Test format & strategy
| Domain | % of Test | Key Topics |
|---|---|---|
| Heart of Algebra | ~35% | Linear equations, systems, inequalities, absolute value |
| Quadratics & Polynomials | ~15% | Factoring, completing the square, quadratic formula, polynomial operations |
| Functions & Graphs | ~15% | Function notation, transformations, composition, inverse |
| Ratios, Rates & Data | ~10% | Proportions, unit rates, percentages, data interpretation |
| Statistics & Probability | ~10% | Mean/median, spread, probability, data displays |
| Geometry | ~10% | Area, perimeter, volume, angle relationships, similarity |
| Trigonometry & Additional Topics | ~5% | Right triangle trig, basic identities, complex numbers |
The EST I Mathematics section tests core algebraic reasoning, problem-solving, and quantitative analysis in two parts — one without a calculator and one with. It is the primary university-admissions standardised test for international and American-diploma students across the Middle East.
The EST I Math test runs for 90 minutes across two sections: Section I (30 min, no calculator — focuses on algebra and number sense) and Section II (60 min, calculator permitted — broader topics). The test contains 45 multiple-choice questions and 13 short constructed-response items.
Master linear equations cold for the no-calculator section. On Section II, write down formulas before you compute. For constructed responses, double-check rounding — the EST often asks for exact fractions.
Heart of Algebra
🇬🇧 Heart of Algebra ~35% · Linear equations & inequalities · Systems of equations · Absolute value · Word problems
Linear equations, systems, and inequalities form the largest tested domain on the EST I. Fluency here is non-negotiable for a high score.
Slope-intercept form: \( y = mx + b \). Slope: \( m = \dfrac{y_2-y_1}{x_2-x_1} \). Solve systems by substitution or elimination. For inequalities, flip the sign when multiplying or dividing by a negative.
Solve: \( 2(3x-1)=5x+4 \). \[ 6x-2=5x+4 \implies x=6. \]
Solve the system: \( 2x+y=7 \), \( x-y=2 \). Adding: \( 3x=9 \), so \( x=3 \), \( y=1 \).
Linear inequalities
A linear inequality such as \( 2x-3 > 7 \) is solved like an equation, but the solution is a range: \( x > 5 \). On a number line, use an open circle for strict inequalities (\( > \), \( < \)) and a closed circle for \( \geq \) or \( \leq \).
Quadratics & Polynomials
🇬🇧 Quadratics & Polynomials ~15% · Factoring · Completing the square · Quadratic formula · Discriminant
Quadratic expressions appear frequently on the EST I — in standard, vertex, and factored forms. Knowing how to move between forms and when to use each is a key skill.
Standard: \( ax^2+bx+c \). Vertex: \( a(x-h)^2+k \) with vertex at \( (h,k) \). Factored: \( a(x-r_1)(x-r_2) \) with roots \( r_1, r_2 \). Quadratic formula: \( x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \). Discriminant \( \Delta = b^2-4ac \): positive → two real roots, zero → one root, negative → no real roots.
| x | y |
|---|
Roots solve \(a(x-h)^2+k=0\), i.e. \(x=h\pm\sqrt{-k/a}\) — the vertex-form shortcut to the quadratic formula above.
Solve \( x^2-5x+6=0 \) by factoring. \[ (x-2)(x-3)=0 \implies x=2 \text{ or } x=3. \]
Polynomial operations
To multiply polynomials, distribute each term of the first factor across every term of the second. The key identity \( (a+b)(a-b)=a^2-b^2 \) shortcuts many EST questions. Dividing a polynomial by a linear factor \( (x-k) \) uses synthetic or long division; the Remainder Theorem states the remainder equals \( f(k) \).
Find the values of \(k\) for which \(x^2 + kx + 9 = 0\) has two equal roots. Equal roots when \(\Delta = 0\): \(k^2 - 4(1)(9) = 0 \Rightarrow k^2 = 36 \Rightarrow k = \pm 6\).
Functions & Graphs
🇬🇧 Functions & Graphs ~15% · Function notation · Graph transformations · Composite & inverse · Piecewise
A function assigns exactly one output to each input. The EST I tests function notation, composition, inverses, and transformations.
Notation: \( f(x) \) means the output of \( f \) at input \( x \). Composition: \( (f\circ g)(x)=f(g(x)) \). Inverse: \( f^{-1} \) reverses \( f \); find it by swapping \( x \) and \( y \) and solving for \( y \). Transformation: \( f(x)+k \) shifts up, \( f(x-h) \) shifts right, \( -f(x) \) reflects over \( x \)-axis.
