GCSE Mathematics
GCSE Mathematics revision note sheet NUMBER
One Billion The billion now used in the UK is 1 000 000 000
Number 1 (Directed Numbers) Using a number line is one way of dealing with adding and subtracting negative numbers.
Eg 1 2 + +5 = 2 + 5 = 7 ------ +5 ----→ Like signs
2 - -5 = 2 + 5 = 7 0|_1|_2|_3|_4|_5|_6|_7| change to +
2 + -5 = 2 - 5 = -3 <-------- -5 --------- Unlike signs
2 - +5 = 2 - 5 = -3 -3|_-2|_-1|__0|__1|__2|__3|__4| change to –
When multiplying and dividing negative numbers the same rules about the signs apply
+
Eg 2 2 x +5 = 10 Like signs +
3 x -5 = -15 Unlike signs
-8 ÷ -2 = 4 answer is + -8 ÷ +4 = -2 answer is –
Number 2
Highest Common Factor (HCF) and Lowest Common Multiple (LCM)
Eg 3 What are the HCF and LCM of 8 and 28 ?
Factors of 8 are 1, 2, 4, 8 28 are 1, 2, 4, 7, 14, 28
Common factors are 1,2 and 4 The HCF is 4
Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112 28 are 28, 56, 84, 112
Common multiples are 56 and 112 The LCM is 56
Products of primes
A prime number has only 2 factors itself and 1. 1 is therefore NOT a prime number.
Some of the prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …
8 and 24 when written as products of their primes are
8 = 2 x 2 x 2 = 23 28 = 2 x 2 x 7 = 22 x 7 Note HCF 4 = 2 x 2 and LCM 56 = 2 x 2 x 2 x 7
Squares, cubes, square roots and cubed roots
Eg 4 What are the following: 32, 52, 33, 53, √9, √25, 3√27, 3√125 ?
3 squared = 32 = 3 x 3 = 9 52 = 5 x 5 = 25 3 cubed = 33 = 3 x 3 x 3 = 27 53 = 5 x 5 x 5 = 125
Square root of 9 = √9 = 3 √25 = 5 Cubed root of 27 = 3√27 = 3 3√125 = 5
Indices and standard form
Multiplication and division of indices
Ax x A y = Ax + y and Ax ÷ Ay = Ax – y
Eg 5 Find (2 x 103) x (3 x 104) and (9 x 107) ÷ (3 x 105) ?
2 x 103 x 3 x 104 = 2 x 3 x 103 x 104 = 6 x 107 9 x 107 ÷ 3 x 105 = 9 ÷ 3 x 107 ÷ 105 = 3 x 102
Standard form
All numbers can be written as M x 10E where M is the mantissa and 1 ≤ M < 10 and E is the exponent and is an integer.
Eg 6 Write the following in standard form 299 792 458 and 0.007 297 350 6 ?
299 792 458 = 2.997 924 58 x 100 000 000 = 2.997 924 58 x 108
0.007 297 350 6 = 7.297 350 = 7.297 350 6 x 10-3
Fractions
Common denominator Writing fractions with a common denominator allows the size to be compared. The common
denominator is the lowest common multiple of the denominators
Eg 7 Which fraction is large 2/3 or 3/5 ?
The easiest way to find a common denominator is to multiply them together 3 x 5 = 15
Numerators → 2 x 5 = 10 3 x 3 = 9 2/3 is larger than 3/5
Denominators → 3 5 15 5 3 15
© Timothy John Tyne Page 1 of 25
,GCSE Mathematics
Adding and subtracting fractions
First rewrite the fractions with a common denominator then add (or subtract) the numerators.
Eg 8 Subtract 2/3 from 3/5 ?
Numerators → 2 - 3 = 10 - 9 = 1
Denominators → 3 5 15 15 15
- = - =
Multiplication and division of fractions
To multiply fractions the numerators and denominators are multiplied separately.
To divide one fraction by another the divisor is inverted and the resulting fractions multiplied together.
Eg 9 Multiply 2/3 and 5/7 ?
