Year 7 Algebraic Expressions Objectives: Appreciate the purpose of algebraic variables and simplify algebraic expressions. Substitute into algebraic expressions. Form algebraic expressions from worded information. Last modified: 4th May 2024 INTRO :: What is algebra? Algebra concerns representing missing information. Put simply, we use letters, known as variables, to (usually) represent numbers. Usually the value of variables are not initially known, but we hope to combine available information to find their value. Examples: 𝒂 might represent someone’s age this year. Source: Google 𝜽 might represent an unknown angle. INTRO :: Examples of algebraic expressions Suppose 𝑎 represented your current age. What would these expressions represent? 𝑎+4 Your age in 4? years time. 2𝑎 𝑎 3 ? Twice your age. ? age. A third of your A few variable naming conventions: We tend to use a single lower-case letter, either using the English alphabet (a to z) or using the greek alphabet (𝛼, 𝜃, 𝜆, 𝜇) INTRO :: Two stages of algebraic problems Worded problem [JMC 2008 Q18] Granny swears that she is getting younger. She has calculated that she is four times as old as I am now, but remember that 5 years ago she was five times as old as I was at that time. What is the sum of our ages now? We won’t solve this now, but how would we approach such a problem? Stage 1: Represent problem algebraically Stage 2: ‘Solve’ equation(s) to find value of variables. Let 𝑎 be my age and 𝑔 be Granny’s age. 𝑔 − 5 = 5𝑎 − 25 𝑔 = 5𝑎 − 20 5𝑎 − 20 = 4𝑎 𝑎 = 20 ∴ 𝑔 = 80 𝑔 = 4𝑎 𝑔 − 5 = 5(𝑎 − 5) These next few lessons we’ll be looking at Stage 1. Stage 2, ‘solving’, we’ll do later this year. Being able to do these two stages for difficult problems is a vital skill for Maths Challenges/Olympiads. Algebraic Simplification – Adding/Subtracting How does this ‘simplify’? Why conceptually does it work? 4𝑎 + 3𝑎 → 𝟕𝒂? If you had “4 lots of 𝑎” and added “3 lots of 𝑎”, we’d clearly have “7 lots of 𝑎”, i.e. 7𝑎 More Examples: Bro Note: An algebraic 𝑥 is written using two back-to-back c’s. Do NOT write as a × symbol. 