Strand 3 of 5 - Number

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1 3 rd Year Maths Ordinary Level Strand 3 of 5 - Number "The essence of mathematics is not to make simple things complicated, but to make complicated things simple." ~Stanley Gudder, University of Denver, USA. No part of this publication may be copied, reproduced or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission from The Dublin School of Grinds. (Notes reference: 3-mat-o-Strand 3 of 5 Number

2 EASTER REVISION COURSES Looking to maximise your CAO points? Easter is a crucial time for students to vastly improve on the points that they received in their mock exams. To help students take advantage of this valuable time, The Dublin School of Grinds is running intensive, examfocused Easter Revision Courses. Each course runs for five days (90 minutes per day). All courses take place in Stillorgan, Co. Dublin. The focus of these courses is to maximise students CAO points. EASTER REVISION COURSE FEES: 6TH YEAR & 5TH YEAR COURSES PRICE TOTAL SAVINGS 1st Course nd Course rd Course FREE th Course th Course th Course ,025 7th Course ,220 8th Course ,465 9th Course ,710 3RD YEAR COURSES PRICE TOTAL SAVINGS SPECIAL OFFER 1st Course nd Course To avail of this offer, early booking is required as courses were fully booked last year. 3rd Course FREE th Course th Course th Course th Course th Course BUY 2 COURSES GET A 3 RD COURSE FREE What do students get at these courses? minutes of intensive tuition per day for five days, with Ireland s leading teachers. 99 Comprehensive study notes. 99 A focus on simple shortcuts to raise students grades and exploit the critically important marking scheme. 99 Access to a free supervised study room. NOTE: These courses are built on the fact that there are certain predicable trends that reappear over and over again in the State Examinations. DSOG Easter pg A4 FINAL PRINT.indd 2 25% SIBLING DISCOUNT AVAILABLE. Please call to avail of this discount. FREE DAILY BUS SERVICE For full information on our Easter bus service, see 3 pages ahead. Access to food and beverage facilities is also available to students. To book, call us on or book online at 20/02/ :25

3 Timetable An extensive range of course options are available over a two-week period to cater for students timetable needs. Courses are held over the following weeks:»» Monday 10th Friday 14th April 2017»» Monday 17th Friday 21st April 2017 All Easter Revision Courses take place in The Talbot Hotel, Stillorgan (formerly known as The Stillorgan Park Hotel). 6th Year Easter Revision Courses SUBJECT LEVEL DATES TIME Accounting H Monday 10th - Friday 14th April 12:00pm - 1:30pm Agricultural Science H Monday 10th - Friday 14th April 10:00am - 11:30am Applied Maths H Monday 10th - Friday 14th April 8:00am - 9:30am Art History H Monday 10th - Friday 14th April 2:00pm - 3:30pm Biology Course A* H Monday 10th - Friday 14th April 8:00am - 9:30am Biology Course A* H Monday 17th - Friday 21st April 10:00am - 11:30am Biology Course B* H Monday 10th - Friday 14th April 10:00am - 11:30am Biology Course B* H Monday 17th - Friday 21st April 8:00am - 9:30am Business H Monday 10th - Friday 14th April 12:00pm - 1:30pm Business H Monday 17th - Friday 21st April 8:00am - 9:30am Chemistry Course A* H Monday 17th - Friday 21st April 8:00am - 9:30am Chemistry Course B* H Monday 17th - Friday 21st April 10:00am - 11:30am Classical Studies H Monday 10th - Friday 14th April 2:00pm - 3:30pm Economics H Monday 10th - Friday 14th April 8:00am - 9:30am Economics H Monday 17th - Friday 21st April 10:00am - 11:30am English Paper 1* H Monday 17th - Friday 21st April 8:00am - 9:30am English Paper 2* H Monday 10th - Friday 14th April 8:00am - 9:30am English Paper 2* H Monday 17th - Friday 21st April 10:00am - 11:30am French H Monday 10th - Friday 14th April 10:00am - 11:30am French H Monday 17th - Friday 21st April 8:00am - 9:30am Geography H Monday 10th - Friday 14th April 8:00am - 9:30am Geography H Monday 10th - Friday 14th April 2:00pm - 3:30pm German H Monday 17th - Friday 21st April 12:00pm - 1:30pm History (Europe)* H Monday 17th - Friday 21st April 2:00pm - 3:30pm History (Ireland)* H Monday 17th - Friday 21st April 12:00pm - 1:30pm Home Economics H Monday 10th - Friday 14th April 12:00pm - 1:30pm Irish H Monday 10th - Friday 14th April 10:00am - 11:30am Irish H Monday 17th - Friday 21st April 12:00pm - 1:30pm Maths Paper 1* H Monday 10th - Friday 14th April 8:00am - 9:30am Maths Paper 1* H Monday 10th - Friday 14th April 12:00pm - 1:30pm Maths Paper 1* H Monday 17th - Friday 21st April 8:00am - 9:30am Maths Paper 2* H Monday 10th - Friday 14th April 10:00am - 11:30am Maths Paper 2* H Monday 17th - Friday 21st April 10:00am - 11:30am Maths Paper 2* H Monday 17th - Friday 21st April 12:00pm - 1:30pm Maths O Monday 10th - Friday 14th April 10:00am - 11:30am Maths O Monday 17th - Friday 21st April 12:00pm - 1:30pm Physics H Monday 17th - Friday 21st April 10:00am - 11:30am Spanish H Monday 10th - Friday 14th April 12:00pm - 1:30pm Spanish H Monday 17th - Friday 21st April 10:00am - 11:30am Note: 5th Year students are welcome to attend any of the 6th Year courses above. * Due to large course content, these subjects have been divided into two courses. For a full list of topics covered in these courses, please see 3 pages ahead. To book, call us on or book online at 5th Year Easter Revision Courses SUBJECT LEVEL DATES TIME English H Monday 10th - Friday 14th April 12:00pm - 1:30pm Maths H Monday 10th - Friday 14th April 10:00am - 11:30am Note: 4th Year students are welcome to attend any of the 5th Year courses listed above. 3rd Year Easter Revision Courses SUBJECT LEVEL DATES TIME Business H Monday 17th - Friday 21st April 2:00pm - 3:30pm Studies English H Monday 10th - Friday 14th April 10:00am - 11:30am English H Monday 17th - Friday 21st April 12:00pm - 1:30pm French H Monday 17th - Friday 21st April 12:00pm - 1:30pm Geography H Monday 17th - Friday 21st April 8:00am - 9:30am German H Monday 17th - Friday 21st April 2:00pm - 3:30pm History H Monday 10th - Friday 14th April 8:00am - 9:30am Irish H Monday 10th - Friday 14th April 12:00pm - 1:30pm Maths H Monday 10th - Friday 14th April 8:00am - 9:30am Maths H Monday 17th - Friday 21st April 10:00am - 11:30am Maths O Monday 10th - Friday 14th April 2:00pm - 3:30pm Science H Monday 10th - Friday 14th April 12:00pm - 1:30pm Science H Monday 17th - Friday 21st April 8:00am - 9:30am Spanish H Monday 10th - Friday 14th April 2:00pm - 3:30pm Note: 2nd Year students are welcome to attend any of the 3rd Year courses above. 6th Year Oral Preparation Courses With the Oral marking component worth up to 40%, it is of paramount importance that students are fully prepared for these examinations. These courses will show students how to lead the Examiner towards topics they are prepared for. This will equip students with the information they need to maximise their performance in the State Examinations. FEES: 140 VENUE: The Talbot Hotel, Stillorgan (formerly The Stillorgan Park Hotel) SUBJECT LEVEL DATES TIME French H Sunday 12th March 9:00am - 1:00pm German H Saturday 11th March 9:00am - 1:00pm Irish H Sunday 19th March 9:00am - 1:00pm Spanish H Saturday 11th March 2:00pm - 6:00pm BUY 2 COURSES & GET A 3 RD COURSE FREE!