Numerators → 2 x 5 = 2 x 5 = 10
Denominators → 3 7 3 x 7 21
Eg 10 Divide 1/4 by 2/3 ?
Numerators → 1 ÷ 2 = 1 x 3 = 1x3 = 3
Denominators → 4 3 4 2 4x2 8
Decimals
Writing decimals
3.625 is 3 + (6/10) +(2/100) + (5/1000)
Approximating decimals
102.938 475 6 approximated to
1 significant figure is 100 1 decimal place is 102.9
2 significant figures is 100 2 decimal places is 102.94
3 significant figures is 102 3 decimal places is 102.938
4 significant figures is 102.9 4 decimal places is 102.938 5
5 significant figures is 102.94 5 decimal places is 102.938 48
6 significant figures is 102.938 6 decimal places is 102.938 476
7 significant figures is 102.938 5 7 decimal places is 102.938 475 6
8 significant figures is 102.938 48 8 decimal places is 102.938 475 60
9 significant figures is 102.938 476 9 decimal places is 102.939 475 600
10 significant figures is 102.938 475 6 Note the n + 1th significant figure or decimal place is used to either
round up or down the figure.
Add and subtract decimals The decimal points should be lined up and a decimal point put directly below in the answer.
Eg 11 i) Add 31.354 and 1.017 ? 31.354 ii) subtract 2.718 from 3.142 3. 11 4 12
1.017 + 12. 7 11 8 –
32.371 0. 4 2 4
Multiplying decimals Count the number of digits after the decimal points and add them together. Do the multiplication
ignoring the decimal points. Put a decimal point in the answer so that the number of digits after the decimal point is equal to
the total number of digits added up previously.
Eg 12 Find 6.4 x 1.25 ? 1+2 = 3 digits after the 125 therefore 6.4 x 1.25 = 8.000
decimal points 64 x
500
7500
8000
© Timothy John Tyne Page 2 of 25
, GCSE Mathematics
Dividing decimals
Multiply the dividend and divisor by an integral power of ten so that the divisor is a whole number. Then carry out the
division.
Eg 13 Find 8.544 ÷ 1.2 ? 7.12
8.544 x 10 = 85.44 1.2 x 10 = 12 12)85.44
84 12 x 7 = 84
14
12 12 x 1 = 12
24
24 12 x 2 = 24
00
Percentages
Percentage of a number
Common 5% = 1/20; 10% = 1/10; 20% = 1/5; 25% = 1/4; 33⅓% = 1/3;
percentages 50% = 1/2; 66⅔% = 2/3; 75% = 3/4; 100% = 1
To find the percentage of a number multiply the number by the percentage then divide by 100 (by moving the decimal point
two places to the left).
Eg 14 Find 15% of 40 ? 15 x 40 = 600; 600 ÷ 100 = 6
Increase/decrease by a percentage
Find the percentage as shown above and add it for an increase or subtract it for a decrease.
Eg 15 An item costs £60.00 before VAT of 17.5%. What is the cost including VAT ?
60.00 x 17.5 = 1050.00; 1050.00 ÷ 100 = 10.50; 10.50 + 60 = £70.50
One number as a percentage of another
Find what the number is as a fraction of the other number then multiply by 100
Eg 16 What is 5 as a percentage of 25 ? 5/25 = 0.20; 0.20 x 100 = 20%
Equivalent fractions, decimals and percentages
Converting from:-
Fractions to decimals Decimals to fractions
Divide the numerator by the denominator Multiply the decimal by a (integral) power of ten to get a
whole number then divide the whole number by the same
power of ten and simplify.
Eg 17 Eg 18
5/8 0.625 0.625 x 1000 = 625; 625 = 25 = 5
8)5.000 1000 40 8
48 8 x 6 = 48
20
16 8 x 2 = 16
40
40 8 x 5 = 40
00
Decimals to percentages Percentages to decimals
Multiply by 100 Divide by 100
Eg 19 Eg 20
0.175 0.175 x 100 = 17.5% 17.5% 17.5 ÷ 100 = 0.175
© Timothy John Tyne Page 3 of 25
GCSE Mathematics revision note sheet NUMBER
One Billion The billion now used in the UK is 1 000 000 000
Number 1 (Directed Numbers) Using a number line is one way of dealing with adding and subtracting negative numbers.