If \( f(x)=2x+1 \) and \( g(x)=x^2 \), find \( f(g(3)) \). \[ g(3)=9,\quad f(9)=2(9)+1=19. \]
Domain and range
The domain is all valid inputs; the range is all possible outputs. For \( f(x)=\sqrt{x-3} \), the domain is \( x\geq 3 \) (argument under the root must be non-negative). For \( f(x)=\dfrac{1}{x-2} \), exclude \( x=2 \) from the domain.
Find \(f^{-1}(x)\) for \(f(x) = \dfrac{3x-1}{2}\). Let \(y = \dfrac{3x-1}{2}\). Swap \(x\) and \(y\): \(x = \dfrac{3y-1}{2} \Rightarrow 2x+1 = 3y \Rightarrow y = \dfrac{2x+1}{3}\). So \(f^{-1}(x) = \dfrac{2x+1}{3}\).
Ratios, Rates & Data
🇬🇧 Ratios, Rates & Data ~10% · Proportional reasoning · Percentages · Unit rates · Data table interpretation
The Problem-Solving and Data Analysis domain asks you to interpret tables, graphs, and real-world contexts — no formula memorisation needed, but careful reading is essential.
A car travels 240 km in 3 hours. How long does it take to travel 400 km at the same speed? \[ \text{Speed}=\frac{240}{3}=80\ \text{km/h},\quad t=\frac{400}{80}=5\ \text{hours}. \]
Interpreting graphs
On the EST I, you may be given a scatterplot, bar chart, or two-way table and asked to read a value, find a percentage, or determine whether a conclusion is justified. Always identify the units on each axis before computing.
Ratio: compare two quantities, e.g. \(a:b\). Rate: ratio with different units (speed, price per unit). Percentage change: \(\dfrac{\text{new}-\text{old}}{\text{old}} \times 100\%\). Proportion: two equal ratios \(\dfrac{a}{b} = \dfrac{c}{d}\). Weighted average: \(\bar{x} = \dfrac{\sum w_i x_i}{\sum w_i}\).
A price increases by 20%, then decreases by 20%. Is the final price the same as the original? Original = 100. After 20% increase: \(120\). After 20% decrease: \(120 \times 0.8 = 96\). No — it is 4% less than original.
Statistics & Probability
🇬🇧 Statistics & Probability ~10% · Mean, median, mode · Range & IQR · Probability · Graphs & displays
Statistical questions on the EST I focus on measures of centre and spread, interpreting data displays, and calculating basic probability.
Find the mean of \( 4, 7, 7, 10, 12 \). \[ \bar{x}=\frac{4+7+7+10+12}{5}=\frac{40}{5}=8. \]
Counting and probability
The probability of event \( A \) AND event \( B \) (independent): \( P(A\cap B)=P(A)\cdot P(B) \). The probability of \( A \) OR \( B \): \( P(A\cup B)=P(A)+P(B)-P(A\cap B) \).
Mean \(=\dfrac{\sum x}{n}\); Median = middle value when ordered; Mode = most frequent. Range \(= \max - \min\). IQR \(= Q_3 - Q_1\). Probability: \(P(A) = \dfrac{\text{favourable}}{\text{total}}\); \(P(A \cup B) = P(A)+P(B)-P(A \cap B)\). Mutually exclusive: \(P(A \cap B)=0\).
A bag has 4 red and 6 blue balls. Two are drawn without replacement. Find \(P(\text{both red})\). \(P = \dfrac{4}{10} \times \dfrac{3}{9} = \dfrac{12}{90} = \dfrac{2}{15}\).
Geometry
🇬🇧 Geometry ~10% · Area & perimeter · Volume · Angle relationships · Similarity & congruence
Geometry on the EST I covers areas, perimeters, volumes, and coordinate geometry. A reference sheet of key formulas is provided on the test.
A right triangle has legs \( 6 \) and \( 8 \). Find the hypotenuse. \[ c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10. \]
| quantity | value |
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The blue and amber squares' areas always sum to the dark square's area — that's exactly \(a^2+b^2=c^2\).
Coordinate geometry
The midpoint of \( (x_1,y_1) \) and \( (x_2,y_2) \) is \( \left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right) \). A circle with centre \( (h,k) \) and radius \( r \) satisfies \( (x-h)^2+(y-k)^2=r^2 \).
Triangle area \(= \tfrac{1}{2}bh\). Circle area \(= \pi r^2\), circumference \(= 2\pi r\). Trapezoid area \(= \tfrac{1}{2}(a+b)h\). Cylinder volume \(= \pi r^2 h\). Cone volume \(= \tfrac{1}{3}\pi r^2 h\). Sphere volume \(= \tfrac{4}{3}\pi r^3\). Angle sum: triangle 180°; quadrilateral 360°; \(n\)-gon \((n-2) \times 180°\).
