profit, loss, error, etc. 

Percentage of an amount

To find the percentage of an amount, find 1% of the amount and then the required amount.

e.g. An investment of £72 increases by 12%. What is the new amount of the investment?

1% of £72 = £72/100

12% of £72 = 12 x £0.72 = £8.64

The new amount is £72 + £8.64 = £80.64

An alternative method uses the fact that after a 12% increase, the new amount is 100% of the original amount + 12% of the original amount or 112% of the original amount.

The new value of the investment is 112% of £72 1% of £72 = £0.72
112% of £72 = 112
¥ £0.72 = £80.64

Similarly, a decrease of 12% = 100% of the original amount – 12% of the original amount or 88% of the original amount. 

Reverse percentages

To find the original amount after a percentage change, use reverse percentages.

e.g. A television is advertised at £335.75 after a price reduction of 15%. What was the original price?

£335.75 represents 85% of the original price (100% – 15%)

So 85% of the original price = £335.75

1% of the original price = £335.75/85 = £3.95 85

100% of the original price = 100 x £3.95 = £395

The original price of the television was £395. 

e.g. A telephone bill costs £101.05 including VAT at 17.5%. What is the cost of the bill without the VAT?

£101.05 represents 117.5% of the bill (100% + 17.5%)

117.5% of the bill = £101.05 

1% of the bill = £101.05/117.5 = £0.86

100% of the bill =100 ¥ £0.86 = £86

The telephone bill was £86 without the VAT. 

You should check the answer by working the numbers back the other way. 

Ratio and proportion

A ratio allows one quantity to be compared to another quantity in a similar way to fractions.

e.g. In a box there are 12 lemons and 16 oranges. The ratio of lemons to oranges is 12 to 16, written as 12 : 16.

The order is important in ratios as the ratio of oranges to lemons is 16 to 12 or 16 : 12.

Equivalent ratios

Equivalent ratios are ratios which are equal to each other. The following ratios are all equivalent to 2 : 5.

2 : 5 = 4 : 10 = 6 : 15 = 8 : 20 =... 

Equal ratios can be found by multiplying or dividing both sides of the ratio by the same number.

e.g. Express the ratio 40p to £2 in its simplest form.

You must ensure that the units are the same. (Remember £2 = 200p.)
The ratio is 40 : 200 = 1 : 5 in its simplest form. (
Dividing both sides of the ratio by 40.)

e.g. Two lengths are in the ratio 4 : 5. If the first length is 60 cm, what is the second length?

The ratio is 4 : 5 = 4 cm : 5 cm = 1 cm : 5/4 cm (Writing as an equivalent ratio with 1 cm on the left-hand side.)

                                                 = 60 cm : 60 x 5/4 cm (Writing as an equivalent ratio with 60 cm on the left-hand side.)

                                                 = 60cm : 75cm 

So the second length is 75 cm. 

Proportional parts

To share an amount into proportional parts, add up the individual parts and divide the amount by this number to find the value of one part.

e.g. £50 is to be divided between two sisters in the ratio 3 : 2. How much does each get?

Number of parts = 3 + 2 = 5

Value of each part = £50 ÷ 5 = £10

The two sisters receive £30 (3 parts at £10 each) and £20 (2 parts at £10 each). 

Check that the amounts add up correctly (i.e. £30 + £20 = £50). 

Standard form

Standard form is a short way of writing very large and very small numbers. Standard form numbers are always written as:

A x 10where A lies between 1 and 10 and n is a natural number. 

Very large numbers

e.g. Write 267 000 000 in standard form.

Write down 267 000 000 then place the decimal point so A lies between 1 and 10. 