4 Topics: 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) Adding and Subtracting... 3 Place Value... 6 Highest Common Factor... 8 Prime Numbers Lowest Common Multiple Squares and Square Roots Sets Order of Operations - BPMDAS Commutative & Associative Properties Significant Figures Number Pyramids Writing a number as a Product of its Prime Factors Integers Fractions Percentages Perimeter and Area Ratios Currency Exchange Measure Distance/Time Graphs Distance/Speed/Time Solutions The Dublin School of Grinds Page 2 of 170 David Lewis

5 1) Adding and Subtracting To add, we take steps to the right on the number line. Demonstration A: Simplify = 3 To subtract, we take steps to the left on the number line. Demonstration B: Simplify = 4 Demonstration C: Simplify = 5 The Dublin School of Grinds Page 3 of 170 David Lewis

6 Demonstration D: Simplify = 1 Q1.1. Simplify 8 10 Q1.2. Simplify Q1.3. Simplify Q1.4. Simplify The Dublin School of Grinds Page 4 of 170 David Lewis

7 Q1.5. Simplify 14 7 Q1.6. Simplify Q1.7. Simplify Q1.8. Simplify The Dublin School of Grinds Page 5 of 170 David Lewis

8 2) Place Value The symbols 0 9 are called numerals. We can write any number using these 10 numerals. Consider the following number; 1,234, How would we say the above number aloud? One million, two hundred and thirty-four thousand, five hundred and sixty-seven, point eight, nine, zero, one. What a mouthful! Below is a breakdown of what each value is worth in the above number. Q2.1. Write down the value of the underlined digit in each of these numbers: i) 512: ii) 3988: iii) 1248: iv) 8.2: v) 530,688: vi) 6.413: The Dublin School of Grinds Page 6 of 170 David Lewis

9 Q2.2. Write down each of these numbers using numerals: (i) Two hundred and twenty-five: (ii) Forty thousand, eight hundred and sixty-three: (iii) One million, six hundred thousand: (iv) Nine thousand and eight: Q2.3. The Dublin School of Grinds Page 7 of 170 David Lewis

10 3) Highest Common Factor Factors are numbers that you can multiply together to get another number. Demonstration: 2 and 3 are factors of 6, because 2 3 = 6. Also, 1 and 6 are factors of 6 because 1 6 = 6. Therefore, the factors of 6 are 1, 2, 3 and 6. Q3.1. List the factors of 12. Q3.2. List the factors of 20. Q3.3. List the factors of 100. Q3.4. List the factors of 35. The Dublin School of Grinds Page 8 of 170 David Lewis

11 Demonstration: Find the Highest Common Factor (HCF) of 12 and 18. The factors of 12 are 1, 2, 3, 4, 6 and 12. The factors of 18 are 1, 2, 3, 6, 9 and 18. Therefore, the HCF of 12 and 18 is 6. Q3.5. Find the HCF of 10 and 15. Q3.6. Find the HCF of 14 and 49. The Dublin School of Grinds Page 9 of 170 David Lewis

12 Q3.7. Find the HCF of 39, 52 and 78. Q3.8. Find the HCF of 12, 30 and 48. The Dublin School of Grinds Page 10 of 170 David Lewis

13 4) Prime Numbers A prime number is a number with two factors only. The two numbers are always itself and 1. 2, 3, 5, 7, 11, 13, 17 and 19 are the first eight prime numbers. Q4.1. Is 50 a prime number? Explain. Q4.2. Is 23 a prime number? Explain. Q4.3. Is 1 a prime number? Explain. Q4.4. Is 2 a prime number? Explain. Q4.5. Circle the correct option. (i) All prime numbers are odd numbers. True / False. (ii) Two is the only even prime number. True / False. (iii) There are five prime numbers between 0 and 10. True / False. The Dublin School of Grinds Page 11 of 170 David Lewis

14 5) Lowest Common Multiple We get a multiple of a number when we multiply it by another number. Such as multiplying by 1, 2, 3, 4, 5, etc, but not zero. Just like the multiplication table. Demonstration: The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, What is a Common Multiple? Notice that 20 and 40 appear in both lists? So, the common multiples of 4 and 5 are: 20, 40, 60, etc The Lowest Common Multiple (LCM) is the smallest of the common multiples. In the above demonstration, the LCM of 4 and 5 is 20. Q5.1. Write out the first five multiples of 7. Q5.2. Write out the next three multiples of 8 after 64. Q5.3. Find the LCM of 2 and 7. The Dublin School of Grinds Page 12 of 170 David Lewis

15 Q5.4. Find the LCM of 7 and 8. Q5.5. Find the LCM of 15 and 20. Q5.6. Find the LCM of 12 and 18. Q5.7. Find the LCM of 3, 5 and 10. The Dublin School of Grinds Page 13 of 170 David Lewis

16 Q5.8. Find the LCM of 6, 11 and 33. The Dublin School of Grinds Page 14 of 170 David Lewis

17 6) Squares and Square Roots A Square Number is a number multiplied by itself. Demonstration: Simplify = 16 The small two in the above demonstration is called a Power. Powers are used to reduce the length of time it takes to write a number can be written as 3 6. Q6.1. Simplify the following: (i) 9 2 : (ii) 3 3 : (iii) : (iv) : The Dublin School of Grinds Page 15 of 170 David Lewis

18 Square roots: 25 can be read as the square root of 25. To find 25 we search for a number which, when multiplied by itself, will give = 25 Therefore, 25 = 5 Q6.2 Find the value of: (i) 16: (ii) 36: (iii) 81: For more difficult questions, we may need to use our calculator to find the square root of a number. Demonstration: Find the value of 324 Casio Step 1: Hit the square root button( ) You will find this three buttons above the number 8. Sharp Step 1: Hit the square root button ( ) You will find this two buttons above the number 8. Step 2: Type in the 324. Step 2: Type in the 324. Step 3: Hit the (=) sign. Step 3: Hit the (=) sign. Answer: 18 The Dublin School of Grinds Page 16 of 170 David Lewis

19 7) Sets A set is a collection of distinct objects, these objects are called elements. e.g.: The set of elements in my pencil case are as follows: = {pencil, red pen, black pen, calculator, ruler} e.g.: The set of natural numbers less than 5 are as follows: = {1, 2, 3, 4} We use a special character to say that something is an element of a set. It looks like an odd curvy capital E. For instance, to say that "a red pen is an element of my pencil case", we would write the following: Red pen my pencil case. However, how would I show that a green crayon is not an element of my pencil case? Green crayon my pencil case. A symbol denotes that an element is not a part of a particular set. Mathematical example: If A = {1, 3, 5} then 1 A and 2 A Listing Elements: Q7.1. List the elements of each of these sets: i) {The positive whole numbers less than 7} The Dublin School of Grinds Page 17 of 170 David Lewis

20 ii) {The odd numbers between 11 and 19, inclusive} iii) {The whole numbers between 2 and 9, exclusive} iv) {The first 6 prime numbers} v) {The first four multiples of 12} vi) {The divisors of 21 that are also prime numbers} The Dublin School of Grinds Page 18 of 170 David Lewis

21 Cardinal Number: The Cardinal Number is a fancy way of saying how many elements there are in a set. e.g.: The Cardinal Number of my pencil case which contains {pencil, red pen, black pen, calculator, ruler} is: #My Pencil Case: 5 e.g.: The Cardinal Number of X which contains {1, 3, 5} is: #X = 3 Note: We use the # sign as a quicker way of writing The Cardinal Number ; it s not like how we use it on Twitter! Q7.2. A = {1, 2, 3, 4}, B = {2, 4, 8, 10, 12} Find: i) #A ii) #B iii) #A + #B The Dublin School of Grinds Page 19 of 170 David Lewis

22 Q7.3. Find (i) #A and (ii) #B. Subsets: Sets can be related to each other. If one set is "inside" another set, it is called a "subset". Suppose A = {1, 2, 3} and B = {1, 2, 3, 4, 5, 6}. Then A is a subset of B, since everything in A is also in B. This is written as: is the subset symbol, and is pronounced "is a subset of". To show something is not a subset, you draw a slash through the subset symbol, so the following: is pronounced as "B is not a subset of A". The null set, also called the empty set, is the set that does not contain anything. It is symbolized by using { } or { }. There is only one null set. The null set is a subset of every set. Demonstration: A = {5, 7, 9}. List the subsets of A. Solution: { } {5}, {7}, {9}, (5, 7}, {5, 9}, (7, 9}, {5, 7, 9}. The subsets in bold are known as Proper Subsets. The subsets containing nothing (the null set) and all the elements are known as Improper Subsets. The Dublin School of Grinds Page 20 of 170 David Lewis

23 Demonstration: X = {2, 3, 4} List the subsets of X. { }, {2}, {3}, {4}, {2, 3}, {2, 4}, {3, 4}, {2, 3, 4}. The subsets in bold are known as Proper Subsets. The subsets containing nothing (the null set) and all the elements are known as Improper Subsets. Q7.4. Z = {20, 40, 60}. List the subsets of Z. Underline the Proper Subsets. Union & Intersection of Sets: If two sets are being combined, this is called the "union" of the sets, and is indicated by a large U-type character ( ). If instead of taking everything from the two sets, you're only taking what is common to the two, this is called the "intersection" of the sets, and is indicated with an upside-down U-type character ( ). So if C = {1, 2, 3, 4, 5, 6} and D = {4, 5, 6, 7, 8, 9}, then: These are pronounced as "C union D equals..." and "C intersection D equals..." respectively. As you can see from the above example, we only write each element once if it is in both sets. The Dublin School of Grinds Page 21 of 170 David Lewis