3𝑥 + 7𝑥 − 𝑥 → 𝟗𝒙? 3𝑎 + 4𝑏 + 𝑎 − 2𝑏 → 𝟒𝒂 +? 𝟐𝒃 We ‘collected’ the 𝑎 terms together and the 𝑏 terms together. We say we ‘collected like terms’. Let’s do an activity based on ‘like’ terms. ACTIVITY :: Collecting Like Terms Instructions: In pairs, discuss which terms you think might be ‘like’ terms, i.e. they could be combined together into one when adding/subtracting. 9x x2 3x3 2x2 2x2y x -1 -x3 5xy 4x2y 2 y Therefore, terms are ‘like terms’ if: The involve the same variables and?powers. -3x2 5xy2 +4 Quickfire Examples 𝑥3 + 𝑥2 + 𝑥2 + 𝑥 → 𝒙𝟑 + 𝟐𝒙?𝟐 + 𝒙 𝑥 2 𝑦 + 3𝑥 3 − 𝑥 2 𝑦 + 𝑥𝑦 2 ? 𝟐 → 𝟑𝒙𝟑 + 𝒙𝒚 3𝑥 2 + 4𝑥 − 𝑥 2 + 𝑥 − 3 ? −𝟑 → 𝟐𝒙𝟐 + 𝟓𝒙 3𝑥𝑦 + 3𝑦 + 2 + 1 ? +𝟑 → 𝟑𝒙𝒚 + 𝟑𝒚 8𝑥 2 − 3𝑥 − 𝑥 2 − 4𝑥 ? → 𝟕𝒙𝟐 − 𝟕𝒙 A common Schoolboy ErrorTM 9 − 3𝑥 + 2𝑥 → 𝟗 −? 𝒙 You might be tempted to simplify to 9 − 5𝑥. But we saw earlier with BIDMAS that addition and subtraction have the same precedence. −3 + 2 = −1, so we have −1 lots of 𝑥. Some find it helpful to underline each term (with the + or – symbol on the front) when collecting like terms. ACTIVITY :: Addition Pyramids 𝑎 + 3𝑏 You should have printed the following pyramids. Each block is the sum of the two below it, e.g. as per on the right. Can you fill in the missing blocks? 2 5𝑥 + ? 5𝑦 1 3𝑥?+ 𝑦 𝑥 − 2𝑦 2𝑥 2?− 𝑥 𝑥2 𝑦 𝑎+𝑏 3𝑥 2 −?𝑥 + 3 2𝑥 + ? 4𝑦 2𝑥 + 3𝑦 2𝑏 𝑥2 ? +3 𝑥2 − 𝑥 𝑥+3 4 3 10𝑎 + 2𝑏 𝑎 𝑎+ ?𝑏 −3𝑥 ? 5𝑎 ? +𝑏 5𝑎 + 𝑏 2𝑎 ? +𝑏 0 3𝑎 2𝑎 ? +𝑏 2𝑎 ? −𝑏 −5𝑥 − 2𝑦 2𝑏 ? −5𝑥?− 7𝑦 5𝑦 3𝑥 ? 2𝑥 + ? 2𝑦 𝑥 −?2𝑦 2𝑥 − 3𝑦 −𝑥?+ 𝑦 Multiplying In algebra, we don’t like the × symbol; instead we put things next to each other to indicate they are multiplied. 𝑥×𝑦 𝑦×𝑥 𝑥×2 𝑥×𝑥 𝑥 × 𝑥𝑦 𝑦 × 𝑥𝑦 2𝑥 × 𝑥 2𝑥𝑦 × 3𝑥𝑦 3𝑥𝑦 × 3𝑦𝑧 → → → → → → → → → 𝒙𝒚 ? 𝒙𝒚 ? 𝟐𝒙 ? 𝒙𝟐 ? 𝒙𝟐 𝒚? 𝒙𝒚𝟐? 𝟐𝒙𝟐? 𝟔𝒙𝟐?𝒚𝟐 𝟗𝒙𝒚?𝟐 𝒛 Test Your Understanding (so far) Simplify the following. a b c d e 3𝑥 + 3𝑥 3𝑥 × 3𝑥 9𝑦 2 − 7𝑦 + 4𝑦 − 𝑦 2 7𝑎𝑏 × 𝑏 8𝑞𝑟 2 × 3𝑞𝑟 → 𝟔𝒙 ? → 𝟗𝒙𝟐 ? → 𝟖𝒚𝟐 −? 𝟑𝒚 → 𝟕𝒂𝒃𝟐 ? → 𝟐𝟒𝒒𝟐 𝒓?𝟑 Division 6 9 → 2 ? 3 Fractions are ultimately just divisions. How did we simplify this fraction? Can we apply the same principle to algebraic division? 