# FURTHER MATHEMATICS/MATHEMATICS (ELECTIVE)

## WAEC SYLLABUS ON FURTHER MATHEMATICS/MATHEMATICS (ELECTIVE)

### AIMS OF THE SYLLABUS

*The aims of the syllabus are to test candidates*

(i) development of further conceptual and manipulative skills in Mathematics;

(ii) understanding of an intermediate course of study which bridges the gap betweenElementary Mathematics and Higher Mathematics;

(iii) acquisition of aspects of Mathematics that can meet the needs of potentialMathematicians, Engineers, Scientists and other professionals.

(iv) ability to analyse data and draw valid conclusion

(v) logical, abstract and precise reasoning skills.

### EXAMINATION SCHEME

*There will be two papers, Papers 1 and 2, both of which must be taken.*

### PAPER 1:

will consist of forty multiple-choice objective questions, covering the entiresyllabus. Candidates will be required to answer all questions in 1½ hours for 40marks. The questions will be drawn from the sections of the syllabus as follows:

Pure Mathematics - 30 questions

Statistics and probability - 4 questions

Vectors and Mechanics - 6 questions

### PAPER 2:

will consist of two sections, Sections A and B, to be answered in 2½ hours for 100marks### Section A:

will consist of eight compulsory questions that areelementary in type for 48marks. The questions shall be distributed as follows:Pure Mathematics - 4 questions

Statistics and Probability - 2 questions

Vectors and Mechanics - 2 questions

### Section B

will consist of seven questions of greater length and difficulty put into threeparts:Parts I, II and III as follows:Part I: Pure Mathematics - 3 questions

Part II: Statistics and Probability - 2 questions

Part III: Vectors and Mechanics - 2 questions

Candidates will be required to answer four questions with at least one from eachpart for 52 marks.

### DETAILED SYLLABUS

*In addition to the following topics, more challenging questions may be set on topics in theGeneral Mathematics/Mathematics (Core) syllabus.*

In the column for CONTENTS, more detailed information on the topics to be tested is givenwhile the limits imposed on the topics are stated under NOTES.

Topics which are marked with asterisks shall be tested in Section B of Paper 2 only.

In the column for CONTENTS, more detailed information on the topics to be tested is givenwhile the limits imposed on the topics are stated under NOTES.

Topics which are marked with asterisks shall be tested in Section B of Paper 2 only.

**KEY: * Topics peculiar to Ghana only.**

** Topics peculiar to Nigeria only

** Topics peculiar to Nigeria only

TOPICS | CONTENTS | NOTES |
---|---|---|

## I. Pure Mathematics(1) Sets | (i) Idea of a set defined by aproperty, Set notations andtheirmeanings.(ii) Disjoint sets, Universal set andcomplement of set (iii) Venn diagrams, Use of setsAnd Venn diagrams to solveproblems. (iv) Commutative and Associativelaws, Distributive propertiesover union and intersection. | (x : x is real), ꓴ, ꓵ, { }, ∉, ∈, ⊂, ⊆,U (universal set) andA’ (Complement of set A).More challenging problemsinvolving union, intersection,the universal set, subset andcomplement of set.Three set problems. Use of DeMorgan’s laws to solve relatedproblems |

(2) Surds | Surds of the he form a/√a , a√𝑏 anda+b√𝑛 where a is rational, b is apositive integer and n is not a perfect square. | All the four operations onsurdsRationalising the denominatorof surds such as a/√a, (a+√b)/(c-√d), (a+√b)/(√c-√d) |

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(3) Binary Operations | Properties:Closure, Commutativity,Associativity and Distributivity,Identity elements and inverses. | Use of properties to solverelated problems. |

(4) Logical Reasoning | (i) Rule of syntax:true or false statements,rule of logic applied toarguments, implications anddeductions.(ii) The truth table | Using logical reasoning todetermine the validity ofcompound statementsinvolving implications andconnectivities. Include use ofsymbols: ~p, pꓥq, pꓦq, p⇒q Use of Truth tables to deduceconclusions of compoundstatements. Include negation. |