Triangles \(ABC\) and \(DEF\) are similar. \(AB = 6\), \(DE = 9\), \(BC = 8\). Find \(EF\). Scale factor \(k = 9/6 = 1.5\). \(EF = 8 \times 1.5 = 12\).
Trigonometry & Additional Topics
🇬🇧 Trigonometry & Additional Topics ~5% · SOH-CAH-TOA · Pythagorean identity · Complex numbers · Vectors
The EST I tests right-triangle trigonometry, radian measure, and basic complex numbers. These appear mostly in Section II.
In a right triangle, the hypotenuse is 13 and one leg is 5. Find \( \sin\theta \) for the angle opposite the leg of length 12. \[ \text{other leg}=\sqrt{13^2-5^2}=12,\quad \sin\theta=\frac{12}{13}. \]
| quantity | value |
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Hypotenuse fixed at 10. SOH-CAH-TOA: \(\sin\theta=\text{opp}/10\), \(\cos\theta=\text{adj}/10\), \(\tan\theta=\text{opp}/\text{adj}\).
Complex numbers
A complex number has the form \( a+bi \) where \( i=\sqrt{-1} \). Add by combining real and imaginary parts separately: \( (3+2i)+(1-5i)=4-3i \). Multiply using FOIL and replace \( i^2 \) with \( -1 \).
SOH-CAH-TOA: \(\sin\theta = \text{opp}/\text{hyp}\), \(\cos\theta = \text{adj}/\text{hyp}\), \(\tan\theta = \text{opp}/\text{adj}\). Pythagorean identity: \(\sin^2\theta + \cos^2\theta = 1\). Exact values: \(\sin 30° = \tfrac{1}{2}\), \(\cos 30° = \tfrac{\sqrt{3}}{2}\), \(\sin 45° = \tfrac{\sqrt{2}}{2}\). Complex: \(i^2 = -1\); \(|a+bi| = \sqrt{a^2+b^2}\).
In right triangle, the hypotenuse is 13 and one angle is \(22.6°\). Find the other two sides. Opposite \(= 13\sin 22.6° \approx 13 \times 0.3843 \approx 5\). Adjacent \(= 13\cos 22.6° \approx 12\).
Exponentials & Logarithms
🇬🇧 Exponentials & Logarithms ~5% · Exponential growth/decay · Log laws · Solving exponential equations
Exponential growth and logarithms appear in the Passport to Advanced Math domain and in data-interpretation contexts on the EST I.
Solve \( 2^{x+1}=32 \). \[ 2^{x+1}=2^5 \implies x+1=5 \implies x=4. \]
Growth and decay
Exponential growth: \( A=A_0(1+r)^t \). Exponential decay: \( A=A_0(1-r)^t \). On EST I, these appear in word problems about populations, investments, or radioactive decay — identify \( A_0 \) (initial amount), \( r \) (rate), and \( t \) (time).
\(\log_b x = y \Leftrightarrow b^y = x\). Laws: \(\log(xy) = \log x + \log y\); \(\log(x/y) = \log x - \log y\); \(\log x^n = n\log x\). Exponential growth: \(A = A_0 e^{kt}\) (or \(A_0 b^t\)). Natural log: \(\ln(e^x) = x\); \(e^{\ln x} = x\).
A bacteria culture starts with 500 and doubles every 3 hours. How long until there are 4000? \(4000 = 500 \cdot 2^{t/3} \Rightarrow 2^{t/3} = 8 = 2^3 \Rightarrow t/3 = 3 \Rightarrow t = 9\) hours.
Practice set
Fifty EST I Mathematics problems across all domains — algebra, quadratics, functions, data analysis, geometry and trigonometry — graded by difficulty. Click any problem to reveal the full worked solution.
Solve \( 3x - 7 = 14 \).
BasicIf \( 2x + y = 10 \) and \( x = 3 \), find \( y \).
BasicFind the slope of the line through \( (2,\,5) \) and \( (6,\,13) \).
BasicSolve the system \( x+y=8 \), \( x-y=2 \).
IntermediateAdd the equations: \( 2x=10 \), so \( x=5 \), \( y=3 \).
\[ x=5,\quad y=3. \]Write the equation of the line with slope \( -2 \) through \( (0,\,4) \).
IntermediateSolve \( \dfrac{x}{3}+2=5 \).
IntermediateSolve \( |2x-3|=7 \).
Intermediate\( 2x-3=7 \Rightarrow x=5 \) or \( 2x-3=-7 \Rightarrow x=-2 \).
\[ x=5\text{ or }x=-2. \]Solve the system \( 2x+3y=12 \), \( x-y=1 \).
