All the basics you need to know for the Level 3/4 course....... Level 3/4 Mathematics All the basics you need to know for the Level 3/4 course.......

I can .... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 Created by Mr. Lafferty

multiply small decimal numbers together ? 0.04 x 0.7 Question 1 multiply small decimal numbers together ? 0.04 x 0.7 answer

0.028 multiply numbers together then count number of decimal places. Answer to Question 1 multiply numbers together then count number of decimal places. 0.028

divide small decimal numbers Question 2 divide small decimal numbers 15.3 ÷ 0.03 answer

Answer to Question 2 Do not divide by a decimal scale it up to a whole number Scale the number to be divided by the same amount 1530 ÷ 3 = 510

tell you the meaning of the words Question 3 tell you the meaning of the words Profit Loss Salary Overtime answer

Answer to Question 3 Profit: When you sell something for MORE than you bought it. Loss : When you sell something for LESS than you bought it. Salary : How much a person is paid to do a job. Can be measured weekly, monthly or annually. Overtime : Extra work outside normal work time. Usually at night or weekends.

tell you the meaning of the words Question 4 tell you the meaning of the words Hire Purchase Foreign Exchange answer

Answer to Question 4 Hire Purchase : When you pay for goods by putting down a deposit and then paying the rest up in small fixed payments over a period of time. Foreign Exchange : Converting one currency to another.

explain the meaning of Significant Figures Question 5 explain the meaning of Significant Figures answer

Answer to Question 5 A measure of both: Quantity Accuracy

explain the term double negative. Question 6 explain the term double negative. (-4) – (-6) answer

Double negative changes to a positive Answer to Question 6 Double negative changes to a positive (-4) – (-6) = (-4) + 6 = 2

tell you the instrument that helps us to add and subtract integers. Question 7 tell you the instrument that helps us to add and subtract integers. answer

Answer to Question 7 thermometer

explain the rules of multiplication and division of integers. Question 8 explain the rules of multiplication and division of integers. answer

Different signs = negative answer Same signs = positive answer Answer to Question 8 Different signs = negative answer Same signs = positive answer

explain BODMAS 3 – ( 2 x (-4))2 Question 9 explain BODMAS 3 – ( 2 x (-4))2 answer

B - Brackets O – Other M - Multiplication A - Addition S - Subtraction Answer to Question 9 3 – ( 2 x (-4))2 B - Brackets O – Other M - Multiplication A - Addition S - Subtraction 3 – (-8)2 3 – 64 – 61

explain what an integer is. Question 10 explain what an integer is. answer

A positive or negative number including zero. Answer to Question 10 A positive or negative number including zero.

scientific notation / standard form Question 11 explain the term scientific notation / standard form and convert to it. 5680000 0.00623 answer

(a) 5.68 x 106 (b) 6.23 x 10-3 Answer to Question 11 a x 10n a between 1 and 10 n is an integer (a) 5.68 x 106 (b) 6.23 x 10-3

write numbers in normal form Question 12 write numbers in normal form (a) 5.93 x 104 (b) 7.43 x 10-5 answer

Answer to Question 12 Normal Form (a) 59300 (b) 0.0000743

Question 13 write these numbers to 3 significant figures 92418400 0.04057090100 answer

Answer to Question 13 3 significant figures 92400000 0.0406

round decimals to any significant figures Question 14 round decimals to any significant figures (a) 93.0108 ( to 2 s.f.) (b) 0.0706194 ( to 4 s.f.) answer

Answer to Question 14 (a) 93 ( to 2 s.f.) (b) 0.07062 ( to 4 s.f.)

Question 15 write numbers to any decimal place (a) 29.0864 ( to 2 dp) (b) 0.0109964 ( to 4 dp) answer

Higher Mathematics: ©HappySmile Productions Answer to Question 15 (a) 29.09 ( to 2 dp) (b) 0.0110 ( to 4 dp) Higher Mathematics: ©HappySmile Productions

Question 16 explain how to get triangular numbers and write down the first 6 triangular numbers. answer

Answer to Question 16 1, 3, 6, 10, 15, 21

explain how to get square numbers and write down the first 10 Question 17 explain how to get square numbers and write down the first 10 answer

Answer to Question 17 1, 4, 9, 16, 25, 36 49, 64, 81, 100

decompose a number into product of primes e.g. 18 Question 18 decompose a number into product of primes e.g. 18 answer

Answer to Question 18 18 3 6 2 = 3 x 6 = 3 x 3 x 2 All Prime !

