Simplifying fractions Remember A fraction has been simplified when the numerator (the top) and the denominator (the bottom) are expressed as whole numbers cancelled dow as far as possible. important when working with probabilities and ratios. They are also used in + calculations in everyday life Remember * Tochangea fraction or a decimal into a percentage, multiply it by 100. © To change a percentage into. fraction, rewrite % as division by 100. ¢ To write a fraction asa decimal, divide the top number by the bottom number. Exercise 1 Simplify these ate Boe ae Copy and complete this table, eivi ions in their lowest terms. 2 the fra Fraction Decimal | Percentage Change each of these toa mixed number. 8° 92 Change each of these to.an improper fraction 10 24 dla Exercise 1* and write each of these as a single fraction 1s 28 3 92 + 64 7 Bxo08 8 03%K_ Simplify and write each of these as an ordinary number. 9 68 = 01 10 765 x 0,001 u 1274 13 25x} 14 46x 2Bastions How many equal parts of a whole Slice a pizza, and we get fractions: Ge @ (One-Half) — (One-Quarter)_ (Three-Eighths) ‘The top number says how many slices we have. The bottom number says how many equal slices the whole pizza was cut into. * PaaTs oF A WHOLE NuMBER L rs = EW - Fractions. S PaRTs OF A WHOLE NUMBER Numerator 1“ ‘we ay or Denominator Numerator / Denominator We call the top number the Numerator, itis the number of parts we have. We call the bottom number the Denominator, it is the number of parts the whole is divided into. Numerator Denominator You just have to remember those names! (If you forget just think "Down"-ominator)What is numerator and denominator? The numerator and denominator of a fraction are the top number and bottom number parts of a fraction. 3 4~ Numerator Denominator 7 4 * A fraction visually displays a division. The numerator represents the dividend and the denominator represents the divisor. 2 3 2+ t Dividend Divisor Equivalent Fractions Some fractions may look different, but are really the same, for example 4g = a = 4p (Four-Eighths) (Two-Quarters) (One-Half) It is usually best to show an answer using the simplest fraction ( '/2 in this case ). That is called Simplifying, or Reducing the FractionSimplifying Fractions To simplify a fraction, divide the top and bottom by the highest number that 1 can divide into both numbers exactly. Simplifying Fractions ‘Simplifying (or reducing) fractions means to make the fraction as simple as possible. Why say foureighths (2) when we realy mean half) 2 4 x 2 4 8 4 2 (Four-Eighths) —(Two-Quarters)__ (One-Half) What is simplifying fractions? Simplifying fractions is reducing fractions to their simplest form. To do this we look at the numerator (the top number) and the denominator (the bottom number) and find a common factor that we can use to cancel the fraction down to its lowest terms. The numerator and denominator are always whole numbers. For example, the fraction J, can be simplified to 4 as both the numerator and denominator have 5 asa factor. The common factor of 5 cancels on the top and the bottom, leaving us with the simplified version. 