Eg 1 2 + +5 = 2 + 5 = 7 ------ +5 ----→ Like signs
2 - -5 = 2 + 5 = 7 0|_1|_2|_3|_4|_5|_6|_7| change to +
2 + -5 = 2 - 5 = -3 <-------- -5 --------- Unlike signs
2 - +5 = 2 - 5 = -3 -3|_-2|_-1|__0|__1|__2|__3|__4| change to –
When multiplying and dividing negative numbers the same rules about the signs apply
+
Eg 2 2 x +5 = 10 Like signs +
3 x -5 = -15 Unlike signs
-8 ÷ -2 = 4 answer is + -8 ÷ +4 = -2 answer is –
Number 2
Highest Common Factor (HCF) and Lowest Common Multiple (LCM)
Eg 3 What are the HCF and LCM of 8 and 28 ?
Factors of 8 are 1, 2, 4, 8 28 are 1, 2, 4, 7, 14, 28
Common factors are 1,2 and 4 The HCF is 4
Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112 28 are 28, 56, 84, 112
Common multiples are 56 and 112 The LCM is 56
Products of primes
A prime number has only 2 factors itself and 1. 1 is therefore NOT a prime number.
Some of the prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …
8 and 24 when written as products of their primes are
8 = 2 x 2 x 2 = 23 28 = 2 x 2 x 7 = 22 x 7 Note HCF 4 = 2 x 2 and LCM 56 = 2 x 2 x 2 x 7
Squares, cubes, square roots and cubed roots
Eg 4 What are the following: 32, 52, 33, 53, √9, √25, 3√27, 3√125 ?
3 squared = 32 = 3 x 3 = 9 52 = 5 x 5 = 25 3 cubed = 33 = 3 x 3 x 3 = 27 53 = 5 x 5 x 5 = 125
Square root of 9 = √9 = 3 √25 = 5 Cubed root of 27 = 3√27 = 3 3√125 = 5
Indices and standard form
Multiplication and division of indices
Ax x A y = Ax + y and Ax ÷ Ay = Ax – y
Eg 5 Find (2 x 103) x (3 x 104) and (9 x 107) ÷ (3 x 105) ?
2 x 103 x 3 x 104 = 2 x 3 x 103 x 104 = 6 x 107 9 x 107 ÷ 3 x 105 = 9 ÷ 3 x 107 ÷ 105 = 3 x 102
Standard form
All numbers can be written as M x 10E where M is the mantissa and 1 ≤ M < 10 and E is the exponent and is an integer.
Eg 6 Write the following in standard form 299 792 458 and 0.007 297 350 6 ?
299 792 458 = 2.997 924 58 x 100 000 000 = 2.997 924 58 x 108
0.007 297 350 6 = 7.297 350 = 7.297 350 6 x 10-3
Fractions
Common denominator Writing fractions with a common denominator allows the size to be compared. The common
denominator is the lowest common multiple of the denominators
Eg 7 Which fraction is large 2/3 or 3/5 ?
The easiest way to find a common denominator is to multiply them together 3 x 5 = 15
Numerators → 2 x 5 = 10 3 x 3 = 9 2/3 is larger than 3/5
Denominators → 3 5 15 5 3 15
© Timothy John Tyne Page 1 of 25
,GCSE Mathematics
Adding and subtracting fractions
First rewrite the fractions with a common denominator then add (or subtract) the numerators.
Eg 8 Subtract 2/3 from 3/5 ?
Numerators → 2 - 3 = 10 - 9 = 1
Denominators → 3 5 15 15 15
- = - =
Multiplication and division of fractions
To multiply fractions the numerators and denominators are multiplied separately.
To divide one fraction by another the divisor is inverted and the resulting fractions multiplied together.
Eg 9 Multiply 2/3 and 5/7 ?