(5) Functions | (i) Domain and co-domain of afunction. | The notation e.g. f : x →3x+4; g : x → x ^{2} ;where x ∈ R . |

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(ii) One-to-one, onto, identity andconstant mapping; | Graphical representation of afunction ; Image and therange. | |

(iii) Inverse of a function. | Determination of the inverseof a one-to-one function e.g. Iff: x →sx + 4/3, the inverserelation f-1: x → (1/3)x - 4/9is also afunction. | |

(iv) Composite of functions. | Notation: fog(x) =f(g(x))Restrict to simple algebraicfunctions only. | |

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(6) PolynomialFunctions | (i) Linear Functions, EquationsandInequality | Recognition and sketching ofgraphs of linear functions andequations. Gradient and intercepts formsof linear equations i.e. ax + by + c = 0; y = mx + c; y/a + x/b = k. Parallel andPerpendicular lines. Linear Inequalities e.g. 2x + 5y ≤ 1, x + 3y ≥ 3 Graphical representation oflinear inequalities in twovariables. Application to LinearProgramming. |

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(ii) Quadratic Functions,Equations and Inequalities | Recognition and sketchinggraphs of quadratic functionse.g. f: x → ax2 +bx + c, where a, band c Є R. Identification of vertex, axis ofsymmetry, maximum andminimum, increasing anddecreasing parts of a parabola. Include values of x for which f(x) >0 or f(x) < 0. Solution of simultaneousequations: one linear and onequadratic. Method ofcompleting the squares forsolving quadratic equations. Express f(x) = ax ^{2} + bx + c inthe formf(x) = a(x + d) ^{2} + k,where k is the maximum orminimum value. Roots ofquadratic equations – equalroots (b ^{2} - 4ac = 0),real andunequal roots (b ^{2} – 4ac > 0),imaginary roots (b ^{2} – 4ac <0);sum and product of rootsof a quadratic equation e.g. ifthe roots of the equation 3x ^{2}+ 5x + 2 = 0 are 𝛼 and β,form the equation whose roots are 1/𝛼 and 1/β Solving quadraticinequalities. | |

(iii) Cubic Functions and Equations | Recognition of cubic functionse.g. f: x → ax^{3} + bx^{2} + cx + d.Drawing graphs of cubicfunctions for a given range. Factorization of cubicexpressions and solution ofcubic equations. Factorizationof a ^{3} ± b^{3}.Basic operations onpolynomials, the remainder and factor theorems i.e. theremainder when f(x) is dividedby f(x – a) = f(a). When f(a) iszero, then (x – a) is a factor off(x). | |

## We provide educational resources/materials, curriculum guide, syllabus, scheme of work, lesson note & plan, waec, jamb, O-level & advance level GCE lessons/tutorial classes, on various topics, subjects, career, disciplines & department etc. for all the Class of Learners | ||

(7) Rational Functions | (i) Rational functions of the formQ(x) = f(x)/g(x),g(x) ≠ 0. where g(x) and f(x) arepolynomials. e.g. f:x → (ax + b)/(px ^{2} + qx + r) | g(x) may be factorised intolinear and quadratic factors(Degree of Numerator lessthan that of denominatorwhich is less than or equal to4). |

(ii) Resolution of rationalfunctions into partialfractions. | The four basic operations. Zeros, domain and range,sketching not required. | |

(8) Indices andLogarithmicFunctions | (i) Indices | Laws of indices.Application of the laws ofindices to evaluating products,quotients, powers and nthroot. Solve equations involvingindices. |

(ii) Logarithms | Laws of Logarithms. Application of logarithms incalculations involving product,quotients, power (log a ^{n}),nthroots (log √𝑎, log a ^{1/n}).Solve equations involvinglogarithms (including changeof base). Reduction of a relation such as y = ax ^{b},(a,b are constants) toa linear form: log _{10}y = b log_{10}x+log_{10}a.Consider other examples suchas log ab ^{x} = log a + x log b;log (ab) ^{x} = x(log a + log b)= x log ab*Drawing and interpretinggraphs of logarithmic functionse.g. y = ax ^{b}.Estimating thevalues of the constants a andb from the graph | |