5 ix 1 10 2x5 2How do | Simplify a Fraction ? ‘There are two ways to simplify @ fraction Method 1 Try to exactly divide (only whole number answers) both the top and bottom of the fraction by 2, 3, 5, 7 pu ete, until we can't go any further. ' r 24 Example: Simplify the fraction 7 : +2 +2 +3 x FS see 2 2 = 2 108 ~ 54 ~ 27 7 9 ew ew QW we) ep. a3. That is as far as we can go. The fraction simplifies to 4 Example: Simplify the fraction 22 ; Dividing by 2 doesn’t work because 35 can't be exactly divided by 2 (35/2 = 17%2) Likewise we can't divide exactly by 3 (10/3 = and also 35/3=114) No need to check 4 (we checked 2 already, and 4 Is just 2x2). But 5 does work! That is as far as we can go. The fraction simplifies to >© SimpLiFY A FRACTION ¢ MAKE THE NUMERATOR AND DENOMINATOR AS SMALL AS POSSIBLE “w= v4 ‘Le 3 * SIMPLIFY A FRACTION * MAKE THE NUMERATOR AND DENOMINATOR AS SMALL AS POSSIBLE cimeury 2 22 1 UP PLE —_—_ — — 12 6 3 20 | ‘Simeuiry = —— a 2 5How to simplify fractions In order to simplify fractions: (@ Look at the numerator and the denominator and find the highest common factor. @ Divide both the numerator and the denominator by the HCF. © write the fraction in its simplest terms. Greatest Common Factor ein The highest number that divides exactly into two or more numbers. It is the "greatest" thing for simplifying fractions! Let's start with an Example ... Greatest Common Factor of 12 and 16 Factors of 12: @ 3, @ 6, 12 + Find all the Factors of each number, + Clecle the Common factors, Factors of 6: Q 8, 16 + Choose the Greatest of those oman Factors Gye eects commer trecte Method 2 Divide both the top and bottom of the fraction by the (Greatest Common Factor] (you have to work it out first!) Example: Simplify the fraction > + The largest number that goes exactly into both § and 12 is 4, so the Greatest Common Factor is 4. Divide both top and bottom by 4: That is as far as we can go. The fraction simplifies to 2Three Types of Fractions There are three types of fraction: x @aun 2 25 | Smaller —> Larger —» Larger —> Smaller—> (or equal) Proper Improper Mixed Fraction cient Fraction ‘The numerator is less than the denominator Examples: 1/3, 9/4, 7/7 A whole number and proper fraction together Examples: 1 1/3, 2/4, 16 7/5 The numerator is greater than (or equal to) the denor Examples: “/, 4/4, 7/7 ——— (or equal to) the bottom number. It is usually “top-heavy” €é ‘An Improper Fraction has a top number larger than B. 4 (seven-fourths or seven-quarters) More Examples 3 z 16 15 100 z 3 15 15 x See how the top number is bigger than (or equal to) the bottom number? ‘That makes it an Improper Fraction, (but there is (nothing wrong about Improper Fractions).Fractions A Fraction (such as 7/4) has two numbers Numerator Denominator ‘The top number (the Numerator) is the number of parts we have. The bottom number (the Denominator) is the number of parts the whole is divided into. Example: 7/4 means: * We have 7 parts + Each part is a quarter (1/4) of @ whole Improper Fraction So an improper fraction is a fraction where the top number (numerator) is greater than or equal to the bottom number (denominator): it is top-heavy. Can be Equal What about when the numerator is equal to the denominator? For example 4/4 ? Well itis the same as @ whole, but itis written as a fraction, so most people agree itis a type of improper fraction ale Improper Fractions or Mixed Fractions We can use either an improper fraction or a mixed fraction to show the same amount. For example 13 = 2, as shown here: 3 x @G -@6Fanctions: improper Mixed. How many wnoves 13 oa Converting Improper Fractions to Mixed Fractions To convert an improper fraction to @ mixed fraction, follow these steps: * Divide the numerator by the denominator. a * Write down the whole number answer + Then write down any remainder above the denominator. Example: Convert +4 to a mixed fraction. Divide: em 11 + 4 = 2 with a remainder of 3 Write down the 2 and then write down the remainder (3) above the denominator (4) Answer: That example can be written like this: atExample: Convert 3 to a mixed fraction. ae 3 «1033 a Rf Answer: Improper <—— Convert ——> MIXED. DIVIDE