Question 19 explain what the LCM is between a set of numbers e.g. LCM for 3, 4 and 5 answer

Answer to Question 19 the lowest number that 3, 4, 5 all divide into without leaving a remainder is 60

explain what the HCF is between a set of numbers Question 20 explain what the HCF is between a set of numbers e.g. HCF for 25, 50 and 100 answer

Answer to Question 20 the highest number that divides into 25, 50, 100 without leaving a remainder is 25

know the basic angle properties at Level E Question 21 know the basic angle properties at Level E answer

Answer to Question 21 115o 95o 120o Two angles making a Angles round a point Add up to 360o 115o Two angles making a straight line add to 180o angles opposite each other at a cross are equal. 34o 3 angles in a triangle ALWAYS add up to 180o. 50o 40o 65o 90o 146o 145o

explain the term alternate angles Question 22 explain the term alternate angles answer

two angles that form a Z - shape Answer to Question 22 two angles that form a Z - shape = equal angles

explain the term corresponding angles Question 23 explain the term corresponding angles answer

two angles that form a F - shape Answer to Question 23 two angles that form a F - shape = equal angles

explain the term interior angle of a shape Question 24 explain the term interior angle of a shape answer

Answer to Question 24 Pentagon 72o Hexagon 60o Octagon 45o Interior Angles Interior Angle

explain the term exterior angle of a shape Question 25 explain the term exterior angle of a shape answer Higher Mathematics: ©HappySmile Productions

Answer to Question 25 This is called the “Exterior angle” Pentagon B A

find the exterior angle of a shape Question 26 find the exterior angle of a shape Pentagon Find Exterior angle B A E O Q answer

Exterior angle = 180o – interior angle Answer to Question 26 Exterior angle = 180o – interior angle Pentagon This is called the “Exterior angle” B A E C D O Q Exterior angle = 180o – 108o = 72o

find the interior angle of a shape Question 27 find the interior angle of a shape Octagon 45o Find the interior angles answer

Interior angle = (180o – 45o) = 135o Answer to Question 27 Octagon 45o Interior angle = (180o – 45o) = 135o

use angle properties to calculate missing angles Question 28 use angle properties to calculate missing angles Find all missing angles ao , bo and co d = 115o co ao bo answer

Answer to Question 28 ao = co = 115o bo = 65o

remove a single bracket (a) -2 ( 8y – 3) (b) 4b ( – 5b + a) Question 29 remove a single bracket (a) -2 ( 8y – 3) (b) 4b ( – 5b + a) answer

Answer to Question 29 (a) -16y + 6 (b) -20b2 + 4ab

Question 30 remove a single bracket and simplify (a) 2 (x + 5) - 7 (b) 3(a + 4) + 2(a – 1) answer

(a) 2x + 10 -7 = 2x + 3 (b) 3a + 12 + 2a – 2 = 5a + 10 Answer to Question 30 (a) 2x + 10 -7 = 2x + 3 (b) 3a + 12 + 2a – 2 = 5a + 10

evaluate algebraic expressions Question 31 evaluate algebraic expressions a = 3 ; b = 4 and c = -1 answer

Answer to Question 31

factorise algebraic expressions Question 32 factorise algebraic expressions (a) 4y + 80 (b) 9x – 6x2 answer

(a) 4y + 80 = 4(y + 20) (b) 9x – 6x2 = 3x(3 – 2x) Answer to Question 32 (a) 4y + 80 = 4(y + 20) (b) 9x – 6x2 = 3x(3 – 2x)

Question 33 simplify algebraic expressions (a) -6x + 7y – 16x + 11y (b) 2b – 3b2 – 5b + 5b2 answer

Answer to Question 33 (a) -22x + 18y (b) -3b + 2b2

find a percentage of quantity (with a calculator) Question 34 find a percentage of quantity (with a calculator) What is 17.5 % of £450 answer

Answer to Question 34 = £76.50 17.5 % of £450

explain the term appreciation with reference to percentages Question 35 explain the term appreciation with reference to percentages The average house prices in Glasgow have appreciated by 50% over the past 10 years. If you bought the house for £80 000 ten years ago. How much would the house be worth now? answer

when an item has increased in value by a certain percentage. Answer to Question 35 when an item has increased in value by a certain percentage. Appreciation = 50% x £ 80 000 = 0.50 x £80 000 = £ 40 000 New value = Old Value + Appreciation = £ 80 000 + £40 000 = £ 120 000

explain the term depreciation with reference to percentages Question 36 explain the term depreciation with reference to percentages A Mini Cooper cost £15 000 in 2005. At the end 2006, it will have depreciated by 20% What will the mini cooper worth at end 2006? answer

when an item has decreased in value by a certain percentage. Answer to Question 36 when an item has decreased in value by a certain percentage. End 2006 Depreciation = 20% x £15 000 = 0.2 x £15 000 = £3 000 New value = Old value - Depreciation = £15 000 - £3000 = £12 000

express a value as a percentage of another value. Question 37 express a value as a percentage of another value. Frances scored 13 out of 20 in a Maths test. What was her percentage score? answer