2𝑥 → 2 5𝑥 𝑥 𝒙? → 𝟓? 2𝑥 2 𝟐𝒙 ? → 3𝑥 𝟑 10𝑥𝑦 15𝑥 𝟐𝒚 → ? 𝟑 𝑥 2𝑦 𝒙 → ? 2 𝑥𝑦 𝒚 12𝑥𝑦 2 𝑧 16𝑥𝑦𝑧 2 𝟑𝒚 → ? 𝟒𝒛 Test Your Understanding 6𝑧 → 3 7𝑥 2 𝑦 2 𝑥𝑦 2 2 3 5𝑎 𝑏 𝑐 20𝑎𝑏 2 𝑐 𝟐𝒛 ? → 𝟕𝒙𝒚 ? 𝟐 𝒂𝒄 ? → 𝟒 9𝑥𝑦 2 8𝑥 2 𝑦 2 𝑥 −𝑦 𝑦 2𝑦 2 → 𝟓𝒙𝟐? 𝒚 Exercise 1 1 a b c d e f g h i j 2 a c e g i j k l Simplify the following, or write ‘already simplified’. 3 8𝑥 + 9𝑥 − 𝑥 + 𝑦 → 𝟏𝟔𝒙?+ 𝒚 4𝑧 2 + 3𝑧 → 𝑨. 𝑺.? 9𝑞 + 5𝑟 2 − 3𝑞 → 𝟔𝒒 +?𝟓𝒓𝟐 8𝑥𝑦 + 𝑥𝑦 2 → 𝑨. 𝑺.? 10𝑎 + 10𝑏 + 𝑏2 → 𝑨. 𝑺.? 10𝑎𝑏 + 𝑎 + 𝑎𝑏 → 𝟏𝟏𝒂𝒃?+ 𝒂 9𝑥𝑦 − 11𝑥𝑦 → −𝟐𝒙𝒚 ? 2 𝟐 1 + 𝑥 − 2 − 𝑥 + 𝑥 → 𝒙 −?𝟏 𝑥 − 2𝑦 + 2𝑥 → 𝟑𝒙 −?𝟐𝒚 𝑥 2 𝑦 2 + 𝑥𝑦 2 + 𝑥𝑦 − 2𝑥𝑦 2 𝟐 + 𝒙𝒚 → 𝒙𝟐 𝒚𝟐 − 𝒙𝒚 ? a c e g i k 8𝑣 → 𝒗 8 𝑐2 → 𝑨. 𝑺. 2 9𝑥𝑦 → 𝟑𝒙 3𝑦 9𝑞𝑟 𝟑𝒒 → 6𝑟 2 𝟐𝒓 2𝑗𝑘 𝟏 → 4𝑗 2 𝑘 𝟐𝒋 5 2 ? ? ? ? ? b d f h j 8𝑣 → 𝟖 𝑣 8𝑣 2 → 𝟐𝒗 4𝑣 14𝑥 2 𝑦 → 𝟏𝟒𝒙 𝑥𝑦 20𝑥 2 𝑦 2 𝑧 2 → 𝟓𝒚𝒛 4𝑥 2 𝑦𝑧 𝑥2𝑦2 𝒙𝒚 → 2𝑥𝑦 𝟐 ? ? ? ? ? 45𝑥 𝑦 𝟓𝒙𝟐 → ? 54𝑥 3 𝑦 7 𝟔𝒚𝟓 𝑥2 4 a 2𝑥 + → 𝟑𝒙 ? 𝑥 𝟐 3𝑎𝑏 × 3 → 𝟗𝒂𝒃? b 2𝑏 × 𝑏 → 𝟐𝒃? 12𝑥 2 𝑦 𝟐 𝟐 d 2𝑎𝑏 × 𝑏 → 𝟐𝒂𝒃 3𝑎 × 3𝑎 → 𝟗𝒂 ? ? 𝟐 b 3𝑥 − 3𝑥𝑦 → 𝒙𝒚? 𝟐 f 7𝑎𝑏𝑐 × 3𝑏 → 𝟐𝟏𝒂𝒃 2𝑎𝑏 × 𝑎 → 𝟐𝒂 ? 𝒃 𝒄 ? 2 2 12𝑥 𝑦 𝟐 𝟐 𝟐 h 8𝑎𝑐 × 2𝑎𝑏 → 𝟏𝟔𝒂 2𝑎𝑏 × 𝑎𝑏 → 𝟐𝒂 ?𝒃 ? 𝒃𝒄 c − 𝑥𝑦 + 𝑦 → 𝟑𝒙𝒚?+ 𝒚 2 2 𝟑 𝟑 3𝑥𝑦 3𝑥 𝑦 × 4𝑥𝑦 → 𝟏𝟐𝒙? 𝒚 2 3 3 𝟑 𝒃𝟐 𝒄𝟑 3𝑥 𝑦𝑧 𝑥 𝑦 4𝑎𝑏𝑐 × 4𝑎2 𝑏𝑐 2 → 𝟏𝟔𝒂? 2 − d 2𝑥 𝑦 + 𝟔 𝑦𝑧 𝑥𝑦 2 5𝑎 3 2 → 𝟐𝟓𝒂? 2𝑎 2 2 × 3𝑎3 3 → 𝟏𝟎𝟖𝒂 → 𝒙𝟐 𝒚 + ?𝟏𝟑 ? 𝟑𝒙𝟐 Substitution 2 What is the value of 3𝑥 when 𝑥 = 2? (Click answer) 12? 36? We saw by BIDMAS earlier that because indices come first, the 𝑥 is first squared, THEN multiplied by 3. 3𝑥 2 means 3 𝑥 2 not 3𝑥 2 . If you think of 3𝑥 2 as “3 lots of 𝑥 2 ” you’re less likely to make an error. Substitution If 𝑎 = 2, 𝑏 = −3 and 𝑐 = −4, what is the value of: Terms 𝑎2 + 2𝑏 = 𝟒 + (−𝟔) ? = −𝟐 𝑏𝑐 − 𝑐 = 𝟏𝟐 − −𝟒 ? = 𝟏𝟔 𝑏 2 − 𝑎𝑐 = 𝟗 − −𝟖 ? = 𝟏𝟕 Bro Tips: Start by working out each of the terms first (mentally if you can) leaving the +/- symbols between as they are. Don’t try to do all at once. Another Example If 𝑥 = −1, 𝑦 = 5 and 𝑧 = −3, what is the value of: ? 6𝑧 2 = 𝟓𝟒 ? 10𝑥 2 = 𝟏𝟎 4𝑧 = −𝟏𝟐 ? 