THE NUMERATOR ‘8¥ THE OewarmaroR 2 1 —_—_—_—_—_——> _— q=4 = An 4 23) bur 2 Dl |p +|»Mixed Fractions or Improper Fractions We can use either an improper fraction or a mixed fraction to show the same amount. For example 12 =<, as shown here! 3 ty Converting Mixed Fractions to Improper Fractions To convert a mixed fraction to an improper fraction, follow these steps: * Multiply the whole number part by the fraction’s denominator. A + Add that to the numerator * Then write the result on top of the denominator. Example: Convert 32 to an improper fraction. Multiply the whole number part by the denominator: wh 3x 5=15 {Add that to the numerator: —m i5+2=17 Then write that result above the denominator: We can do the numerator in one go:Improper <— Convert ——> Example: Convert 2 to an improper fraction. if tam mony 32 Convert 3 —Z- ro an improper FaACTION axs (as+3) Ss 18 Are Improper Fractions Bad ? NO, they aren't bad! Mixed Fraction: what is: 14240 7 2, ois 14244 : 4 orisits 1424 Improper Fraction: whats: 1+ 2 av But, for everyday use, people understand mixed fractions better. at 7 ttiss + Example: It is eas For mathematics they are actually better than mixed fractions. Because mixed fractions can be ‘confusing when we write them in a formula: should the two parts be added or multiplied?{Vhbesvetion will remind you of how to ne do calculations in the correct order. Exercise 2 Caloulate these, 1 (-5)+ 10 219 =(=3} 3 a OT) (48), B(-4) x3 6 12+(-2) (~16) a x22 . 94 Exercise 2” Caleulate these. P4-2%3 2208143 3 16+452 4 ie 4 16+ 5 16+ 244 sop 7 (2x3) 8 yay 92+INE Percentages Percentages are used to compare quantities. The unit of comparison is LOO, and this by term ‘per cent’ is used (‘per’ means divide, and ‘cent? means LOO), Example 1 Pucilie Airlines increased ticket prices by 8%. Calculate the new peice of a $2450 ticket. ‘Doaraing Cloven 18 mine Woaraing PACEFLC: zRENE ha Tl fs stss 6 Wty a Tomar | wana 1 wut TRAE MAY Len at mie ae a ec 36AR = 36K Nace OR Ne 8 ORE is $2646,00 Tho G | megerenme PACIFIC AIRLINES 00%Calculate these. ‘and write each answer in standard form, 7156 10%) + (5.6 x 10%) 8 (3.6 104) = 9.x 102) Calculate these, and write each answer in standard form 9 (45x 104 10. 10)? + (4 107) 11 10% = (3.47% 107) ou! will niged the tinformation in this table to answer Questions.12 and 13 1 Galaxy (neare Copy and complete these sentences 12 The Andromeda Galaxy is... times further away from the Earth than Saturn. 13 Tomakeascale model showing the distances earth sun ofthe four bodies from the Earth, a sindent F=f ————_______+5 1 stance ftom the Exth (km) marks the Sun 1 em from the Barth, hole aca OS How far along the line should the other three celestial bodies be placed Significant figures and decimal places {tis often useful to simplify numbers by writing them either correct to so many signifi figures (s.£) or correct to so many decimal places (d.p.) ees Example 5 Example 6 Write 672.900 correct to 3 significant figures. Write 1.007645 correct to 2 significant figures. 672900 = 673000 (to 3s.) 0.007645 = 0.0076 (to 2s.) ths. =9:9 > 5, S02 rounds up to 3 3rd sil = 4:4 < 5. $0 6 isnot rounded up to 7. (672.900 is closer in value to 673.000 than to (0.007645 is closer in value to 0.0076 than to (1.0077 672.000.) Example 7 Example 8 Write 6.4873 correct to 2 decimal places. Write 23.428 correct to 1 decimal place, 6.4873 = 649 (to 2d.p.) 23428 = 234 (toldp) > 5. So 8 rounds upto 9. 