Answer to Question 37 scored 13 out of 20

work backwards to find initial value. Tricky Question 38 work backwards to find initial value. After a 10% increase the price of a house is £88 000. What was the price before the increase. answer

110 % = £88 000 We have : 1 % : Price before is 100% : Answer to Question 38 100 % + 10 % = £88 000 Deduce from question : 110 % = £88 000 We have : 1 % : Price before is 100% : £800 x 100 = £80 000

work backwards to find initial value. Tricky Question 39 work backwards to find initial value. The value of a car depreciated by 15%. It is now valued at £2550. What was it’s original price. answer

Answer to Question 39 85 % = £2 550 We have : 1 % : 100 % - 15 % = £2 550 Deduce from question : 85 % = £2 550 We have : 1 % : Price before is 100% : £30 x 100 = £3 000

find a percentage of quantity (with out a calculator) Question 40 find a percentage of quantity (with out a calculator) What is 17.5 % of £600 answer

10%  600 ÷ 10 = £ 60 5%  half of 10% = £ 30 2.5%  half of 5% = £ 15 Answer to Question 40 10%  600 ÷ 10 = £ 60 5%  half of 10% = £ 30 2.5%  half of 5% = £ 15 17.5% £105

identify the main parts of a circle. Question 41 identify the main parts of a circle. Name the main parts of the circle O answer

Answer to Question 41 O radius Circumference Diameter

Question 42 explain the connections between the radius, diameter and the circumference of a circle. O answer

Answer to Question 42 radius O Circumference Diameter

Question 43 explain the difference between the circumference and the area of a circle. answer

Circumference is the name given to the perimeter of the circle Answer to Question 43 Circumference is the name given to the perimeter of the circle (the length round the outside of the circle) The area is the inside of the circle

calculate the perimeter and area for a circle. Question 44 calculate the perimeter and area for a circle. Find the area and perimeter of the circle ? 4cm answer

Answer to Question 44 4cm

explain how to calculate the area of a composite shape. Question 45 explain how to calculate the area of a composite shape. 20cm 5 cm answer

Answer to Question 45 Area = rectangle + semicircle 20cm 5 cm

calculate the area of any triangle. Find the area below. Question 46 calculate the area of any triangle. Find the area below. 10cm 4cm answer

Altitude h outside triangle this time. Answer to Question 46 Altitude h outside triangle this time. 10cm 4cm

calculate the area of a rhombus. Find the area below. Question 47 calculate the area of a rhombus. Find the area below. 5cm 2cm answer

Answer to Question 47 5cm 2cm

calculate the area of a kite and inverted kite. Find the area below. Question 48 calculate the area of a kite and inverted kite. Find the area below. 9cm 4cm 7cm answer

Answer to Question 48 9cm 4cm 7cm

calculate the area of a parallelogram. Find the area below. Question 49 calculate the area of a parallelogram. Find the area below. 9cm 3cm answer

Answer to Question 49 9cm 3cm

calculate the area of a trapezium. Find the area below. Question 50 calculate the area of a trapezium. Find the area below. 5cm 6cm 4cm answer

Answer to Question 50 5cm 6cm 4cm

Question 51 calculate the area of composite shapes. Find the area shaded grey below. 10cm 11cm 8cm 4cm answer

Answer to Question 51 10cm 11cm 8cm 4cm

write down the area formula for Question 52 write down the area formula for Any triangle Rhombus Parallelogram Kite Trapezium answer

The area formula for Any triangle Rhombus Parallelogram Kite Trapezium Answer to Question 52 The area formula for Any triangle Rhombus Parallelogram Kite Trapezium

write down in words Pythagoras Theorem for right-angled triangles. Question 53 write down in words Pythagoras Theorem for right-angled triangles. answer

c b a Answer to Question 53 Two shorter sides squared and added together equal to the longest side c b a

calculate the longest side given the two shorter sides. Question 54 calculate the longest side given the two shorter sides. 8 12 c answer

Answer to Question 54 8 12 c

Question 55 calculate a shorter side given the longest side and the other shorter side. 20cm 12cm a cm answer