24 Q7.5. A {1, 2, 3, 4, 5, 6} and B {3, 5, 7, 9} are two sets. i) List the elements of (A B). ii) List the elements of (A B). Q7.6. List the elements of (M N) and (M N) when M = {4, 5, 6}, N = {3, 4, 5}. The Dublin School of Grinds Page 22 of 170 David Lewis

25 Venn Diagrams: Venn diagrams were invented by a guy named John Venn (not messing; that was really his name) as a way of picturing relationships between different groups of things. Inventing this type of diagram was, apparently, pretty much all he ever accomplished. Cheers John. e.g. 1: (A B): e.g. 2: (A B): The Dublin School of Grinds Page 23 of 170 David Lewis

26 Demonstration: A = {6, 8, 10, 12, 14} and B = {11, 12, 13, 14, 15}. Complete the Venn diagram. Step 1: Highlight the elements that are common in both sets. A = {6, 8, 10, 12, 14} and B = {11, 12, 13, 14, 15} Step 2: Write these elements into the diagram. Step 3: Write the elements left in A. Step 4: Write in the elements left in B. The Dublin School of Grinds Page 24 of 170 David Lewis

27 Q7.7. Draw a Venn diagram containing the sets X = {4, 6, 8, 10}, Y = {8, 10, 12}. i) List the elements of (X Y) ii) List the elements of (X Y) iii) What is #(X Y)? The Dublin School of Grinds Page 25 of 170 David Lewis

28 iv) What is #(X Y)? Q7.8. A = {2, 4, 6, 8, 10, 12) and B = (10, 12) are two sets. i) Complete the Venn diagram. ii) List the elements of (A B) iii) List the elements of (A B) The Dublin School of Grinds Page 26 of 170 David Lewis

29 iv) Write down #A. Real life scenario: Let s think about our classmates interests, Adam likes soccer but not rugby, Brian likes soccer and rugby, Carla likes rugby but not soccer and Dena doesn t like either of these sports. How would we show this on a Venn diagram? The big rectangle is known as the Universal Set, this contains all the elements. Every set is a subset of the Universal Set. It is easiest to think of the Universal Set as the entire class, and the different subsets as different interests/hobbies of people in the class. The Dublin School of Grinds Page 27 of 170 David Lewis

30 Complement of a Set: The complement of a set A is the set of elements in the universal set that are NOT elements of A. The complement of the set A in this case would be: {4, 5}. The mathematical way of writing this is A = {4, 5} Set Difference: A\B is the set of elements that are in A but not in B. Think of the \ as a shield keeping B away from A. The Dublin School of Grinds Page 28 of 170 David Lewis

31 Q7.9. U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8} and B = {6, 8, 9}. i) Represent the sets on a Venn diagram ii) List the elements of (A B). iii) List the elements of A. iv) List the elements of B\A. The Dublin School of Grinds Page 29 of 170 David Lewis

32 v) List the elements of (A B). Q7.10. In a survey of 50 people, 20 said they watched Breaking Bad, 40 said they watched The Simpsons and 15 said they watched both. i) Represent this information on a Venn diagram. ii) How many people watched only Breaking Bad? iii) How many people watched neither of these shows? The Dublin School of Grinds Page 30 of 170 David Lewis

33 Q7.11. The Dublin School of Grinds Page 31 of 170 David Lewis

34 Q7.12. The Dublin School of Grinds Page 32 of 170 David Lewis

35 Q7.13. The Dublin School of Grinds Page 33 of 170 David Lewis

36 Q7.14. Q7.15. The Dublin School of Grinds Page 34 of 170 David Lewis

37 Q7.16. Q7.17. The Dublin School of Grinds Page 35 of 170 David Lewis

38 Q7.18. The Dublin School of Grinds Page 36 of 170 David Lewis

39 Q7.19. The Dublin School of Grinds Page 37 of 170 David Lewis

40 Q7.20. Q7.21. Q7.22. The Dublin School of Grinds Page 38 of 170 David Lewis

41 The Dublin School of Grinds Page 39 of 170 David Lewis

42 Q7.23. The Dublin School of Grinds Page 40 of 170 David Lewis

43 8) Order of Operations - BPMDAS When we are asked to add or subtract certain numbers, we can work from left to right to find the answer. Demonstration: Simplify = 5 Multiplication and Division are equally important operations. They are more important than Addition and Subtraction. Therefore, we Multiply/Divide before we Add/Subtract. Demonstration: Simplify = 9 Powers and Square Roots are the most important operations. Therefore, we carry out these operations before we Multiply/Divide/Add/Subtract. Demonstration: = 72 BPMDAS is a buzz word that helps us understand the Order of Operations. B = Brackets P = Powers M = Multiplication D = Division A = Addition S = Subtraction The Dublin School of Grinds Page 41 of 170 David Lewis

44 Q8.1. Simplify 3 16 Q8.2. Simplify Q8.3. Simplify!""!!!! (!"!!) The Dublin School of Grinds Page 42 of 170 David Lewis

45 Q8.4. Simplify!"!(!"!)!!!! Q8.5. The Dublin School of Grinds Page 43 of 170 David Lewis

46 9) Commutative & Associative Properties Commutative Laws: The "Commutative Laws" say we can swap numbers over and still get the same answer when we add: a + b = b + a Example:... or when we multiply: a b = b a Example: Associative Laws: The "Associative Laws" say that it doesn't matter how we group the numbers (i.e. which we calculate first) when we add: (a + b) + c = a + (b + c)... or when we multiply: (a b) c = a (b c) The Dublin School of Grinds Page 44 of 170 David Lewis

47 Demonstration: This: (2 + 4) + 5 = = 11 Has the same answer as this: 2 + (4 + 5) = = 11 This: (3 4) 5 = 12 5 = 60 Has the same answer as this: 3 (4 5) = 3 20 = 60 Q9.1. Explain, with the aid of an example, what the Commutative Laws means. Q9.2. Explain, with the aid of an example, what the Associative Laws means. The Dublin School of Grinds Page 45 of 170 David Lewis

48 10) Significant Figures The significant figures are the "interesting" or "important" digits. Demonstration: At the last Irish soccer game in the Aviva Stadium, the attendance was 48,724. When reporting the attendance, some news broadcasters round off the exact number of people. If they rounded the attendance to ONE significant figure, they would have said there was 50,000 at the game. If they rounded the attendance to TWO significant figures, they would have said there was 49,000 at the game. If they rounded the attendance to THREE significant figures, they would have said there was 48,700 at the game. If they rounded the attendance to FOUR significant figures, they would have said there was 48,720 at the game. Q10.1. The attendance at a One Direction concert at the 3Arena was 14,075. Write this number to: i) one significant figure: ii) two significant figures: iii) three significant figures: iv) four significant figures: Q10.2. The attendance at Young Scientist & Technology Exhibition 2014 at the RDS was 7,285. Write this number to: i) one significant figure: ii) two significant figures: iii) three significant figures: The Dublin School of Grinds Page 46 of 170 David Lewis

49 Q10.3. Rounded off to nearest 100 people, there were 600 people at a UCD soccer match. What is (i) the largest and (ii) the smallest number of people that could have attended? Q10.4. Rounded off to nearest 10 people, there were 90 people at a party. What is (i) the largest and (ii) the smallest number of people that could have attended? The Dublin School of Grinds Page 47 of 170 David Lewis

50 11) Number Pyramids Number Pyramids are like puzzles to test your addition and subtraction ability. All you need to do is add the two numbers in the lower blocks to find the number in the block above. Q11.1. Complete the Number Pyramid by filling in the empty boxes. Q11.2. Complete the Number Pyramid by filling in the empty boxes. The Dublin School of Grinds Page 48 of 170 David Lewis

51 Q11.3. Complete the Number Pyramid by filling in the empty boxes. The Dublin School of Grinds Page 49 of 170 David Lewis

52 12) Writing a number as a Product of its Prime Factors To write a number as a product of its prime factors, you must divide the given number by the lowest prime number that goes in evenly. Finally, you must multiply these numbers to equal the given number. Demonstration: Write 1050 as the product of its prime factors. So now we multiply the Prime Factors; our answer should equal 1050: = 1050 Q12.1. Write 60 as a product of its prime factors. The Dublin School of Grinds Page 50 of 170 David Lewis