2nd dp. = 2:2 <5, So4 isnot rounded upto 5. in value (0 6.49 than to 6.48.) (23.428 is closer in value to 23.4 than to 23.5.) Exercise 5 1 Write 783 correct to | signilicant figure. } correct 16 3 significant figures 2 Write 3 Correct 0.439 to 2 significant figures: 4 Correct 1.5057 to 3 significant figures.2+ 5=20) 36 53-48 8-38.97 1 xe 97=111 WA 13,085 — w= 12.1 Exercise 10* Solve these for r, giving each ans 1235 4x= 1234 445. ver correct to 3 significant figure 3 396 6 40.9 ——————_AAn$R a Example 9 (Add 5 to both sides) A(y + 3) = 20 (Divide both sides by 4) (Divide both sidesby 3) y+3 (Subtract 3 from both sides) (Check: 3X 4 = x=2 > (Check; 4(2 + 3) = 20) Example 10 2(x+ 3)=9 (Multiply out the bracket) 2x +6 =9 (Subtract 6 from both sides) 3. (Divide both sides by 2) =} (Check; 2( + 3) =9) ———————— Exercise 11 Salve these for x 1 2x+4=10 Sas + 2x4 3)=10 5 5(x—2)=30 7 9=3- 8 26—3x)=6 2 3(10 — 3x)= 30 9 4(2-x)=16 13 7(2=5x)=49 14 The sum of two consecutive numbers is 477. What are the numbers? (Let the first number be x.) 1311 9%@+4)=4r 1 15 Find x and the size of each angle in this triangle, Find x and the Size of each angle. se Asi) 17 ‘The formula for converting degrees Fahrenheit (F) to degrees Celsius (C)is P = 32 + 1.8C. Pind Cwhen Fis 5.Equipment needed: a cylinder with a diameter of 5~ 11) cm (a drinks can or cardbodrd tube is ideal), « length of string about 30 times as long asthe diameter of the cylinder, a ruler and graph paper: © Wrap the string tightly around the cylinder. keeping the turns close toyether, Ask ifriend to draw a straight line across the string while you hold the ends of the string © Unwind the string, Measure the distance of each mark from the first mark. Enter your esults in a table, * Plot these points on a graph of D against M. (Plot D on the vertical axis and Mon the horizontal axis.) Draw the best straight line through these points. * Calculate the gradient of the line, and then write down the equation of the line * The gradient should equal ad, where dis the diameter of the eylindet: Use your gradient towork out an estimate for m. # Repeat the activity with various eylinders, and obtain farther estimates for 7. Graphs of the form ax + bi The graph of 3x + 4y = 12 isa straight line Vhe equation can be rearranged as y = —4r + 3, showing ht line with gradient ~# and gintervept (0, 3) that the graph isa stra An easy way {6 draw or sketch this graph isto find where the graph crosses the axes, Example 4 Skeich the graph of x + 2y = 8. Substituting y = 0 gives x = 8, which shows that (8, 0) lies on the line. Substituting a= (0 gives y ~4. which shows that (0, 4) ics on the line. Exercise 17 For Questions 1~7: find where the graph crosses the axes und sketch the graph, lrty=5 2 ixty=6 3x ty=6 4 3x +2y=12 5 4x + 54/= 20 6 y-2y>4 7 Ay 3y= 24 % A firm selling Cis finds that the number sold (N thousand) is telated to the price (£P) by the formula OP + N= 90, 4 Draw the graph of N against I forO = N= 90 the vertical axty sh and the horlgontal axis should be the N axis), b Lise your graph to find the price when 10000 CDs are sold Use your graph to tind the number sold Af the priee of «C1 is sot at U8. Use your praph to find the price 190.000 ¢ uld be the Paxis, i are vold, Is this a sensible valiveExercise 18* (Revision) 1 Find the gradients of the fines parallel to AB when @ Ais(-2,=1), Bis (4. 2) B Ais(—1.4),Bis(1.-1 2 The Leaning Towerof Pisais 55 m high. and the gradient Oo its Jean is 11. By haw much does the top everhang the bottom? 