𝑥 + 2𝑧 = −𝟏 + −𝟔 ? = −𝟕 𝑦 2 − 𝑧 2 = 𝟐𝟓 − 𝟗 ? = 𝟏𝟔 𝑥𝑧 + 𝑦𝑧 = 𝟑 + −𝟏𝟓 = −𝟏𝟐 ? Test Your Understanding If 𝑥 = −6, 𝑦 = −3 and 𝑧 = 2, what is the value of: ? 5𝑧 2 = 𝟐𝟎 ? 4𝑦 2 = 𝟑𝟔 𝑦𝑧 + 𝑥 2 = −𝟔 + 𝟑𝟔 ? = 𝟑𝟎 𝑧 − 𝑥𝑦 = 𝟐 − 𝟏𝟖 ? = −𝟏𝟔 2𝑥 − 𝑦 2 𝑧 = −𝟏𝟐 − 𝟏𝟖 ? = −𝟑𝟎 Formulae A formula (plural: formulae) is a rule to generate one value of interest from others. For example, the following formula allows you to find the temperature in Fahrenheit given the temperature in Celsius: 9 𝐹 = 𝐶 + 32 5 The variable of interest goes on the LHS of the equals. What is 𝐹 when: 𝐶 = 10 𝐶=0 𝐶 = 30 ? 𝟒𝟎 → 𝑭 = 𝟏𝟖 + 𝟑𝟐 = ? → 𝑭 = 𝟑𝟐 → 𝑭 = 𝟓𝟒 + 𝟑𝟐 = ? 𝟖𝟔 Exercise 2 1 If 𝑥 = 3, 𝑦 = 4 and 𝑧 = 5 what is the value of: 𝑦 2 + 2𝑧 = 𝟏𝟔 + 𝟏𝟎 a ? = 𝟐𝟔 b 4𝑥 2 = 𝟑𝟔 ? c 𝑥 − 𝑦𝑧 = 𝟑 − 𝟐𝟎?= −𝟏𝟕 2 d 𝑦𝑧 = 𝟏𝟎𝟎 ? 4 If 𝑥 = −3, 𝑦 = −5, 𝑧 = 7, what is: = 𝟏𝟓𝟎 a 6𝑦 2 ? b 𝑥𝑦 + 𝑥𝑧 = 𝟏𝟔 + −𝟐𝟏 = −𝟓 ? ? = 𝟗𝟖 c 𝑥 2 𝑧 − 𝑦𝑧 = 𝟔𝟑 − −𝟑𝟓 d 2𝑥 2 𝑦 2 = 𝟒𝟓𝟎 ? 2 3 e 𝑦 − 𝑥 = 𝟐𝟓 − −𝟐𝟕 ? = 𝟓𝟐 2 If 𝑎 = −2, 𝑏 = −3, 𝑐 = 4, what is: a 2𝑏2 = 𝟏𝟖 ? b 𝑐 2 − 𝑏2 = 𝟏𝟔 − 𝟗 = ? 𝟕 c 𝑏𝑐 + 𝑎 = −𝟏𝟐 + −𝟐 ? = −𝟏𝟒 d 𝑎𝑏 − 𝑏𝑐 = 𝟔 − −𝟏𝟐? = 𝟏𝟖 e 8𝑎2 + 4𝑏 = 𝟑𝟐 + −𝟏𝟐 ? = 𝟐𝟎 5 3 The profit 𝑝 of PippinCo can be calculated using the number of sold items 𝑠 and the number of hours open ℎ and the 1.3𝑠 𝑝= ℎ−5 Calculate the profit if 1420 items are sold and the shop was open 8 hours that day. 𝒑 = £𝟔𝟏𝟓. ? 𝟑𝟑 The smell intensity 𝑆 given the distance 𝑑 in metres from the source is given by 6 If 𝑝 = 5, 𝑞 = −10, 𝑟 = −1, what is: the formula: 2 100 =𝟗 ? a 9𝑟 𝑆 = 3 + 24.1 2 = −𝟐𝟓𝟎 b 𝑝 𝑞 ? 𝑑 What is the smell intensity when the c 𝑝𝑞 − 𝑞𝑟 + 𝑝𝑟 = −𝟓𝟎 − 𝟏𝟎 +? −𝟓 = −𝟔𝟓 distance is 4 metres? 𝟐𝟓. 𝟔𝟔 = 𝟏𝟓 ? d 𝑝 𝑞 − 𝑟 + 𝑞 𝑟 − 𝑝 = −𝟒𝟓 + 𝟔𝟎 ? 2 10 e 𝑞 − 5𝑟 − 𝑟 = 𝟐𝟓 − 𝟏 =?𝟐𝟒 Forming Expressions Worded problem [JMC 2008 Q18] Granny swears that she is getting younger. She has calculated that she is four times as old as I am now, but remember that 5 years ago she was five times as old as I was at that time. What is the sum of our ages now? Stage 1: Represent problem algebraically Stage 2: ‘Solve’ equation(s) to find value of variables. Let 𝑎 be my age and 𝑔 be Granny’s age. 