53 Q12.2. Write 150 as a product of its prime factors. Q12.3. Write 860 as a product of its prime factors. Q12.4. Write 92 as a product of its prime factors. The Dublin School of Grinds Page 51 of 170 David Lewis

54 13) Integers Integers are the set of numbers that are both positive and negative whole numbers, including zero. We deal with integers throughout our daily lives, e.g.: to measure temperature. Q13.1. The temperature in a fridge is 5 C and the temperature in the kitchen is 19 C. How many degrees difference is this? The Dublin School of Grinds Page 52 of 170 David Lewis

55 When multiplying integers, we must remember the following rules: Rule 1: A Positive Number (+) multiplied by a Positive Number (+) equals a Positive Number (+). Demonstration: 5 8 = 40 (+) (+) = (+) Rule 2: A Positive Number (+) multiplied by a Negative Number ( ) equals a Negative Number ( ). Demonstration: 4 ( 2) = 8 (+) ( ) = ( ) Rule 3: A Negative Number ( ) multiplied by a Negative Number ( ) equals a Positive Number (+). Demonstration: ( 6) ( 5) = 30 ( ) ( ) = (+) Q13.2. Write as a single integer: i) ( 3) 6: ii) 2 ( 9): iii) ( 12) ( 10): iv) ( 11) ( 9): The Dublin School of Grinds Page 53 of 170 David Lewis

56 The same rules apply to division. Demonstration: Express the following as a single integer.! (!!")!!" Step 1: Multiply the top line:!!"!!" Step 2: Tidy up:!!!! Step 3: Follow the above rules:! (or 3)! Q13.3. Express the following as a single integer. Q13.4. Express the following as a single integer. The Dublin School of Grinds Page 54 of 170 David Lewis

57 Q13.5. Express the following as a single integer. Q13.6. In a multiple choice exam there are fifteen questions. Each correct answer gains three points. Each wrong answer loses one point. (i) Carl gave 9 correct answers and 6 incorrect answers. How many points did he get altogether? (ii) Bren gave 12 correct answers and 3 incorrect answers. How many points did he get altogether? The Dublin School of Grinds Page 55 of 170 David Lewis

58 (iii) Marcos answered all fifteen questions. Explain how Marcos could get 41 points altogether? Q13.7. Carrauntoohil is Ireland s highest mountain. It is about 1,000 metres above sea level. When you climb a mountain, the temperature drops by 1 C for every 100 metres you ascend. When the temperature at sea level is 18 C, what will the temperature be on the top of Carrauntoohil? Q13.8. John checks his thermometer when he reaches the highest peak of Carrauntoohil. It reads 12 C. According to John s estimations, what is the temperature at sea level? The Dublin School of Grinds Page 56 of 170 David Lewis

59 14) Fractions There are three different types of fractions. i) Proper Fractions If the top number is smaller than the bottom number, this is called a proper fraction. e.g.:!,!,!.!!! ii) Improper Fractions If the top number is bigger than the bottom number, the fraction is called an improper fraction or a top-heavy fraction and is greater than 1. e.g.:!,!!" and.!!! iii) Mixed Number A number that consists of a whole number and a proper fraction is called a mixed number or a mixed fraction. e.g.: 3!,! 5! and! 8!.! Q14.1. What type of fraction are the following? i) 12! :! ii)! :! iii)! :! Slice a pizza, and you will have fractions: The Dublin School of Grinds Page 57 of 170 David Lewis

60 Equivalent Fractions: Equivalent Fractions look differently, but they represent the same the same amount: Q14.2. Simplify!!" Q14.3. Simplify!!" Q14.4. Simplify!"!" The Dublin School of Grinds Page 58 of 170 David Lewis

61 Adding & Subtracting Fractions: We can add fractions easily if the bottom number (the denominator) is the same: Demonstration: Demonstration: Q14.5. Simplify!!" +!!" Q14.6. Simplify!!!! The Dublin School of Grinds Page 59 of 170 David Lewis

62 Q14.7. Simplify!!!! The Dublin School of Grinds Page 60 of 170 David Lewis

63 However, in more difficult questions, we will need to find the Lowest Common Multiple of the bottom number (the denominator). Alternatively, you can simply let your calculator do all the work! Demonstration A: Simplify: Step 1: Find the Lowest Common Multiple of 3 and 5. Multiples of 3: 3, 6, 9, 12, 15, 18, 21, Multiples of 5: 5, 15, 20, 25, The Lowest Common Multiple of 3 and 5 is 15. Step 2: Change each fraction so that they both have the same denominator (bottom number).! +!!"!" Step 3: Add the fractions. Alternatively, use your calculator: Casio: Press the fraction button, ( ), which is three buttons above the number 7. Your screen should have a fraction with empty flashing boxes. Press 1, press down, press 3, press right, type +, press the fraction button, press 1, press down, press 5, press equals. Your answer should be!.!" Sharp: Press the fraction button, (a/b), which is just above the number 8. Your screen should have a fraction with empty flashing boxes. Press 1, press down, press 3, press right, type +, press the fraction button, press 1, press down, press 5, press equals. Your answer should be!!". The Dublin School of Grinds Page 61 of 170 David Lewis

64 Demonstration B: Simplify!!!! Step 1: Find the Lowest Common Multiple of 5 and 8. Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, Multiples of 8: 8, 16, 24, 32, 40, The Lowest Common Multiple of 5 and 8 is 40. Step 2: Change each fraction so that they both have the same denominator (bottom number).!"!!"!" Step 3: Subtract the fractions. Alternatively, use your calculator (follow instructions from Demonstration A). Demonstration C: Simplify!!!! Step 1: Find the Lowest Common Multiple of 7 and 5. Multiples of 7: 7, 14, 21, 28, 35, Multiples of 5: 5, 10, 15, 20, 25, 30, 35, Step 2: Change each fraction so that they both have the same denominator (bottom number).!"!"!"!" Step 3: Subtract the fractions. Alternatively, use your calculator (follow instructions from Demonstration A). The Dublin School of Grinds Page 62 of 170 David Lewis

65 Q14.8. Simplify!!!! Q14.9. Simplify!!!! The Dublin School of Grinds Page 63 of 170 David Lewis

66 Converting Mixed Numbers into Improper Fractions: Demonstration A: Express 5! as an improper fraction.! Step 1: Multiply the bottom number (denominator) by the big number on the left (the whole number). 4 5 = 20 Step 2: Add this number to the top number (numerator) = 21 Step 3: Rewrite as an improper fraction. The Dublin School of Grinds Page 64 of 170 David Lewis

67 Demonstration B: Express 2! as an improper fraction.! Step 1: Multiply the bottom number (denominator) by the big number on the left (the whole number). 3 2 = 6 Step 2: Add this number to the top number (numerator) = 8 Step 3: Rewrite as an improper fraction. The Dublin School of Grinds Page 65 of 170 David Lewis

68 Q Express 8! as an improper fraction.! Q Express 3! as an improper fraction.! Q Express 2! as an improper fraction.! The Dublin School of Grinds Page 66 of 170 David Lewis

69 Q Express 2! + 3! as one fraction in its simplest form.!! Q Express 5! 4! as one fraction in its simplest form.!! The Dublin School of Grinds Page 67 of 170 David Lewis

70 Q Cristiano Ronaldo, Gareth Bale and Falcao are the only candidates for the FIFA Ballon d'or (Best Player in the World Award). Ronaldo got!! of the votes. Bale got of the votes.!!" (i) What fraction of the votes did Falcao get? (ii) Which player won the award? Justify your answer. Multiplying Fractions: To multiply two fractions, we multiply top by top and bottom by bottom. After, we must simplify the fraction if there is a common factor. Demonstration: Simplify Step 1: Multiply top by top, bottom by bottom. Step 2: Simplify the fraction by dividing each number by the Highest Common Factor. In this case, the HCF is 4. The Dublin School of Grinds Page 68 of 170 David Lewis

71 Q Simplify!!!! Q Simplify!!"!! Demonstration: Simplify 3!! 4!! Step 1: Convert the Mixed Fractions into Improper Fractions. Step 2: Multiply top by top, bottom by bottom. Step 3: Simplify the fraction by dividing each number by the Highest Common Factor. In this case, the HCF is 5. The Dublin School of Grinds Page 69 of 170