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(9) PermutationAnd Combinations. | (i) Simple cases of arrangements | Knowledge of arrangementand selection is expected. Thenotations: ^{n}C_{r},( ^{n}_{r}) and^{n}P_{r} for selection and arrangementrespectively should be notedand used. e.g.arrangement ofstudents in a row, drawingballs from a box with orwithout replacements. ^{n}P_{r} = n!/(n-r)!^{n}C_{r}= n!/r!(n-r)! |

(ii) Simple cases of selection ofobjects. | ||

(10) BinomialTheorem | Expansion of (a + b)^{n}.Use of (1+x) ^{n} ≈ 1 + nxfor anyrational n, where x is sufficientlysmall. e.g (0.998) ^{1/3} | Use of the binomial theoremfor positive integral index only. Proof of the theorem notrequired. |

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(11) Sequencesand Series | (i) Finite and Infinite sequences. | e.g. (i) U_{1}, U_{2},…, U_{n}.(ii) U _{1}, U_{2},…. |

(ii) Linear sequence/ArithmeticProgression (A.P.) andExponentialsequence/GeometricProgression (G.P.) | Recognizing the pattern of asequence. e.g.(i) U_{n} = U_{1} + (n-1)d, whered is the common difference.(ii) U_{n}= U_{1}r^{n-1}where r is thecommon ratio. | |

(iii) Finite and Infinite series. | ||

(iv) Linear series (sum of A.P.)and | (i) U_{1} + U_{2} + U_{3} + … + U_{n}(ii)U _{1} + U_{2} + U_{3} + …. | |

exponential series (sum ofG.P.) | (i) S_{n} = n/2(U_{1} + U_{n})(ii) S _{n} = 1/2 [2a + (n – 1)d](iii) S _{n} = U_{1} (1-r^{n})/(1 - r),for r < 1 (iv) S _{n} = U_{1}(r^{n} - 1)/r – 1,for r > l. (v) Sum to infinity (S) = a/(1 - r) forr < 1 | |

*(v) Recurrence Series | Generating the terms of arecurrence series and findingan explicit formula for thesequence e.g. 0.9999 = 9/10 + 9/10^{2} +9/10^{3} + 9/10^{4} + .... | |

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(12) Matrices and LinearTransformation | (i) Matrices | Concept of a matrix – statethe order of a matrix andindicate the type. Equal matrices – If twomatrices are equal, then theircorresponding elements areequal. Use of equality to findmissing entries of givenmatrices Addition and subtraction ofmatrices (up to 3 x 3matrices). Multiplication of a matrix by ascalar and by a matrix (up to 3x 3 matrices) |

(ii) Determinants | Evaluation of determinants of2 x 2 matrices. **Evaluation of determinantsof 3 x 3 matrices. | |

(iii) Inverse of 2 x 2 Matrices | Application of determinants tosolution of simultaneous linearequations. | |

(iv) Linear Transformation | e.g. If A = (^{𝑎}_{𝑐} ^{𝑑}_{𝑏}) , thenA ^{-1} = 1/_{𝑎𝑐 - 𝑑𝑏} (^{𝑑}_{-𝑐}^{-𝑏}_{𝑎})Finding the images of pointsunder given lineartransformationDetermining the matrices ofgiven linear transformation.Finding the inverse of a lineartransformation (restrict to 2 x2 matrices). Finding the composition oflinear transformation.Recognizing the Identitytransformation. (i) ( ^{1}_{0} ^{0}_{-1}) reflection in thex - axis(ii) ( ^{-1}_{0} ^{0}_{1}) reflection in they - axis(iii) ( ^{0}_{1} ^{1}_{0}) reflection in the liney = x(iv) ( ^{cos θ}_{sin θ} ^{-sin θ}_{cos θ} )for anticlockwiserotation through θabout the origin. (v) ( ^{cos 2θ}_{sin 2θ} ^{sin 2θ}_{-cos 2θ} )thegeneral matrix for reflection ina line through the originmaking an angle θ with thepositive x-axis. *Finding the equation of theimage of a line under a givenlinear transformation | |