3 Sketch the following graphs meas 2) ae ee © ly=5—x d 5x4 3y=10 nto (h. 3) 4 Tid bsuch that the ine from the origin to (3, 4h is parallel to the ine from the o 5 ind theegiiation of the lines passing through (6, 4) that are parallel and pecpendicular to By =x + 21. © A temperature F in degrees Fahrenheit is related to the temperature C in degrees Celsius by the forinula F =2€ + 32. a Drs a graph of F against Cfor 50 = C40. and —22 “Find .es Celstus and b Use your graph to estimate 80 Fahrenheit cl -aph to find which temperature fas the same value in both degrees our Fahrenheit nd degrees Celsius, w down the equations that will produce these patterns.Basic principles Triangles (A dashed line indicates ain axis of symme!"Y) Isosceles triangle Pees Acute, obtuse and right angles are possible, Angle properties a+b Equitateral triangle La ‘The rotational symmetry isof order & (Angle sum of triangle) (Angles on straight line are supplementary) (Vertically opposite angles) 180' e+ d= 180° etd+e+f = 360° (Angles at a point) since d= 180°— (Angles on straight line) and @+b=180°~c (Angle sum of triangle) Parallel lines Alternate angles are equal Co-imerior angles a+ b= 180" Quadrilaterals Square Ps © Rotational symmetry ‘of onder 4 Rhombus Rotational symmetry of order? Kite d=a+b (Exterior angle of triangle) ‘Corresponding angles a areequal Rectangle RoThe diagram represents a rectangular tase A” sprinkler at the point, halfway’ between © 3 ‘Thesprinkler wets the area within 15m from Using a scale of Lem (o 5m, draw a diagram anid shade the area wetted by the spritikler A child is playing on the lawn. She starts al and then runs across the lawn. keeping the Same distance from the sides AD end AB Until she is 10 m from the side DC She then rns straight to the corse" B Draw the path that the child tabes ‘onto your diagram, What length of her path is wet? he garden: Calculate the size of each lettered ang! S ZA 6. Z; ve < o>Basic ideas of sets ‘the concept oF sel is a simple but powerful idea, The theory of sets is mainly duc to the work of the Geenian mathematician. Cantor. [ted to arguments and controversy. but by the 1920s his ideas were generally accepted and led (0 reat advances in mathematics. The objects can be numbers, animals, ideas, colours, in fact anything you ean imag’ all the members of the set. or by giving arule to, describe the members, ‘The list or rul A set ean be described by listin is euclosed by braces | | —__ Example 1 Example 2 described by a list A set described by a ra {Maureen, John, Louice} is the set {even numbers between 1 and 11) consisting of the three people called is the set consisting of the five Maureen, John and Lonice tumbers 2, 4, 6.8, 10, ———.) Sets are offen labelled by a single capital letter. A = fodd numbers between 2 and 10} means A isthe set consisting of the four n 33,5, 7.9. Seis cant be infinite in size, for example the set of prime numbers Example 3 IPB = {2.8.4.6 10} andi = {even numbers between 1 and 11}, then: nif) = 5, n(P) = 3:in other words both F and F have the same number of elements 3 € B means 3 is nota member of the set E 6 SF means 6 fs 4 member ol the set F E = F because both E and F have the same members. The order in which thie members are listed does not matter. the concept of the empty set mighit seem strange, but it is very useful Example 4 Give two examples of the empty set, a Theset of people you know over + m tal b. The set of odd numbers divisible by two, Exercise 22 1 Write down lwo more members of each of these sets, ‘arrot, potalo, ped, «.} b fred, green, blue, .} To he) (13 8.2 Key Point Asetisa collection of objects. which are called ‘the elements or member: ‘of the set. —=>-__—_ Key Point The number of elements in the set Ai written as nA), —-_— Key Point Membership of aseti indicated by the symbo © and non-membershi bythe symbol €. ———— Key Point ‘The empty set is thes with no members. Itis denoted by the symbol ort}. ——2 List these sets ’ a {days of the week} @ {pnme nt © {subjects vou study at school! 