72 Q Simplify 1!! 3!! Q Simplify 3!! 5 The Dublin School of Grinds Page 70 of 170

73 Dividing Fractions: To divide a fraction, we flip the second term upside down and multiply as we did in the last section. Demonstration: Simplify!!!! Step 1: Flip the second term. Step 2: Multiply top by top, bottom by bottom. Q Simplify!!!! Q Simplify!!"!! The Dublin School of Grinds Page 71 of 170

74 Q Simplify 2!! Demonstration: Simplify 6!! 1!! Here, we will have an extra step in the beginning as there are Mixed Fractions that we need to convert to Improper Fractions. There will also be an added step at the end, as the number can be simplified. Step 1: Convert the Mixed Fractions into Improper Fractions. Step 2: Flip the second term. Step 3: Multiply top by top, bottom by bottom. Step 4: Simplify the fraction by dividing each number by the Highest Common Factor. In this case, the HCF is 20. The Dublin School of Grinds Page 72 of 170

75 Q Simplify 4!! 1!! Q Simplify 3!! 1!! Q Simplify 7!! 2!! The Dublin School of Grinds Page 73 of 170

76 Q Simplify 10 4!! Q The Dublin School of Grinds Page 74 of 170

77 Word Problems involving fractions: Demonstration: A journey to Stillorgan from a student s house in Dun Laoghaire consists of a! km cycle, then a! 4! km bus journey, and finally a! km walk from the bus stop.!! Find the total length of the journey to Stillorgan. Step 1: Write the distances down in a mathematical way: Step 2: Convert the Mixed Fractions into Improper Fractions. Step 3: Find the Lowest Common Multiple of the bottom numbers (denominators). Multiples of 5: 5, 10, 15, 20, 25, 30, Multiples of 4: 4, 8, 12, 16, 20, 24, Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, The LCM of 5, 4 and 2 is 20. Step 4: Change each fraction so that they both have the same denominator (bottom number). Step 5: Add the top numbers. =!"#!" The Dublin School of Grinds Page 75 of 170

78 The question asked us how long the journey is from Dun Laoghaire to Stillorgan. A Mixed Fraction would be more suitable than an Improper Fraction in this case. Step 6: Convert!"#!" = 5!!" km. into a Mixed Fraction. Q Olivia spends! of her savings in Dundrum Town Centre and! of her savings on!! buying a laptop. She saves the rest. What fraction of her savings does Olivia save? Q Andy spent 2! hours studying on Saturday morning. He spent 60 minutes studying! Irish, 50 minutes studying Maths and the rest of the time studying Science. What fraction of the time did he spend: (i) Studying Irish (ii) Studying Maths (iii) Studying Science The Dublin School of Grinds Page 76 of 170

79 Q At a wedding! of the guests are female. The remaining 180 guests are male. How! many guests are there altogether? Q Tom spent!! of his money. If he had 65 left, how much had he at first? The Dublin School of Grinds Page 77 of 170

80 Q The Dublin School of Grinds Page 78 of 170

81 Q The Dublin School of Grinds Page 79 of 170

82 Q The Dublin School of Grinds Page 80 of 170

83 15) Percentages Finding a Percentage of a Quantity: To find a percentage of a quantity, we will use our calculator. This is the quickest and most efficient way of solving these questions. Being completely comfortable using the different calculator functions is extremely important for students leading up to State Examinations and end of term exams. Demonstration: Find 25% of 48. Casio Type 48 Sharp Type 48 Hit Multiply sign ( ) Type 25 Hit SHIFT (top left blue button) Hit the Left Bracket Button (just above the number 8) Hit Multiply sign ( ) Type 25 Hit 2ndF (top left orange button) Hit 1 Hit the Equals (=) sign Answer: 12 If we were to do this question manually, it would take a few extra lines of workings: Find 25% of 48. Step 1: Express 25% as a fraction!"!"" Step 2: Simplify fraction. The HCF of 25 and 100 is 25.!! Step 3: Multiply this fraction by 48.!! 48!"! which tidies up to 12. The Dublin School of Grinds Page 81 of 170

84 Q15.1. Find 75% of 16. Q15.2. Find 11% of 600. Q15.3. Find 12.5% of The Dublin School of Grinds Page 82 of 170

85 Q15.4. Find 32% of Converting a Fraction into a Percentage: To convert a fraction into a percentage, we multiply the fraction by. Demonstration: Convert! into a percentage.! Casio Sharp Hit the Fraction button ( ). This is three buttons above the number 7 Type 5 Hit Down Type 8 Hit Right Hit Multiply ( ) Hit the Fraction button ( ) Type 100 Hit Down Type 1 Hit the Equals (=) sign Hit the fraction button (a/b). This button is just above the number 8 Type 5 Hit Down Type 8 Hit Right Hit Multiply ( ) Hit the Fraction button ( ) Type 100 Hit Down Type 1 Hit the Equals (=) sign Hit S D (One button above DEL) Answer: 62.5% The Dublin School of Grinds Page 83 of 170

86 Q15.5. Convert! into a percentage.! Q15.6. Convert! into a percentage.!" Q15.7. Convert! into a percentage.! Q15.8. Convert! into a percentage. Round your answer to one place of decimals.!" The Dublin School of Grinds Page 84 of 170

87 Q15.9. In a Maths exam, John got 336 marks out of 600. In a Science exam, he got 44 marks out of 80. In which exam did he achieve the higher percentage mark? Q A cinema has 200 seats. At a new movie, there were only two empty seats. What percentage of the seats were full? Q Mike has saved 365. He wants to buy a Playstation 4. He sees that it is 400 in his local store. He bargains with the shop owner who eventually says that he will reduce the price of the console by 10%. Can Mike now afford to buy it? The Dublin School of Grinds Page 85 of 170

88 Finding a whole number when given a percentage of it: If we are given a percentage of a quantity, we find 1% and then then multiply that amount by 100 to find the total quantity which is 100%. This is best explained with an example. Demonstration: 15% of Darragh s savings amounts to 30. How much savings does Darragh have? Step 1: Find 1% of Darraghs savings 15% = 30 To find 1%, we divide the amount of money by the given percentage; Therefore, 1% = 2 Step 2: To find his total savings, we multiply 2 by = 200 Q If 20% of a sum of money is 70, find the total sum of money. The Dublin School of Grinds Page 86 of 170

89 Q Bernie spent 50% of her money in Penneys and 25% in TK Maxx. If she had left, how much had she at first? Q Dimitri earns 520 a week. He recieves a pay rise of 4%. How much does he earn each week after the pay rise? Q Aaron got a wage increase of 5%. If his new wage is , find his wage before the increase. The Dublin School of Grinds Page 87 of 170

90 Q A price of a trip to Paris is increased by 8% to What was the original cost of the trip? Q A one-year old motorbike is worth 24,000. This is a decrease in value of 25% of its value from new. What was the original price of the car? Q A research claimed that one egg provides 14% of a male s Recommended Daily Allowance (RDA) for protein. One egg provides 7g of protein. What is a male s RDA for protein? The Dublin School of Grinds Page 88 of 170

91 16) Perimeter and Area The perimeter of a shape is the distance around the edge of the shape. The area is the amount of space covered by a shape. Demonstration: (i) What is the perimeter of this soccer pitch? (ii) What is the area of this soccer pitch? (i) Pretend you are walking from the top left corner to the top right corner; this length is 100 m. Now walk from the top right corner to the bottom right corner; this is 45 m long. After, walk from the bottom right to the bottom left; this length is 100 m. Finally, walk from the bottom left to the top left; this length is 45 m. i.e. 100 m + 45 m m + 45 m = 290 m (ii) The soccer pitch is in the shape of a rectangle. The area of a rectangle is found by multiplying the length by the width. 100m 45m = 450m 2 The Dublin School of Grinds Page 89 of 170

92 Demonstration: (i) What is the perimeter of this shape? (ii) What is the area of this shape? (i) Step 1: Follow the edge of the shape from the top left in a clock-wise rotation. 15 mm + 7 mm + 8 mm + (12 7) mm + (15 + 8) mm + 12 mm Step 2: Simplify by sorting out the brackets. 15 mm + 7 mm + 8 mm + 5 mm + 23 mm + 12 mm Step 3: Tidy up = 70 mm (ii) Step 1: Split the shape into two rectangles. Label the two rectangles. The Dublin School of Grinds Page 90 of 170

93 Step 2: Focus on Rectangle A. Area of Rectangle A: Length Width 15 mm 12 mm = 180 mm 2 Step 3: Focus on Rectangle B. Area of Rectangle B: Length Width 8 mm 5 mm = 40 mm 2 Step 4: Add the two areas of the two rectangles together. 180 mm mm 2 = 220 mm 2 The Dublin School of Grinds Page 91 of 170