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(13) Trigonometry | (i) Trigonometric Ratios and Rules | Sine, Cosine and Tangent ofgeneral angles (0^{o}≤θ≤360^{o}).Identify trigonometric ratios ofangles 30^{o}, 45^{o}, 60^{o} withoutuse of tables.Use basic trigonometric ratiosand reciprocals to prove giventrigonometric identities. Evaluate sine, cosine andtangent of negative angles. Convert degrees into radians and vice versa. Application to real lifesituations such as heights anddistances, perimeters, solutionof triangles, angles ofelevation and depression,bearing (negative and positiveangles) including use of sineand cosine rules, etc,Simple cases only. sin (A ± B), cos (A ± B), tan(A ± B). |

(ii) Compound and MultipleAngles. | Use of compound angles insimple identities and solutionof trigonometric ratios e.g. finding sin 75 ^{o}, cos 150^{o} etc,finding tan 45 ^{o} without usingmathematical tables orcalculators and leaving youranswer as a surd, etc.Use of simple trigonometricidentities to find trigonometricratios of compound andmultiple angles (up to 3A). | |

(iii) Trigonometric Functions andEquations | Relate trigonometric ratios toCartesian Coordinates ofpoints (x, y) on the circle x^{2} +y^{2} = r^{2}.f:x →sin x,g: x → a cos x + b sin x = c. Graphs of sine, cosine,tangent and functions of theformasinx + bcos x. Identifyingmaximum and minimum point, increasing and decreasingportions. Graphical solutions ofsimple trigonometric equationse.g. asin x + bcos x = k. Solve trigonometric equationsup to quadratic equations e.g. 2sin ^{2}x – sin x – 3 =0, for 0^{o} ≤x ≤ 360^{o}.*Express f(x) = asin x + bcosx in the form Rcos (x ± 𝛼) or Rsin (x ± 𝛼) for 0 ^{o} ≤ 𝛼 ≤90^{o} and use the result tocalculate the minimum andmaximum points of a givenfunctions. | |

## We provide educational resources/materials, curriculum guide, syllabus, scheme of work, lesson note & plan, waec, jamb, O-level & advance level GCE lessons/tutorial classes, on various topics, subjects, career, disciplines & department etc. for all the Class of Learners | ||

(14) Co-ordinateGeometry | (i) Straight Lines | Mid-point of a line segmentCoordinates of points whichdivides a given line in a givenratio. Distance between two points; Gradient of a line; Equation of a line: (i) Intercept form; (ii) Gradient form; Conditions for parallel andperpendicular lines. Calculate the acute anglebetween two intersecting linese.g. if m1 and m2 are thegradients of two intersectinglines, then tan θ = (m _{1} - m_{2})/(1 + m_{1}m_{2}).Ifm _{1}m_{2} = -1,then the lines areperpendicular. *The distance from anexternal point P(x _{1}, y_{1}) to agiven lineax + by + c using the formulad = |(ax _{1} + by_{1} + c)/√(a^{2} + b^{2}) |.Loci of variable points whichmove under given conditionsEquation of a circle: (i) Equation in terms ofcentre, (a, b), andradius, r, (x - a) ^{2} + (y - b)^{2} = r^{2};(ii) The general form: x ^{2} + y^{2} + 2gx + 2fy + c = 0,where(-g , -f ) is the centre andradius, r = √(𝑎 ^{2} + 𝑏^{2} - 𝑐).Tangents and normals tocircles |

(ii) Conic Sections | Equations of parabola inrectangular Cartesian coordinates (y ^{2} = 4ax, includeparametric equations (at^{2},at)).Finding the equation of atangent and normal to aparabola at a given point. *Sketch graphs of givenparabola and find the equationof the axis of symmetry. | |

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(15) Differentiation | (i) The idea of a limit | (i) Intuitive treatment of limit.Relate to the gradient ofa curve. e.g. f ^{1}_{x} = lim_{h→o}{[f(x + h) - f(x)]/h} |