Check answer ! Always smaller than hypotenuse Answer to Question 55 20cm 12cm a cm Check answer ! Always smaller than hypotenuse

write down the three different versions of Pythagoras Theorem Question 56 write down the three different versions of Pythagoras Theorem c b answer a

c b a Answer to Question 56 Finding hypotenuse c Finding Finding shorter side a Finding shorter side b a c b

write down the rule for a linear pattern. Question 57 write down the rule for a linear pattern. answer

Answer to Question 57 b = 3s + 8

write down the steps for finding the rule for a linear pattern. Question 58 write down the steps for finding the rule for a linear pattern. answer

2. Write down part of formula Answer to Question 58 1. Find the difference 2. Write down part of formula 3. Find the correction factor. 4. Write down the full formula

convert hour and minutes to decimal hours. Question 59 convert hour and minutes to decimal hours. 48 minutes to decimal is 2hr 15 minutes to decimal is answer

= 0.8 hr = 2.25 hr Answer to Question 59 To change minutes to a decimal divide by 60 = 0.8 hr = 2.25 hr

convert decimal hours to hours and minutes. Question 60 convert decimal hours to hours and minutes. 0.7 hrs to minutes is 3.4 hrs to hours and minutes is answer

3.4 hrs to hours and minutes is Answer to Question 60 To change decimal time to minutes ‘multiply by 60’ 0.7 hrs to minutes is = 42 mins 3.4 hrs to hours and minutes is = 3 hrs 24 mins

calculate the speed of an object. Question 61 calculate the speed of an object. Daniel drove from his house to the Blackpool, a distance of 135 miles. It took him 2hrs 15mins. What was his average speed? answer

Answer to Question 61

calculate the time taken for an object. Question 62 calculate the time taken for an object. How long did the bus journey take if it travelled a total distance of 60 km at an average speed of 40 km/hr. answer

Answer to Question 62

calculate the distance travelled for an object. Question 63 calculate the distance travelled for an object. A racing car travelled at 50 km/hr. What is the distance covered in 6 hours ? answer

Answer to Question 63

write down the three formulae linking distance, speed and time. Question 64 write down the three formulae linking distance, speed and time. answer

D S T D = S T Answer to Question 64 Simple way to remember the 3 formulae !

Question 65 simplify ratios answer

Answer to Question 65

express ratios in unity form Question 66 express ratios in unity form answer

Answer to Question 66

solve questions involving ratio The ratio of boys to girls is 4:5. If there are 16 boys, how many girls are there. answer

boys girls 4 5 x 4 x 4 16 20 Answer to Question 67 The ratio of boys to girls is 4:5. If there are 16 boys, how many girls are there. boys girls 4 5 x 4 x 4 16 20

work out portions using ratios. Question 68 work out portions using ratios. Ryan and Kerry share a raffle win of £400 in the ratio 3:5. How much does each get ? answer

Answer to Question 68 Step 1 : Since the ratio is 3:5, there are : Ryan and Kerry share a raffle win of £400 in the ratio 3:5. How much does each get ? Step 1 : Since the ratio is 3:5, there are : 3+5 = 8 shares Step 2 : Each share is worth : Step 3 : Ryan gets 3 x 50 = £150 Check ! 150 + 250 = 400 Kerry gets 5 x 50 = £250

Question 69 draw a graph of direct proportion and state the key points of the graph. answer

Answer to Question 69 Two quantities which are in DIRECT PROPORTION always lie on a straight line passing through the origin.

solve problems involving Question 70 solve problems involving direct proportion The cost of 6 cakes is £4.20. Find the cost of 5 cakes. answer

Are we expecting more or less Answer to Question 70 Are we expecting more or less Easier method Cakes Pence 6  420 5 (less) 19-Apr-17

Question 71 solve problems involving indirect (inverse) proportion It takes 10 worker 12 months to build a house. How long should it take 8 men. answer

Are we expecting more or less Answer to Question 71 Are we expecting more or less Easier method Workers months 10  12 8 10  12 1  12 x 10 = 120 (more)

add two simple fractions Question 72 add two simple fractions answer

Answer to Question 72 20 + 18 24 ÷2 ÷2

subtract two simple fractions Question 73 subtract two simple fractions answer

Answer to Question 73 25 - 6 30

multiply two simple fractions Question 74 multiply two simple fractions answer

Answer to Question 74

divide two simple fractions Question 75 divide two simple fractions answer

Answer to Question 75

add two mixed fractions Question 76 add two mixed fractions answer

Answer to Question 76 3 + 4 6

subtract two mixed fractions Question 77 subtract two mixed fractions answer

Answer to Question 77 21 - 16 24

multiply two mixed fractions Question 78 multiply two mixed fractions answer

Answer to Question 78

divide two mixed fractions Question 79 divide two mixed fractions answer

Answer to Question 79

solve simple equations Question 80 solve simple equations answer

Answer to Question 80

solve simple equations Question 81 solve simple equations answer

Answer to Question 81

solve more complex equations Question 82 solve more complex equations answer

Answer to Question 82

solve equations with brackets Question 83 solve equations with brackets answer

Answer to Question 83

solve equations with double brackets Question 84 solve equations with double brackets answer