𝑔 − 5 = 5𝑎 − 25 𝑔 = 5𝑎 − 20 5𝑎 − 20 = 4𝑎 𝑎 = 20 ∴ 𝑔 = 80 𝑔 = 4𝑎 𝑔 − 5 = 5(𝑎 − 5) Remember this problem? We’ll be looking how we can turn worded information into algebraic expressions. Forming Expressions Suppose 𝑎 represents your age. How would you represent: Your age in 5 years time? 𝒂+𝟓 ? Twice what your age was 5 years ago? 𝟐(𝒂 − 𝟓) ? 5 years younger than twice your age? 𝟐𝒂 − 𝟓? Half what your age was 3 years ago? 𝟏 𝟐 𝒂−𝟑 𝒂 − 𝟑? 𝒐𝒓 𝟐 Anyone called Bob is four times you age. Anyone called Charles is two years younger than you. What is (in terms of 𝑎): The age of one person called Bob: The total age of a Bob and a Charles: The total age of you, a Bob and two Charles: ? 𝟒𝒂 𝟒𝒂 + 𝒂 − ? 𝟐 = 𝟓𝒂 − 𝟐 𝒂 + 𝟒𝒂 + 𝟐 𝒂 − 𝟐 = 𝟕𝒂 − 𝟒 ? Later this year you’ll properly learn how to ‘expand brackets’. A Harder One “The sum of 5 consecutive whole numbers is 285. What is the smallest of these numbers?” Supposed we used one variable 𝑛. What unknown thing could it represent? Option 1 Option 2 Let 𝒏 be the smallest number. ? Then the five numbers would be: Let 𝒏 be the middle number. ? Then the five numbers would be: 𝑛, 𝑛 + 1, 𝑛 + 2, ? 𝑛 + 3, 𝑛 + 4 𝑛 − 2, 𝑛 − 1,? 𝑛, 𝑛 + 1, 𝑛 + 2 Then the sum of these numbers would be: 5𝑛 +?10 Then the sum of these numbers would be: 5𝑛 ? Why might Option 2 might make the later ‘solving’ stage easier? Check Your Understanding A A cat costs £𝑐 and a dog £2 less. What is the cost (in £) of: a) 4 cats? 𝟒𝒄 b) 3 dogs? 𝟑 𝒄−𝟐 ? ? B There is a queue of 𝑛 people. If there are 𝑞 people in front of me, how many people are behind me? 𝒏−𝒒−𝟏 ? C The average mark of people in a class was 60. a) By introducing a suitable variable for the number of people in the class, what would be the total mark of everyone in the class? Let 𝑛 be the number of people. Total mark = 𝟔𝟎𝒏 b) If a new person joins the class, and gets a mark of 80, what is the total mark now? 𝟔𝟎𝒏 + 𝟖𝟎 c) If this person’s mark made the average mark rises to 62, give another expression for the total mark of all the people. 