AdvancedFrom the second: \( x=y+1 \). Substitute: \( 2(y+1)+3y=12 \Rightarrow 5y=10 \Rightarrow y=2 \), \( x=3 \).
\[ x=3,\quad y=2. \]A line is perpendicular to \( y=3x-1 \). What is its slope?
IntermediateSolve the inequality \( 4x - 5 > 11 \).
IntermediateSolve \( x^2 - 7x + 12 = 0 \) by factoring.
IntermediateFind the vertex of \( y = x^2 - 4x + 7 \).
Intermediate\( h=-\dfrac{b}{2a}=2 \), \( k=4-8+7=3 \).
\[ (2,\,3). \]Use the quadratic formula to solve \( x^2 + 2x - 8 = 0 \).
IntermediateHow many real solutions does \( x^2 + x + 1 = 0 \) have?
IntermediateDiscriminant \( \Delta=1-4=-3<0 \).
\[ \text{No real solutions.} \]A parabola has vertex \( (3,-2) \) and passes through \( (5,6) \). Find \( a \) in \( y=a(x-3)^2-2 \).
AdvancedSubstitute \( (5,6) \): \( 6=a(4)-2 \).
\[ a=2. \]Expand and simplify \( (2x-3)^2 \).
IntermediateFind the roots of \( 2x^2 - 8 = 0 \).
BasicFactor \( 3x^2 - 12 \).
IntermediateIf \( f(x) = 2x + 5 \), find \( f(3) \).
BasicIf \( f(x) = x^2 \) and \( g(x) = x + 1 \), find \( f(g(2)) \).
IntermediateFind the inverse of \( f(x) = 3x - 6 \).
IntermediateSwap \( x \) and \( y \): \( x=3y-6 \).
\[ f^{-1}(x)=\dfrac{x+6}{3}. \]The graph of \( f(x) \) is shifted 3 units right. Write the new function.
IntermediateIf \( g(x) = \dfrac{1}{x-2} \), what is the domain?
AdvancedIf \( h(x)=-f(x)+2 \), describe the transformation.
AdvancedReflect over the \( x \)-axis, then shift up 2 units.
A taxi charges a flat fee of 5 SAR plus 2 SAR per km. Find the cost for 12 km.
BasicDivide 120 in the ratio \( 3:5 \). Find the larger part.
IntermediateA price increased from 80 to 100 SAR. Find the percentage increase.
IntermediateA car travels 240 km at 80 km/h. How long is the trip?
IntermediateIf \( 5 \) workers finish a task in \( 12 \) days, how many days for \( 3 \) workers?
AdvancedTotal work \( =60 \) worker-days.
\[ t=\dfrac{60}{3}=20\text{ days}. \]A shop offers a 15% discount on a 200 SAR item. Find the sale price.
IntermediateFind the mean of \( 6,\,9,\,11,\,14,\,10 \).
BasicFind the median of \( 3,\,7,\,8,\,12,\,15,\,20 \).
IntermediateA bag has 4 red, 3 blue, 5 green balls. Find \( P(\text{blue}) \).
IntermediateA data set has mean 12 and range 10. If all values are doubled, find the new mean.
AdvancedFind the mode of \( 2,\,5,\,7,\,5,\,3,\,5,\,8 \).
BasicFind the area of a triangle with base 10 and height 7.
BasicFind the hypotenuse of a right triangle with legs 7 and 24.
IntermediateFind the area of a circle with diameter 10.
IntermediateTwo angles of a triangle are \( 55^\circ \) and \( 75^\circ \). Find the third.
BasicA rectangle has perimeter 36 and length 10. Find the width.
IntermediateFind \( \sin\theta \) if the opposite side is 5 and the hypotenuse is 13.
IntermediateA right triangle has \( \tan\theta=\dfrac{3}{4} \). Find \( \sin\theta \).
AdvancedHypotenuse \( =\sqrt{9+16}=5 \).
\[ \sin\theta=\dfrac{3}{5}. \]Simplify \( 2^3 \times 2^4 \).
BasicSolve \( 2^x = 64 \).
IntermediateA population of 500 grows at 10% per year. Find it after 2 years.
AdvancedEvaluate \( \log_2 32 \).
IntermediateSolve \( \log_3(x+1)=2 \).
AdvancedSimplify \( \log 5 + \log 20 \).
AdvancedFind the volume of a rectangular prism \( 4\times 3\times 5 \).
BasicA fair die is rolled twice. Find \( P(\text{both show 6}) \).
AdvancedTest generator
Build a randomised EST I Math mini-test from the question bank, or generate a complete full-length paper that mirrors the official EST I Mathematics exam format — Section I (no calculator) and Section II (calculator permitted).
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Officially released exams and mark schemes from the examining body.
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8 original ClipSAT worksheets with full answer keys, organized by unit — free to download and print.