Answer to Question 84

solve equations with fractions Question 85 solve equations with fractions answer

Answer to Question 85

solve equations with brackets & fractions Question 86 solve equations with brackets & fractions answer

Answer to Question 86 Multiply EVERY term by 12

solve simple inequations (inequalities) Question 87 solve simple inequations (inequalities) answer

Answer to Question 87

solve harder inequations (inequalities) Question 88 solve harder inequations (inequalities) answer

Answer to Question 88

explain the meaning of the terms Question 89 explain the meaning of the terms Mean Median Mode Range answer

Answer to Question 89 The Mean Sum of all the data ÷ by the number of data values The Median (put the data in order then find the MIDDLE value) The Mode (the number that appears the most) 4. Range Highest value - the lowest value

construct a frequency table Question 90 construct a frequency table 12 23 41 55 77 15 32 40 51 69 21 12 16 43 56 71 32 75 34 42 55 76 21 73 22 56 41 19 20 47 78 17 answer

Answer to Question 90 10-19 6 20-29 5 30-39 3 40-49 50-59 60-69 70-79 Choose suitable Class interval Class Intervals Tally Frequency 10-19 6 20-29 5 30-39 3 40-49 50-59 60-69 70-79 6 5 1 6

and find the mean from it. Question 91 No of Siblings (S) Freq. (f) add a third column to a frequency table and find the mean from it. 9 1 13 2 6 3 1 5 1 Totals 30 answer

Answer to Question 91 S x f 9 0 x 9 =0 1 13 1 x 13 = 13 2 6 2 x 6 = 12 No of Siblings (S) Freq. (f) S x f 9 0 x 9 =0 1 13 1 x 13 = 13 2 6 2 x 6 = 12 3 1 3 x 1 = 3 5 1 5 x 1 = 5 Totals 30 33

cumulative frequency graph. Question 92 Day Freq. (f) Cum. Freq. Total so far 1 2 2 find the median from a cumulative frequency graph. 2 3 5 3 1 6 4 6 12 5 5 17 6 8 25 7 4 29 answer

Answer to Question 92 Day Freq. (f) Cum. Freq. Total so far 1 2 2 2 3 5 3 1 6 4 6 12 5 5 17 6 8 25 7 4 29

construct a Pie-chart Question 93 Rugby Football Cricket Ice Hockey 75 90 45 60 Favourite Sport Squash 30 answer

Answer to Question 93 Rugby Football Cricket Ice Hockey 75 90 45 60 Favourite Sport Squash 30 Total 300 Rugby Football Cricket Ice Hockey Squash 90o 108o 54o 72o 36o

construct a stem-leaf diagram Question 94 construct a stem-leaf diagram 12 40 57 54 55 13 15 32 41 21 23 29 51 answer

Answer to Question 94 12 40 57 54 55 13 15 32 41 21 23 29 51 1 1 3 9 2 4 5 7 2 2 2 3 4 5 stem leaves n = 20 Key : 2 3 means 23

construct a scattergraph and describe the main features Question 95 construct a scattergraph and describe the main features answer

Answer to Question 95 When two quantities are strongly connected we say there is a strong correlation between them. Best fit line x x Best fit line x x Strong positive correlation Strong negative correlation No correlation

explain the term probability Question 96 explain the term probability in words and numbers answer

The chance of an event happening Answer to Question 96 The chance of an event happening Probability can be thought of as a fraction or decimal. It always lies between 0 and 1. 0 meaning impossible ( could not happen) 1 meaning certain ( will definitely happen)

Question 97 evaluate a probability There are 3 red and 4 green balls in a bag. What is the probability a green ball is picked. answer

Answer to Question 97 There are 3 red and 4 green balls in a bag. What is the probability a green ball is picked.

Online Revision Exercises Integers Circle Angles Fractions Type of numbers Ratio & Proportion Scientific Notation 1 2 Equations & Inequalities Algebra DST Decimals Area of Quadrilaterals Statistics Linear Patterns 1 2 Percentages Pythagoras 1 2 Level F exam type question