Numerators → 2 x 5 = 2 x 5 = 10
Denominators → 3 7 3 x 7 21
Eg 10 Divide 1/4 by 2/3 ?
Numerators → 1 ÷ 2 = 1 x 3 = 1x3 = 3
Denominators → 4 3 4 2 4x2 8
Decimals
Writing decimals
3.625 is 3 + (6/10) +(2/100) + (5/1000)
Approximating decimals
102.938 475 6 approximated to
1 significant figure is 100 1 decimal place is 102.9
2 significant figures is 100 2 decimal places is 102.94
3 significant figures is 102 3 decimal places is 102.938
4 significant figures is 102.9 4 decimal places is 102.938 5
5 significant figures is 102.94 5 decimal places is 102.938 48
6 significant figures is 102.938 6 decimal places is 102.938 476
7 significant figures is 102.938 5 7 decimal places is 102.938 475 6
8 significant figures is 102.938 48 8 decimal places is 102.938 475 60
9 significant figures is 102.938 476 9 decimal places is 102.939 475 600
10 significant figures is 102.938 475 6 Note the n + 1th significant figure or decimal place is used to either
round up or down the figure.
Add and subtract decimals The decimal points should be lined up and a decimal point put directly below in the answer.
Eg 11 i) Add 31.354 and 1.017 ? 31.354 ii) subtract 2.718 from 3.142 3. 11 4 12
1.017 + 12. 7 11 8 –
32.371 0. 4 2 4
Multiplying decimals Count the number of digits after the decimal points and add them together. Do the multiplication
ignoring the decimal points. Put a decimal point in the answer so that the number of digits after the decimal point is equal to
the total number of digits added up previously.
Eg 12 Find 6.4 x 1.25 ? 1+2 = 3 digits after the 125 therefore 6.4 x 1.25 = 8.000
decimal points 64 x
500
7500
8000
© Timothy John Tyne Page 2 of 25
, GCSE Mathematics
Dividing decimals
Multiply the dividend and divisor by an integral power of ten so that the divisor is a whole number. Then carry out the
division.
Eg 13 Find 8.544 ÷ 1.2 ? 7.12
8.544 x 10 = 85.44 1.2 x 10 = 12 12)85.44
84 12 x 7 = 84
14
12 12 x 1 = 12
24
24 12 x 2 = 24
00
Percentages
Percentage of a number
Common 5% = 1/20; 10% = 1/10; 20% = 1/5; 25% = 1/4; 33⅓% = 1/3;
percentages 50% = 1/2; 66⅔% = 2/3; 75% = 3/4; 100% = 1
To find the percentage of a number multiply the number by the percentage then divide by 100 (by moving the decimal point
two places to the left).
Eg 14 Find 15% of 40 ? 15 x 40 = 600; 600 ÷ 100 = 6
Increase/decrease by a percentage
Find the percentage as shown above and add it for an increase or subtract it for a decrease.
Eg 15 An item costs £60.00 before VAT of 17.5%. What is the cost including VAT ?
60.00 x 17.5 = 1050.00; 1050.00 ÷ 100 = 10.50; 10.50 + 60 = £70.50
One number as a percentage of another
Find what the number is as a fraction of the other number then multiply by 100
Eg 16 What is 5 as a percentage of 25 ? 5/25 = 0.20; 0.20 x 100 = 20%
Equivalent fractions, decimals and percentages
Converting from:-
Fractions to decimals Decimals to fractions
Divide the numerator by the denominator Multiply the decimal by a (integral) power of ten to get a
whole number then divide the whole number by the same
power of ten and simplify.
Eg 17 Eg 18
5/8 0.625 0.625 x 1000 = 625; 625 = 25 = 5
8)5.000 1000 40 8
48 8 x 6 = 48
20
16 8 x 2 = 16
40
40 8 x 5 = 40
00
Decimals to percentages Percentages to decimals
Multiply by 100 Divide by 100
Eg 19 Eg 20
0.175 0.175 x 100 = 17.5% 17.5% 17.5 ÷ 100 = 0.175
© Timothy John Tyne Page 3 of 25