94 Q16.1. i) What is the perimeter of this shape? ii) What is the area of this shape? The Dublin School of Grinds Page 92 of 170

95 Q16.2. This diagram is an architect s floor plan for her client. Can you: i) Find the perimeter of this floor? ii) Find the area of this floor? 8m *More space for working on the following page. The Dublin School of Grinds Page 93 of 170

96 The Dublin School of Grinds Page 94 of 170

97 To find the area of a triangle we need to remember the following formula: The perpendicular height is the vertical height of the triangle. Another version of this formula can be found on page 9 of the Formulae and Tables booklet which is approved for use in the State Examinations; Demonstration: Find the area of this triangle. Step 1: Write out the formula: Step 2: Substitute the base and the perpendicular with the given measurements. = 20 cm 2 The Dublin School of Grinds Page 95 of 170

98 Q16.3. Find the area of this triangle. The Dublin School of Grinds Page 96 of 170

99 Q16.4. John wants to print the YouTube emblem. Find the area that will use red ink. The Dublin School of Grinds Page 97 of 170

100 Q16.5. A square garden consists of four triangular lawns and a square patio. The house owner wants to know (i) the area covered by lawn and (ii) the area of the square patio. The Dublin School of Grinds Page 98 of 170

101 Q16.6. Find the area of the blue region in this flag. Extra points if you know what flag it is! The Dublin School of Grinds Page 99 of 170

102 17) Ratios A ratio is just a comparison between two different things. For example, you could look at a group of people and refer to the ratio of men to women in the group. Suppose there are thirty-five people, fifteen of whom are men. Then the ratio of men to women is 15 to 20. Notice that, in the expression the ratio of men to women, men came first. This order is very important, and must be respected. Whichever word came first, its number must come first. If the expression had been ratio of women to men, then the numbers would have been 20 to 15. Expressing the ratio of men to women as 15 to 20 is expressing the ratio in words. There are two other notations for this: Odds notation: 15:20 Fractional notation:!"!" Ratios (and fractions) can be tidied up: 15:20 tidies to 3:4 and!"!" tidies to!! NOTE: If you want or need to, you can use the fraction button on your calculator to tidy up. When given a ratio it is best to think of it in bits. Step 1: Decide what you want to find and put that on the right. Step 2: Write down the given information. Step 3: Find 1 bit. Step 4: Find what s asked. This is best illustrated with examples. The Dublin School of Grinds Page 100 of 170

103 Demonstration: Divide 40 in the ratio 3 : 2 Solution: Step 1: We want to find Euros, so we put that on the right. Step 2: There are 5 bits (3 + 2 = 5) and we know these 5 bits equal 40 so we put this in a table: Bits 5 Euros 40 Step 3: Ask yourself, how do we get from 5 bits to 1 bit? To get from 5 to 1 we divide by 5 Note: We must get to 1 bit by dividing or multiplying. We must not use addition or subtraction in this section. Whatever we do on the left we must do on the right Our table would now look as follows: Bits Euros 5 ( 5) = 1 40 ( 5) = 8 The Dublin School of Grinds Page 101 of 170

104 Step 4: We must now find how much 3 bits is and how much 2 bits is. How do we get from 1 bit to 3 bits? We multiply by 3. How do we get from 1 bit to 2 bits? We multiply by 2. Bits 5 ( 5 ) = 1 Euros 40 ( 5 ) = 8 ( 3) = 3 ( 3) = 24 ( 2) = 2 ( 2) = 16 Answer = 24 : 16 Q17.1. Q17.2. The Dublin School of Grinds Page 102 of 170

105 Demonstration: 770 is shared between Paul, David and Stephen. Stephen receives double what David receives and Paul receives double what Stephen receives. How much does each boy receive? Solution: David : Stephen : Paul Bits 7 ( 7 ) = 1 Euro 40 ( 7 ) = 110 ( 2 ) = 2 ( 4 ) = 4 ( 2 ) = 220 ( 4 ) = 440 Answer: David receives 110 Stephen receives 220 Paul receives 440 The Dublin School of Grinds Page 103 of 170

106 Demonstration: The ratio of home fans to away fans at a recent Leinster versus Munster game, played in Leinster was 7 3. There were 15,000 Munster fans at the game. How many Leinster fans were at the game? Solution: Bits 3 ( 3 ) = 1 Fans 15, 000 ( 3 ) = 5, 000 ( 7 ) = 7 ( 7 ) = 35, 000 Answer = 35,000 Notice that I didn t start with the total number of bits. I put down what was given (as Step 2 says) which was the number of Munster fans, which was 3 bits. Q17.3. The Dublin School of Grinds Page 104 of 170

107 Q17.4. NOTE: If there are fractions in the ratio, you should get rid of the fractions by multiplying each number by the biggest number on the bottom. For example, if the examiner gave you the ratio you could multiply everything by 4 Now multiply again by 3 = 6: 4: 3 And then you could continue the question without awkward and ugly fractions. Q17.5. The Dublin School of Grinds Page 105 of 170

108 Q17.6. NOTE: If there is mixed number fractions in the ratio, change them to normal fractions and then use the rule above For example, 4! :! 2! could be written as! 13:7. which we would then multiply by 3 to get The Dublin School of Grinds Page 106 of 170

109 Q17.7. Peter and Anne shared a lotto prize in the ratio 3! 2!. Peter s share was 3,500. What was!! Anne s share? Q17.8. The Dublin School of Grinds Page 107 of 170

110 18) Currency Exchange We will use the exact same steps as in ratios. Demonstration: If the exchange rate is 1 $1.21, find the value of a) 7 in dollars b) $10 in Euros a) Step 1: Rewrite the exchange rate and put in the value of the currency which we were told to convert ( 7). Call the missing figure x. Euros 1 Dollars $ $x Step 2: Cross-multiply (disregard currency signs until the final answer). 1x = 8.47 Step 3: Put the correct currency before your answer The Dublin School of Grinds Page 108 of 170

111 b) Euros 1 Dollars $1. 21 x $10 Step 2: Cross-multiply (disregard currency signs until the final answer). 10 = 1.21x!"!.!" = x = x Step 3: Put the correct currency before your answer. NOTE: Money must be rounded off to two decimal places. Answer = 8.26 The Dublin School of Grinds Page 109 of 170

112 Q18.1. If 1 = US$0.75 = 85 a) How many US dollars would you get for 800? b) How much euro would you get for 595? c) How much euro would you get for $45? d) How much Yen would you get for $67.50? Q18.2. A tourist paid $4620 to a travel agent for a holiday in Ireland, when 1 = $1.32. The cost to the travel agent of organizing the holiday is Calculate, in euro, the profit made by the travel agent. The Dublin School of Grinds Page 110 of 170

113 Q18.3. A family going on holiday to New York exchanged 4,000 to Dollars when the exchange rate was 1 = $1.20. In New York they spend $3,500. When they returned they exchanged the leftover money back to Euros at an exchange rate 1 = $1.25. How much Euro did they get back? 19) Measure In maths and in everyday life, the metric system is the most commonly used method to measure lengths. You must know the following: 1 centimetre = 10 millimetres 1 metre = 100 centimetres 1 kilometre = 1000 metres The Dublin School of Grinds Page 111 of 170

114 Demonstration A: Convert 52 mm to cm. Step 1: Write out how many millimetres equals one centimetre. 10 mm = 1 cm Step 2: We were given millimetres, so put that on the left side, underneath the 10mm. We are looking for centimetres, we will call this x for the moment. We will put this on the right side. 10 mm = 1 cm 52 mm = x cm Step 3: Cross-multiply. (10)(x) = (1)(52) 10x = 52 Step 3: Tidy up x =!"!"! = 5.2 cm Demonstration B: Convert 2.6 kilometres to metres. The Dublin School of Grinds Page 112 of 170

115 Step 1: Write out how many metres equals one kilometre. 1 km = 1000 m Step 2: We were given kilometres, so put that on the left side, underneath the 2.6 km. We are looking for metres, we will call this x for the moment. We will put this on the right side. 1 km = 1000 m 2.6 km = x m Step 3: Cross-multiply. (1)(x) = (2.6)(1000) x = 2600 metres Q19.1. Convert 320 centimetres to metres. The Dublin School of Grinds Page 113 of 170

116 Q19.2. Convert 4.5 kilometres to metres. Q19.3. Convert 98 millimetres to centimetres. Q19.4. Convert 4700 metres to kilometres. The Dublin School of Grinds Page 114 of 170