(iii) Differentiation of polynomials | (ii) Its meaning and itsdetermination from firstprinciples (simple casesonly).e.g. ax^{n} + b, n ≤ 3, (n ∈ I ) e.g. ax ^{m} – bx^{m - 1} + ...+ k,where m ∈ I , k is a constant. | |

(iv) Differentiation oftrigonometricFunctions | e.g. sin x, y = a sin x ± b cosx. Where a, b are constants. | |

(v) Product and quotient rules.Differentiation of implicitfunctions such asax^{2} + by^{2} = c | including polynomials of theform (a + bx^{n})^{m}. | |

**(vi) Differentiation ofTranscendental Functions | e.g. y = e^{ax}, y = log 3x,y = ln x | |

(vii) Second order derivatives andRates of change and smallchanges (Δx),Concept ofMaxima and Minima | (i) The equation of a tangenttoa curve at a point. (ii) Restrict turning points tomaxima and minima. (iii)Include curve sketching (upto cubic functions) andlinear kinematics. | |

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(16) Integration | (i) Indefinite Integral | (i) Integration of polynomialsofthe form ax^{n};n ≠ -1. i.e. ∫x ^{n} dx = (x^{n + 1})/(n + 1) + c, n ≠ -1. |

(ii) Definite Integral | (ii) Integration of sum anddifference of polynomials. e.g. ∫(4x ^{3}+3x^{2}-6x+5) dx | |

(iii) Applications of the DefiniteIntegral | **(iii)Integration ofpolynomialsof the form ax^{n}; n = -1.i.e. ∫x ^{-1} dx = ln x Simple problems onintegration by substitution. Integration of simpletrigonometric functions of theform ∫ _{a}^{b} sin𝑥 𝑑𝑥.(i) Plane areas and Rate ofChange. Include linearkinematics. Relate to the area under acurve. (ii)Volume of solid ofrevolution (iii) Approximation restrictedtotrapezium rule. | |

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II. Statistics andProbability (17) Statistics | (i) Tabulation and Graphicalrepresentation of data | Frequency tables. Cumulative frequency tables. Histogram (including unequalclass intervals). Cumulative frequency curve(Ogive) for grouped data. |

(ii) Measures of location | Central tendency: mean,median, mode, quartiles andpercentiles. Mode and modal group forgrouped data from ahistogram. Median from grouped data. Mean for grouped data (use ofan assumed mean required). | |

(iii) Measures of Dispersion | Determination of: (i) Range, Inter- Quartile andSemi inter-quartile rangefrom an Ogive. (ii) Mean deviation, varianceand standard deviation forgrouped and ungroupeddata. Using an assumedmean or true mean. | |

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(iv)Correlation | Scatter diagrams, use of lineof best fit to predict onevariable from another, meaning of correlation; positive, negative and zerocorrelations,. Spearman’s Rank coefficient. Use data without ties. *Equation of line of best fit byleast square method. (Line ofregression of y on x). | |

(18) Probability | (i) Meaning of probability.(ii) Relative frequency. (iii) Calculation of Probabilityusing simple sample spaces. (iv) Addition and multiplication ofprobabilities. (v) Probability distributions. | Tossing 2 dice once; drawingfrom a box with or withoutreplacement. Equally likely events, mutuallyexclusive, independent andconditional events. Include the probability of anevent considered as theprobability of a set. |

## We provide educational resources/materials, curriculum guide, syllabus, scheme of work, lesson note & plan, waec, jamb, O-level & advance level GCE lessons/tutorial classes, on various topics, subjects, career, disciplines & department etc. for all the Class of Learners | ||

III. Vectors andMechanics (19) Vectors | (i) Definitions of scalar and vectorQuantities. | (i) Binomial distributionP(x=r)= ^{n}C_{r}p^{r}q^{n-r} , whereProbability of success = p,Probability of failure = q, p + q = 1 and n is thenumber of trials. Simpleproblems only. **(ii) Poisson distributionP(x) = (e ^{-λ}λ^{x})/ x! , where λ =np,n is large and p is small. |

(ii) Representation of Vectors. | Representation of vector (^{a}_{b}) inthe form ai + bj. | |