𝟔𝟐 𝒏 + 𝟏 ? ? ? D A 3 x 3 grid contains nine numbers, one in each cell. Each number is doubled to obtain the number on its immediate right and trebled to obtain the number immediately below it. Use suitable expressions to represent the nine numbers. What expression gives the sum of your numbers? 𝟗𝟏𝒙 𝟔 N If the sum of the numbers is 13, what is the middle number? ? 𝟕 ? 𝒙 𝟐𝒙 𝟒𝒙 𝟑𝒙 𝟔𝒙 ? 𝟏𝟐𝒙 𝟗𝒙 𝟏𝟖𝒙 𝟑𝟔𝒙 Exercise 3 1 Questions on provided sheet. A number is represented by 𝑥. How would we represent: a) 2 more than the number. 𝒙+𝟐 b) 5 times the number. 𝟓𝒙 c) 3 less than than twice the number. 𝟐𝒙 − 𝟑 d) Twice as much as 3 less than the number. 𝟐(𝒙 − 𝟑) 𝒙 𝟏 e) A quarter of the number. or 𝒙 ? ? ? 𝟒 4 ? ? ? ? ? 𝟒 5 2 The cost of a badger is 𝑏 pence. A racoon is 5 pence more expensive than a badger and a beaver three times as expensive as a badger. a) What is the cost of a racoon? 𝒃 + 𝟓 b) What is the cost of a beaver? 𝟑𝒃 c) What is the total cost of a racoon and 8 beavers? 𝟐𝟓𝒃 + 𝟓 6 ? ? ? 3 You have 7 consecutive numbers, with the smallest number 𝑛. a) What are the 7 numbers in terms of 𝑛? 𝒏, 𝒏 + 𝟏, 𝒏 + 𝟐, … , 𝒏 + 𝟓, 𝒏 + 𝟔 b) Hence what is the sum of all the numbers? 𝟕𝒏 + 𝟐𝟏 ? ? After tennis training, Andy collects twice as many balls as Roger and five more than Maria. If Roger collected 𝑟 balls, in terms of 𝑟, how many balls did: a) Andy collect? 𝟐𝒓 b) Maria collect? 𝟐𝒓 − 𝟓 c) The total number of balls collected? 𝟓𝒓 − 𝟓 I think of a number, multiply it by 5, subtract 2, subtract the original number, and then halve it. If the starting number was 𝑥, give an expression for the final answer, as simply as possibly. 𝟒𝒙−𝟐 → 𝟐𝒙 − 𝟏 ? 𝟐 In a list of seven consecutive numbers a quarter of the smallest number is five less than a third of the largest number. If 𝑥 is the smallest number, find expressions for: a) “a quarter of the smallest number” 𝟏 𝒙 𝟒 b) “five less than a third of the largest 𝟏 number” 𝒙+𝟔 −𝟓 ? 