117 Q19.5. A tortoise moves 8 millimetres every second. How many metres will it travel in one hour? To measure weight (mass), we most commonly use the following units: 1 gram = 1000 milligrams 1000 grams = 1 kilograms 1000 kilograms = 1 tonne To give you an idea of how heavy these units would feel, here are some examples: 1 gram a paper-clip 1 kilogram a bag of sugar 1 tonne a small/medium car Q19.6. Convert 2.6 kilogrammes to grammes. The Dublin School of Grinds Page 115 of 170

118 Q19.7. Convert 4000 grammes to kilogrammes. Q19.8. Convert 3 grammes to milligrammes. Q19.9. Convert 700 milligrames to grammes. The Dublin School of Grinds Page 116 of 170

119 Q A packet of Persil detergent weighs 1.5 kilogrammes. If each wash in a washing machine takes 75 grams, how many washes can be got from a packet? Q Find the cost of one tonne of coal at 11 per 25 kg bag. The capacity of a container is the amount of liquid that it can hold. To measure the capacity of a container we use litres. Can you think of anything that is sold in 1 litre cartons? Everybody thinks of milk! The Dublin School of Grinds Page 117 of 170

120 We need to know: 1 litre = 1000 cm 3 1 litre = 1000 ml Q Express 3500 millimetres in litres. Q Express 2.4 litres in millimetres. Q Express 500 millimetres in litres. Q Express 0.8 litres in millimetres. The Dublin School of Grinds Page 118 of 170

121 Q A tube of Colgate holds 75 millimetres. How many tubes can be filled from a container which holds 2!! litres? The Dublin School of Grinds Page 119 of 170

122 20) Distance/Time Graphs Q20.1. The Dublin School of Grinds Page 120 of 170

123 Q20.2. The Dublin School of Grinds Page 121 of 170

124 21) Distance/Speed/Time The key to this section is Dad s Silly Triangle Whichever letter you re looking for, just cover it. i.e.: Distance = Speed Time Speed Time = Distance Time = Distance Speed Make sure to include the units in your answer. Demonstration: A car travels 540 kilometres in 6 hours. What speed did it travel at? Solution Speed = Distance Time = = 90 => 90km/h Warning: Be careful when dealing with time. For example 1.2 hours isn t 1 hour and 20 minutes. 1.2 hours is 1 hour and 12 minutes ( = 12). The Dublin School of Grinds Page 122 of 170

125 Q21.1. A train travels at a speed of 32 kilometres per hour and travels a distance of 240 kilometres. how many hours and minutes did it take to complete its journey? Q21.2. A coach travels from the station to the beach, a distance of 576 kilometres away in 6 hours. The coach is only allowed to travel at a maximum speed of 90 km/h. Did the coach break the speed limit? Q21.3. The Dublin School of Grinds Page 123 of 170

126 22) Solutions The Dublin School of Grinds Page 124 of 170

127 Section 1: Adding and Subtracting Q1.1. Q1.2. Q1.3. Q1.4. Q1.5. Q1.6. Q1.7. Q Section 2: Place Value Q2.1. i) ii) iii) iv) v) vi) Units Hundreds Tens Tenths Ten-thousands Tens Q2.2. i) ii) iii) iv) ,863 1,600,000 9,008 Q2.3. The Dublin School of Grinds Page 125 of 170

128 Section 3: Highest Common Factor Q3.1. Q3.2. Q3.3. Q3.4. Q3.5. Q3.6. Q3.7. Q , 2, 3, 4, 6, 12. 1, 2, 4, 5, 10, 20. 1, 2, 4, 5, 10, 20, 25, 50, , 5, 7, Section 4: Prime Numbers Q4.1 No, 50 has more than two factors. (1, 2, 5, 10 etc) Q4.2. Yes, 23 has only two factors: {1, 23} Q4.3. No, 1 only has one factor: {1} Q4.4. Yes, 2 has only two factors: {1, 2} Q4.5. i) False The Dublin School of Grinds Page 126 of 170

129 ii) iii) True False Section 5: Lowest Common Multiple Q5.1. Q5.2. Q5.3. 7, 14, 21, 28, , 80, 88. Multiples of 2: 2, 4, 6, 8, 10, 12, 14, Multiples of 7: 7, 14, 21 LCM = 14 Q5.4. Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, Multiples of 8: 8, 16, 24, 32, 40, 48, 56, LCM = 56 Q5.5. Multiples of 15: 15, 30, 45, 60, Multiples of 20: 20, 40, 60, LCM = 60 Q5.6. Multiples of 12: 12, 24, 36, Multiples of 18: 18, 36, LCM = 36 Q5.7. Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, Multiples of 5: 5, 10, 15, 20, 25, 30, Multiples of 10: 10, 20, 30, LCM = 30 The Dublin School of Grinds Page 127 of 170

130 Q5.8. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, Multiples of 11: 11, 22, 33, 44, 55, 66, Multiples of 33: 33, 66, LCM = 66 Section 6: Squares and Square Roots Q6.1. i) ii) iii) (9)(9) = 81 (3)(3)(3) = 27 (5)(5) (4)(4) = 9 iv) (10)(10) + (9)(9) = 181 Q6.2. i) ii) iii) The Dublin School of Grinds Page 128 of 170

131 Section 7: Sets Q7.1. i) ii) iii) iv) v) vi) {1, 2, 3, 4, 5, 6} {11, 13, 15, 17, 19} { 3, 4, 5, 6, 7, 8} {2, 3, 5, 7, 11, 13} {12, 24, 36, 48} Divisors of 21: {1, 3, 7, 21} So the Prime Numbers that are divisors of 21 = {3, 7} Q7.2. i) ii) iii) iv) = = 1 Q7.3. i) ii) Q7.4. The number of elements in A = 4 The number of elements in B = 5 { }, {20}, {40}, {60}, {20, 40}, {20, 60}, {40, 60}, {20, 40, 60} Q7.5. i) ii) {1, 2, 3, 4, 5, 6, 7, 9} {3, 5} Q7.6. (M N) = {3, 4, 5, 6} The Dublin School of Grinds Page 129 of 170

132 (M N) = {4, 5} Q7.7. i) ii) iii) iv) {4, 6, 8, 10, 12} {8, 10} The number of elements in (X Y) = 2 The number of elements in (X Y) = 5 Q7.8. i) ii) {2, 4, 6, 8, 10, 12} iii) {10, 12} iv) The numbers of elements in A = 6 The Dublin School of Grinds Page 130 of 170

133 Q7.9. i) ii) iii) iv) v) Q7.10. {2, 4, 6, 8, 9} {1, 3, 5, 7, 9, 10} {9} {1, 3, 5, 7, 10} Start in the centre and work outwards. i) ii) 5 The Dublin School of Grinds Page 131 of 170

134 iii) 5 Q7.11. i) ii) iii) Q7.12. i) ii) 21 iii)!"!""!"! = 45% The Dublin School of Grinds Page 132 of 170

135 Q7.13. i) ii) 14 iii) Q7.4.!!""!"! = 12.5% i) ii) The Dublin School of Grinds Page 133 of 170

136 Q7.15. i) {3, 4, 6, 8, 9, 12} ii) {1, 3, 5, 7, 9, 11} iii) The number of elements in R = 3 Q7.16 a) G/B represents the sports that only girls play in a Limerick school. b) G B represents the sports that both boys and girls play in a Limerick school. Q7.17. i) ii) The Dublin School of Grinds Page 134 of 170

137 Q7.18. a) b) c) d) In the above example, A\B represents people who like apples but who do not like bananas. The Dublin School of Grinds Page 135 of 170

138 Q7.19. a) b) ii) A\(B C) represents the elements in A but not B and C = {3, 6} iii) (B C) represents the elements B and C have in common. = {1, 2, 4} iv) Divisors of 6: 1, 2, 3, 6. Divisors of 8: 1, 2, 4, 8. Divisors of 20: 1, 2, 4, 5, 10, 20. Common Divisors are 1 and 2. Q7.20. a) {2, 3, 5, 7, 11, 13, 17, 19} b) {1, 2, 4, 5, 10, 20} The Dublin School of Grinds Page 136 of 170

139 Q7.21. i) ii) {w} {w, x, y} Q7.22. (a) (b) Q7.23. The Dublin School of Grinds Page 137 of 170

140 Section 8: Order of Operations - BPMDAS Q = 12 Q = 48 Q8.3.!""!!!! (!"!!)!"!!!!!!!"!!"!!!"!! =!!! The Dublin School of Grinds Page 138 of 170