3 Describe these sets by a rule. b (Tuesday: thursday} a faba) 4 2.4, Ge © 11.4.9,16} : + Which of these statements are true: & {parallelograms} b Squateé : dd munaberst @ cat © {animals with to legs} © 1€ {prime numbers} 5 Which of these are examples of the empty * @ ‘The set of men with np teeth b ‘The set of months of the year with 32.4eys © Theset of straight nes drawn on the suri The set of prime numbers between 35 and 43 eof asphere Exercise 22* 1. Writedown two more members of each of these s al tenins Bar Maa) b (triangle, square, hexagon ¢ (hydrogen. iron. aluminium. ..) d (1.4.9, 16.1 2 List these sets. a. {all possible means of any two clement b {different digits of 11°} © {allfactors of 35) {powers of 10 tess than one million of 1.3.5} 3 Describe these sets by a rule. a (spring, summer, autumn, winter} b {circle, ellipse. parabola, hyperbola} © {1,2,4.8, 16} 43.4.5), (5,12, 13), (7, 24, 25nd & Which of these statements are true? a Everest € {mountains over 2000 m high} b 2000 tleap years} © 2x + 3y=5 € [straight d —2€ {solutions of x? ~ 26 graphs} ay 5 Which of these are examples of the empty set? fa Theset of three-leaged kangaroos b Theset that has the numeral zeros its only member € The sct of common factors of 11 and 13 The set of solutions of Venn diagramsWhe the umber of elementsin a set is so large that they cannot L all be shown. then a simple closed curve is drawn to indicate the set. 167 = {alltabby eats} then thisis shown ona Venn diagram as 1 C= (allcats in the world), then’T and ¢ ean be shown on g Key Point | a Venn diagram as WTis inside, then Tis . meme 7 called a subset of € This Theset Tis shown inside the set € because every member of eunuereeree: is also a member of C — Example 5 A= {1.2:3.4,5.6.7.8,9) a List the subset © = jodd numbers} b List the subset P= (prime nuribers} © IsQ = {8, 4,6} asubset of A? d isR= {0.1.2 3) asubset of A? a O= {1.515, 7,9) b P= (2.3.5.7) € Qisasubset ol A (OC A) because every member of Qis also.a member of A d Risnotasubset of A (it A) becanse the element is amember of R but isnot a member of A. Uf the problem was only about cats in this world and wasn't concerned about cats outside this, ‘world, then it is more usual to call the set Cthe universal set, denoted by €. The universal set contain all the elements being discussed in a particular problem, and isshown asa rectangle, where = fall tabby cats} Ifthe problem was only about eats in Rome then ® = fall eatsin Romie}; the Veni diagram does not change. The cats outside T are all non-tabby complement of 1 5. Thisset iscenoted by T’ and Is known as the Tahown iid 1 show shad1b How many ice-creams contain all theee ingredients? How many tee-crenins contain just raisins? How many tce-creams contain chocolate and raisins but not nuts? How many different types of tce-creams are there in the shop? Exercise 23 1 € = {all positive tttegers tess than 12}. A = (2.4.6.8, 10].B = (4. 5,6, 7.81 2 Whustrate this ixformation on a Venn di b List AN Band find iA NB) © Dos ANB = BOA 4 Last (ANB © SAN Ba subnet of A 2 © = fall ponitive intege 4..F = (5.6.7.8 strate this information mn RAI © Itt whut does this imply about the sets B and F 3 the alphabet A={ub B u fe Mlustrate thiv information on a Ven r b Listthe scteV MA VOBATR ‘ he set VANE 4 all positive integers). ¥ = (4.8) 12,16 4).8 824 aw Allustrate this information o ut Is the senallest member of F he set of the multiples of 4. S is th ples of 6, What by the LOM af 63 How ts this related to the set FS ‘ pod ofind he LCM of (D bands ay s