GCSE Revision Notes Mathematics Number s and Algebra

GCSE Revision Notes Mathematics Number s and Algebra

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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.

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 8 is greater than 5. Round up to 7.0 9.2 2 is less than 5. Round down to 9.0 5.5 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 1.1 The table shows the cost of a short break at a holiday park. Holiday starts in Adult 1 st and 2 nd Child 3 rd and 4 th Child June 199 each 39 each Free July 299 each 49 each 19 each August 349 each 59 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. 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. 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 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 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 goes to goes to 2 = = 5 1.5 Use approximations to estimate the value of goes to 0.496 goes to 0.5 = = 200

2.Square, Cube and prime numbers Square numbers are numbers multiplied themselves = 2 2 = 4 = 3 3 = 9 = 4 4 = 16 = 5 5 = 25 = 6 6 = 36 = 7 7 = 49 Cube numbers are numbers multiplied by themselves and then multiplied by themselves again = 2 2 2 = 8 = 3 3 3 = 27 = 4 4 4 = 64 = 5 5 5 = 125 = 6 6 6 = 216 = 7 7 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

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. 36 = (ii)work out the Highest Common Factor (HCF) of 36 and 81. 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 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 33 = 11 3 34 = 2 17 35 = 7 5 2.4 and are different prime numbers less than 12. Work out three pairs of numbers and such that is a whole number. For to be a whole number 2 must equal a square number 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

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

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? 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 is more than

3.3 (i) After how many minutes does Pat overtake Vicki? Look at the point where the two lines intersect 49 mins (ii) How far ahead is Vicki when Pat starts again after her rest? Take the point when Pat starts again after her rest = 35mins Pat = 5km Vicki = 6.6km 6.6-5 = 1.6km ahead