141 Q8.4.!!!!"!!!"!" = 1 Q8.5. Section 9: Commutative & Associative Properties Q9.1. Plain English version: When we add numbers, it doesn t matter in which order that they are placed. i.e.: = (Official language: a binary operation is commutative if changing the order of the operands does not change the result.) Q9.2. Plain English version: When we add numbers, it doesn t matter which way we group them. i.e.: (5 + 3) + 6 = (6 + 3) + 5 (Official language: Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed.) The Dublin School of Grinds Page 139 of 170

142 Section 10: Significant Figures Q10.1. i) ii) iii) iv) 10,000 14,000 14,100 14,080 Q10.2. i) ii) Section 11: Number Pyramids Q11.1. Q11.2. The Dublin School of Grinds Page 140 of 170

143 Q11.3. Section 12: Writing a number as a Product of its Prime Factors Q12.1. (2)(2)(3)(5) = 60 Q12.2. (2)(3)(5)(5) = 150 The Dublin School of Grinds Page 141 of 170

144 Q12.3. (2)(2)(3)(43) = 860 Q12.4. (2)(41) = 92 Section 13: Integers Q C + 19 C = 24 C Q13.2. i) ii) iii) iv) The Dublin School of Grinds Page 142 of 170

145 Q13.3.!!"! = 8 Q13.4.!"!! = 28 Q13.5.!!" (!!)!!"!""!!" = 12 Q13.6. i) Correct answers: (9)(3) = 27 points Incorrect answers: (6)( 1) = 6 points Total score = 21 points ii) Correct answers: (12)(3) = 36 points Incorrect answers: (3)( 1) = 3 points Total score: 33 points iii) Correct answers: (14)(3) = 42 points Incorrect answers: (1) ( 1) = 1 point Total score: 41 points The Dublin School of Grinds Page 143 of 170

146 Q13.7. Q C 10 C = 8 C 12 C + 10 C = 22 C Section 14: Fractions Q14.1. i) ii) iii) Mixed Number Improper fraction Proper fraction Q14.2.!! Q14.3.!! Q14.4.!!" Q14.5.!! Q14.6.!! Q14.7.!! Q14.8. LCM = 35!"!"!"!"!!" =!! The Dublin School of Grinds Page 144 of 170

147 Q14.9. LCM = 6!!!! =!! Q Multiply the bottom number (5) by the big number (8) and add the number on top (1). Put your answer over the denominator. 5 8 = = 41 =!"! Q Multiply the bottom number (9) by the big number (3) and add the number on top (4). Put your answer over the denominator (the bottom number). 9 3 = = 31 =!"! Q Multiply the bottom number (5) by the big number (2) and add the number on top (4). Put your answer over the denominator. 5 2 = = 14 Keep the minus sign. =!"! The Dublin School of Grinds Page 145 of 170

148 Q Use the same method as the three previous questions.! +!"!! Find the LCM. LCM = 12!" +!"!"!" =!"!" Q14.14.!"!"!! Find the LCM. LCM = 8!"!"!! =!! Q i) Ronaldo and Bale together:! +!!!" Find the LCM. LCM = 20! +!!"!" =!"!" Therefore, Falcao got! of the votes!" The Dublin School of Grinds Page 146 of 170

149 ii) Gareth Bale won the award as he received the highest amount of votes. Q Multiply top by top, bottom by bottom.!!!!!"!" =!! Q14.17.!!!"!!"!" =!! Q Convert the Mixed Numbers into Improper Fractions:!!"!! Multiply top by top, bottom by bottom.!"#!" Simplify =!"! The Dublin School of Grinds Page 147 of 170

150 Q Convert the Mixed Number into an Improper Fraction:!"!!! =!"! Q Flip the second term and multiply:!!!!!"!" =!"! Q Flip the second term and multiply:!!!"!!"!" =!! Q Flip the second term and multiply:!!!! =!"! Q Convert the Mixed Numbers into Improper Fractions:!"!"!! Flip the second term and multiply:!"!!!"!"!" =!! The Dublin School of Grinds Page 148 of 170

151 Q Convert the Mixed Numbers into Improper Fractions:!"!!! Flip the second term and multiply:!"!!!!"!" =!"! Q Convert the Mixed Numbers into Improper Fractions:!"!"!! Flip the second term and multiply:!"!!!"!"#!" =!"!" Q Convert the Mixed Number into an Improper Fraction:!"!"!! Flip the second term and multiply:!"!!!"!"!" =!"!" The Dublin School of Grinds Page 149 of 170

152 Q14.27.! +!!! Find the LCM. LCM = 15! +!!"!"!!!" So she saves!!" Q He spends 2! hours (or 150 minutes) on study altogether.! i) Studying Irish:!"!"# Simplify: =!! ii) Studying Maths:!"!"# Simplify: =!! iii) He spends! of his time studying Irish and! of his time studying Maths.!! Therefore, he must spend the rest of the time studying Science.! +!!! Find the LCM. LCM = 15! +! =!!!"!"!" The Dublin School of Grinds Page 150 of 170

153 So Andy spent! of his time studying Science.!" Q14.29.!! of the guests = 180 Therefore!! = 90 So!! = 450 people Q14.30.!! of Tom s money = 65!!!! of Tom s money = 13 of Tom s money = 117 Q14.31 (a) So he had 117 at first. (b) 3 8 = (c) (d) The Dublin School of Grinds Page 151 of 170

154 Q The bigger of the two numbers is further to the right. (iv) Answer: Tim is incorrect. Reason: The strip in A is!! whereas the strip in B is!!. Q Section 15: Percentages Q15.1. Q15.2. Q15.3. Q15.4. Q15.5. Q % 40% The Dublin School of Grinds Page 152 of 170

155 Q15.7. Q15.8 Q % 31.3% Maths: 56% Science: 55% We can see that he achieved a higher percentage in Maths. Q = 198!"#!"" = 99%!""! Q % of 400 = 360 He has 365 saved. Therefore, he can afford to buy the Playstation 4. Q % = 70 Divide by 20 to find 1% 1% = % = 350 Q % = Divide by 25 to find 1% 1% = % = Q % of 520 = So now he earns = Q % = The Dublin School of Grinds Page 153 of 170

156 Divide by 105 to find 1% 1% = % = 850 Q % = Divide by 108 to find 1% 1% = % = 540 Q % = 24,000 Divide by 75 to find 1% 1% = % = 32,000 Q % = 7 g Divide by 14 to find 1% 1% = 0.5 g 100% = 50 g The Dublin School of Grinds Page 154 of 170

157 Section 16: Perimeter and Area Q16.1. i) ii) 3 m + 6 m + 5 m + 3 m + 15 m + 3 m + 7 m + 6 m = 48 m Area of top section: Area of a rectangle: Length Width (6 m)(3 m) = 18 m 2 Area of bottom section: Area of a rectangle: Length Width (3 m)(15 m) = 45 m 2 Add area of rectangle A and rectangle B: 18 m m 2 = 63 m 2 Q16.2. i) 8 m + 3 m + 5 m + 4 m + 3 m + 7 m = 30 m ii) The Dublin School of Grinds Page 155 of 170

158 Area of a rectangle: Length Width (8 m)(3 m) = 24 m 2 Area of a rectangle: Length Width (3 m)(4 m) = 12 m 2 Add area of top section and bottom section: 24 m m 2 = 36 m 2 Q16.3. Area of a triangle:!! base height!! 5 cm 6 cm = 30 cm 2 Q16.4. Area of rectangle Area of triangle = Area needed for red ink The Dublin School of Grinds Page 156 of 170

159 Area of a rectangle: Length Width (7 cm)(4 cm) = 28 cm 2 Area of a triangle formula:!!!! base height 2 cm 3 cm = 3 cm 2 So the area of red ink: 28 cm 2 3 cm 2 = 25 cm 2 Q16.5. i) Area of one triangular lawn:!!!! base height 5 m 5 m = 12.5 m 2 Area of all the lawns: 12.5 m 2 4 = 50 m 2 Area of Patio = Area of square garden Area of the 4 lawns. The Dublin School of Grinds Page 157 of 170

160 ii) Area of square garden: (10 m)(10 m) = 100 m m 2 50 m 2 = 50 m 2 Step 1 - Find area of vertical blue region Q16.6. Area of a rectangle: Length Width (20 cm)(70 cm) = 1400 cm 2 Step 2 Find the area of the horizontal blue region Area of a rectangle: Length Width (10 cm)(90 cm) = 900 cm 2 But, we have double counted where the vertical and horizontal regions cross. Therefore, we must find the area of this common section and subtract it from our two totals. Step 3 Find the area of this common region The Dublin School of Grinds Page 158 of 170