(iii) Algebra of Vectors. | Addition and subtraction,multiplication of vectors byvectors, scalars and equationofvectors. Triangle,Parallelogram and polygonLaws. | |

(iv) Commutative, Associative andDistributive Properties. | Illustrate through diagram,Illustrate by solving problemsinelementary plane geometrye.g con-currency of medians anddiagonals. | |

(v) Unit vectors. | The notation:i for the unit vector (^{1}_{0}) andj for the unit vector ( ^{0}_{1})along the x and y axesrespectively. Calculation of unit vector (â) along a i.e. â = a/|a|. | |

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(vi) Position Vectors. | Position vector of A relative toO is _{OA}^{→}Position vector of the midpointof a line segment. Relate tocoordinates of mid-point of aline segment. *Position vector of a point thatdivides a line segmentinternally in the ratio (λ : μ). | |

(vii) Resolution and Compositionof Vectors. | Applying triangle,parallelogram and polygonlaws to composition of forcesacting at a point. e.g. find theresultant of two forces (12N,030 ^{o}) and (8N, 100^{o}) acting ata point.*Find the resultant of vectorsby scale drawing. | |

(viii) Scalar (dot) product and itsapplication.**(ix) Vector (cross) product andits application. | Finding angle between twovectors. Using the dot product toestablish such trigonometricformulae as (i) Cos (a ± b) =cos a cos b ± sin a sin b (ii) sin (a ± b)=sin a cos b sin b cosa (iii) c ^{2} = a^{2} + b^{2} - 2ab cos C(iv) (sin A)/a = (sin B)/b = (sinC)/c | |

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(20) Statics | (i) Definition of a force.(ii) Representation of forces. (iii) Composition and resolution ofcoplanar forces acting at apoint. (iv) Composition and resolution ofgeneral coplanar forces onrigid bodies. (v) Equilibrium of Bodies. (vi) Determination of Resultant. (vii) Moments of forces. (viii) Friction. | Apply to simple problems e.g. suspension of particles bystrings. Resultant of forces, Lami’stheoremUsing the principles ofmoments to solve relatedproblems. Distinction between smoothand rough planes. Determination of thecoefficient of friction. |

(21) Dynamics | (i) The concepts of motion | The definitions ofdisplacement,velocity, acceleration andspeed.Composition of velocities andaccelerations. |

(ii) Equations of Motion | Rectilinear motion.Newton’s laws of motion.Application of Newton’s LawsMotion along inclined planes(resolving a force upon aplane into normal andfrictional forces).Motion under gravity (ignoreair resistance).Application of the equations ofmotions: V = u + at,S = ut + ½ at^{2};v^{2} = u^{2} + 2as. | |

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(iii) The impulse and momentumequations: | Conservation of LinearMomentum(exclude coefficientof restitution). Distinguish betweenmomentum and impulse. | |

**(iv) Projectiles. | Objects projected at an angleto the horizontal. |

### 1. UNITS

*Candidates should be familiar with the following units and their symbols.*

#### ( 1 ) Length

1000 millimetres (mm) = 100 centimetres (cm) = 1 metre(m).1000 metres = 1 kilometre (km)

#### ( 2 ) Area

10,000 square metres (m2) = 1 hectare (ha)#### ( 3 ) Capacity

1000 cubic centimeters (cm3) = 1 litre (l)#### ( 4 ) Mass

1000 milligrammes (mg) = 1 gramme (g)1000 grammes (g) = 1 kilogramme( kg )

1000 ogrammes (kg) = 1 tonne.

#### ( 5) Currencies

The Gambia – 100 bututs (b) = 1 Dalasi (D)Ghana - 100 Ghana pesewas (Gp) = 1 Ghana Cedi ( GH¢)

Liberia - 100 cents (c) = 1 Liberian Dollar (LD)

Nigeria - 100 kobo (k) = 1 Naira (N)

Sierra Leone - 100 cents (c) = 1 Leone (Le)

UK - 100 pence (p) = 1 pound (£)

USA - 100 cents (c) = 1 dollar ($)

French Speaking territories 100 centimes (c) = 1 Franc (fr)

*Any other units used will be defined.*