To find n, count the ‘power of 10’.
Here,
n = 8 so 267000000 = 2.67 x 10

Very small numbers

e.g. Write 0.000 000 231 in standard form.
Write down 0.000 000 321 then place the decimal point so
A lies between 1 and 10

To find n in 0.000 000 321, count the ‘power of 10’.
Here,
n = -7 so 0.000000321 = 3.21 x 10–7 

Adding and subtracting

To add (or subtract) numbers in standard form when the powers are the same you can proceed as follows.

eg. (4.8 x 1011) + (3.1 x 1011) = (4.8 + 3.1) x 1011 = 7.9 x 1011

eg. (4.63 x 10-2) - (2.7 x 10-2) = (4.63 -2.7) x 10-2 = 1.93 x 10-2

To add (or subtract) numbers in standard form when the powers are not the same, convert the numbers to ordinary form.

 e.g. (8.42 x 106) + (6 x 107)

= 8 420 000 + 60 000 000 = 68420000 = 6.842 x 10

Multiplying and dividing

To multiply (or divide) numbers in standard form, use the rules of indices. 

e.g. (7.5 x 104) x (3.9 x 107)
= (7.5
¥ 3.9) x (104 x 107) = 29.25 x 104 + 7  = 29.25 x 1011
= 2.925 x 1012 

e.g. (3 x 105) ÷ ( 3.75 x 108)
= (3 ÷ 3.75) x
 (105 ÷ 108= 0.8 x 105 – 8  = 0.8 x 10–3
= 8 x 10–4 

Compound measures

Compound measures involve more than one unit, such as speed (distance and time) or density (mass and volume).

Speed

Speed = distance/time

e.g. A taxi travels 16 miles in 20 minutes. What is the speed in miles per hour?

As the speed is measured in miles per hour, express the distance in miles and the time in hours.

Time = 20 minutes = 1/3 hour 

Speed = distance/time = 16/(1/3) = 48 mph

The formula for speed can be rearranged:

distance = speed x time or

time = distance/speed

e.g. A cyclist travels 3.6 km at an average speed of 8 kilometres per hour. How long does the journey take?

time = distance/speed = 3.6/8 = 0.45 hours

Remember that 0.45 hours is not 45 minutes as there are 60 minutes in one hour.

To convert hours to minutes, multiply by 60. 

0.45 hours = 0.45 x 60 minutes = 27 minutes 

The journey takes 27 minutes. 

Density

The formula for density:

density = mass/volume

e.g. A piece of lead weighing 170 g has a volume of 15 cm3. Give an estimate for the density of lead.

density = mass/volume = 170/15 = 11.3 g/cm3 (3 s.f.)

The formula for density can be rearranged:

mass = density volume or volume = mass/density

Simple and compound interest

With simple interest, the amount of interest paid is not reinvested.

With compound interest, the amount of interest paid is reinvested and earns interest itself. 

Simple interest formula: 

I = PRT/100

A = P(1 + R/100)T

where I = simple interest, A = total amount, P = principal or original investment,
R = rate (% per annum, or p.a.),
T = time (in years) 

e.g. £4000 is invested for 3 years at 4.5% p.a. Calculate the simple interest and the total amount.

Using the formula I = PRT/100

where P = principal = £4000 

R = rate = 4.5% or 4.5%

T = time = 3 years 

I = (4000 x 4.5 x 3)/100 = £540

A = P + I (Amount = principal + interest)

A = £4000 + £540 = £4540

The simple interest is £540 and the total amount is £4540. 

e.g. £2500 is invested at 6.5% p.a. compound interest. What is the amount after 2 years?

Compound interest can be found by repeatedly applying the simple interest formula. 

A = P + PRT/100

where P = principal = £2500 

R = rate = 6.5% 

T = time = 1 year for each year

Year 1: A = 2500 + (2500 x 6.5 x 1)/100 = 2500 + 162.5 = £2662.50

Year 2: A = 2662.5 + (2662.5 x 6.5 x 1)/100 = 2662.5 + 173.0625 = 2835.5625 = £2835.56

Alternatively, using the compound interest formula:

A = P(1 + R/100)T

A = 2500(1 + 6.5/100)2 = 2835.5625 =  £2835.56


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