𝟑 ? Exercise 3 7 Pippa thinks of a number. She adds 1 to it to get a second number. She then adds 2 to the second number to get a third number, adds 3 to the third to get a fourth, and finally adds 4 to the fourth to get a fifth number. If 𝑛 is the number she started with, what is the sum of her numbers? 𝟓𝒏 + 𝟐𝟎 ? 8 Dilly is 7 years younger than Dally. In 4 years time she will be half Dally’s age. Let Dilly’s age be 𝑎 and Dally’s 𝑏. a) Use the first sentence to write a formula for Dilly’s age in terms of Dally’s: 𝒂 = 𝒃 − 𝟕 b) Use the second sentence to write two different expressions for Dilly’s age in 4 years time, one in terms of 𝑎 and one in terms of 𝑏. 𝟏 𝒂 + 𝟒, 𝒃+𝟒 𝟐 ? ? 9 A woman had 9 children at regular intervals of 15 months. The oldest is now six times as old as the youngest. a) Let 𝑎 be the age of the youngest child (in years). What is the age of the oldest child (be careful!) 𝒂 + 𝟏𝟎 b) Hence use the information to form an equation relating the ages of the oldest and youngest child (you need not solve it). 𝒂 + 𝟏𝟎 = 𝟔𝒂 ? ? 10 Three brothers and a sister shared a sum of money equally among themselves. If the brothers alone had shared the money, then they would have increased the amount they each received by £20. Suppose the total amount of money is 𝑡. a) How much money (in terms of 𝑡) do the brothers each get if they share just 𝟏 between them? 𝒕 𝟑 b) What expression would represent “an increase of £20 from the previous lower 𝟏 amount they would have got” 𝒕 + 𝟐𝟎 ? 𝟒 ? Exercise 3 11 [JMC 2014 Q20] Box P has 𝑝 chocolates and box Q has 𝑞 chocolates, where 𝑝 and 𝑞 are both odd and 𝑝 > 𝑞. What is the smallest number of chocolates which would have to be moved from P to box Q so that box Q has more chocolates than box P? 𝑞−𝑝+2 𝑝−𝑞+2 𝑞+𝑝−2 A B C 2 2 2 D 𝑝−𝑞−2 𝑞+𝑝+2 E 2 2 Solution: ?B 12 [JMO 2007 B1] Find four integers whose sum is 400 and such that the first integer is equal to twice the second integer, three times the third integer and four times the fourth integer. The four numbers could be represented as 𝟏𝟐𝒙, 𝟔𝒙, 𝟒𝒙, 𝟑𝒙 Their sum is 𝟐𝟓𝒙. 𝟒𝟎𝟎 ? Thus 𝒙 = 𝟐𝟒 = 𝟏𝟔 This makes the four numbers 𝟏𝟗𝟐, 𝟗𝟔, 𝟔𝟒, 𝟒𝟖
