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GCSE
Mathematics –
Revision Notes
Topic ‘Number’s and
Algebra’
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1
AQA Unit 2 – Number and Algebra
Contents
1. Rounding and Approximating ............................................................................................................. 3
2.Square, Cube and prime numbers ....................................................................................................... 6
3. Distance-time graphs .......................................................................................................................... 9
4. Fractions ............................................................................................... Error! Bookmark not defined.
8. Best offer questions ............................................................................. Error! Bookmark not defined.
9. Number Machine ................................................................................. Error! Bookmark not defined.
10 Quadratic Equations............................................................................ Error! Bookmark not defined.
11. Substitution ........................................................................................ Error! Bookmark not defined.
12. Rearrange ........................................................................................... Error! Bookmark not defined.
14. Simultaneous equations .................................................................... Error! Bookmark not defined.
15 Inequalities .......................................................................................... Error! Bookmark not defined.
15 Graphing Inequalities .......................................................................... Error! Bookmark not defined.
17. Linear Graphs and equations ............................................................. Error! Bookmark not defined.
2
1. Rounding and Approximating
Firstly it’s important to remember in what order we multiply, add, subtract, etc…
The order is
1. Brackets
2. Exponents (Roots and Powers)
3. Multiply or Divide
4. Add or subtract
Some people like to remember the word BEMDAS to help remember the order
(Brackets Exponents Multiply Divide Add Subtract) .
When rounding to the nearest whole number, we pay attention to the first decimal place. If it is 5 or
greater we round it up to the nearest whole number. If it is less than 5 we round it down to the
nearest whole number.
Look at the first decimal place:
6.8
9.2
5.5
8 is greater than 5. Round up to 7.0
2 is less than 5. Round down to 9.0
If 5 is in the first decimal place approximate up. 5.5 approximates up to 6.0
When rounding correct to one decimal place. We pay attention to the second decimal place.
7.72 2 is less than 5. Round down to 7.70
16.89 9 is greater than 5. Round up to 16.90
4.55 5 is in the second decimal place so we round up. Round up to 4.60
Rounding off to 3 or 4 decimal places is done the exact same way.
When doing approximations we round each number in the calculation to one significant figure and
then work out the calculation.
47.5
11.6 = 50
10 = 500
367.7 + 63 = 400 + 60 = 460
245 ÷ 4.7 = 200 ÷ 5 = 40
667 - 434.65 = 700 - 400 = 300
Questions
3
1.1 The table shows the cost of a short break at a holiday park.
Holiday starts in
June
July
August
Adult
£199 each
£299 each
£349 each
1st and 2nd Child
£39 each
£49 each
£59 each
3rd and 4th Child
Free
£19 each
£39 each
(i) Mr and Mrs Hyde and their three children want a short break starting on 28 July
Use approximations to estimate the cost of this short break.
Solution
Approximate 299 to 300
Approximate 49 to 50
Approximate 19 to 20
300 + 300 + 50 + 50 + 20 = 720
.
(ii) Work out exactly how much more it would cost if they went in August instead of July.
Solution
August 349 + 349 + 59 + 59 + 39 = 855
July = 299 + 299 + 49 + 49 + 19 = 715
855 - 715 = 140
1.2 Use approximations to estimate the value of
Solution
20.02 goes to 20
0.49 goes to 0.5
1.99 goes to 2
=5
1.3 Use approximations to estimate the value of
Solution
4
795.4 goes to 800
2.1 goes to 2
9.8 goes to 10
=
= 20
1.4 Use approximations to estimate the value of
Solution
√
√
goes to √
goes to 2
=
=5
1.5 Use approximations to estimate the value of
Solution
goes to
0.496 goes to 0.5
=
5
= 200
√
2.Square, Cube and prime numbers
Square numbers are numbers multiplied themselves.
=2
=3
=4
=5
=6
=7
2=4
3=9
4 = 16
5 = 25
6 = 36
7 = 49
Cube numbers are numbers multiplied by themselves and then multiplied by themselves
again.
=2
=3
=4
=5
=6
=7
2
3
4
5
6
7
2=8
3 = 27
4 = 64
5 = 125
6 = 216
7 = 343
A prime number is a number that is only divisible by itself and 1.
7÷1=7
7÷1=1
No other whole numbers divide into 7
Every number can be broken down into prime numbers as every number can be broken
down until it can’t be divided into anymore.
12 =
63 =
100 =
440 =
When splitting a number into its prime factors we always begin with the smallest prime
factor and work our way up.
When calculating prime numbers, we work only in integers we do not work in fractions.
The highest common factor (HCF) of two numbers is the biggest number that divides
exactly into the two numbers.
6
The lowest common multiple (LCM)of two numbers is the smallest number in the times
tables of both numbers.
Questions
2.1 (i) Write 36 as the product of prime factors.
Give your answer in index form.
Solution
36 =
(ii)Work out the Highest Common Factor (HCF) of 36 and 81.
Solution
36 ÷ 2 = 18
36 ÷ 3 = 12
36 ÷ 4 = 9
9 also divides into 81
9 is the highest common factor
2.2 Write 44100 as a product of its prime factors.
Solution
2.3 The number 39 can be written as the product of two prime numbers.
39 = 3 × 13
Write three other numbers between 30 and 40 as the product of two prime numbers.
Solution
33 = 11 × 3
34 = 2 × 17
35 = 7 × 5
7
2.4 and are different prime numbers less than 12.
Work out three pairs of numbers and such that √
is a whole number.
Solution
For √
to be a whole number 2
2
=4
2
=9
=2
=5
2
= 16
=7
=2
2
= 25
= 11
=3
2.5 x, y and x - y are all two-digit numbers.
x is a square number
y is a cube number
x - y is a prime number
Find one possible pair of values for x and y
x = 16, 25, 36, 49, 64, 81
y = 27, 64
Test each x,y in the sum x - y
Testing 64 - 27 = 37
37 is a prime number
x = 64 y = 27
8
must equal a square number
3. Distance-time graphs
3.1 Josh drove to a meeting and then back home.
The meeting was 80 miles from his home.
Josh left home at 9 am
He arrived at the meeting after 2 hours
He left for home 4
hours later
He drove 30 miles in half an hour
He then stopped for 1 hour
He arrived home 1
hours later.
Show this information on a distance-time graph
9
3.2 The distance-time graph represents a journey Alf makes.
Alf claims that he stopped for less than one-quarter of his total journey time.
Is he correct?
Solution
10am to 3pm
Total journey time = 5 hours
Each box across represents 15mins
He stops from 11.15 to 12.15 = 1 hour
He stops from 1.30 to 2.00 = 30 mins
Total stopping time = 1.5 hours
=
Alf is wrong as
10
is more than
3.3
(i) After how many minutes does Pat overtake Vicki?
Solution
Look at the point where the two lines intersect
49 mins
(ii) How far ahead is Vicki when Pat starts again after her rest?
Solution
Take the point when Pat starts again after her rest = 35mins
Pat = 5km
Vicki = 6.6km
6.6 - 5 = 1.6km ahead
11
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