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EST Math

Provider · Academic Assessment Ltd. Format · EST I · 90 minutes Chapters · 8 Sections · Calculator + Non-calculator

Test format & strategy

🇬🇧 EST 1 Math — Domain Weightings (44 questions, 80 min)

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.

What to expect

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.

Duration

90 minutes total

Section I

30 min · no calculator

Section II

60 min · calculator OK

Score

200–800

⏱ EST I Math Section Pacing & Calculator Strategy

Section I (No-Calculator): 20 questions in 25 minutes = 75 seconds per question. Prioritize quick mental factoring, linear equation isolation, and slope formulas. Bank easy questions in under 45 seconds to leave buffer time for the final grid-in items.
Section II (Calculator Permitted): 38 questions in 55 minutes = ~87 seconds per question. Use your calculator strategically to verify quadratic solutions, percentage chains, and complex statistical summaries. Never waste time typing arithmetic you can compute mentally.
Zero Guessing Penalty: Since there is no negative marking, never leave any question blank. With 60 seconds left, fill in answers for all remaining blank items.

▲ The 4 Core Commandments for Scoring 750+ on EST I Math

1. Linear Systems by Ratio Inspection: For the system $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$ : if $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ , the lines are parallel (no solution). If all three ratios are equal, they are coincident (infinitely many solutions).
2. Quadratic Discriminant Check: Before applying the full quadratic formula or attempting complex factoring, compute $\Delta = b^2 - 4ac$ . If $\Delta < 0$ , there are no real roots; if $\Delta = 0$ , one repeated rational root; if $\Delta > 0$ and a perfect square, rational roots exist.
3. Percentage Change Denominator Rule: Percentage change is always calculated as $\frac{\text{new} - \text{old}}{\text{old}} \times 100\%$ . Never divide by the new value! A 25% increase followed by a 20% decrease returns to the original value: $1.25 \times 0.80 = 1.00$ .
4. Angle Mode & Circle Geometry: Verify whether angles are in radians or degrees. Remember that arc length $s = r\theta$ (where $\theta$ is in radians) and sector area $A = \frac{1}{2}r^2\theta$ . In degrees, $s = \frac{\theta}{360} 2\pi r$ and $A = \frac{\theta}{360} \pi r^2$ .

EST I Mathematics domain distribution. Heart of Algebra represents the single largest domain at 35%, complemented by Quadratics (15%), Functions (15%), Data/Ratios (10%), Statistics (10%), Geometry (10%), and Trigonometry (5%).

Section I of EST I Math contains 20 questions in 25 minutes. If a student answers the first 12 questions in an average of 50 seconds each, how many seconds per question are available for the remaining 8 questions?

\text{Time for first 12 questions} = 12 \times 50 = 600\text{ seconds}

(10 minutes).

\text{Total time} = 25 \times 60 = 1500\text{ seconds}

\text{Remaining time} = 1500 - 600 = 900\text{ seconds}

. For the remaining 8 questions:

\frac{900}{8} = 112.5\text{ seconds per question}

(1 minute 52.5 seconds per question).

📝 Chapter Quiz

Questions Level

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.

▲ Reference — Lines & systems

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.

Slope-intercept

y=mx+b

Point-slope

y-y_1=m(x-x_1)

Absolute value

|x-a|=b \Rightarrow x=a\pm b

Slope

m=\dfrac{y_2-y_1}{x_2-x_1}

Worked example 1.A

Solve: $2(3x-1)=5x+4$ . $6x-2=5x+4 \implies x=6.$

Worked example 1.B

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

\leq

Graphical representation of the linear system

y = 2x - 1

and

y = -x + 5

. The unique solution is the point of intersection

(2, 3)

▲ EST I Math Traps & Pitfalls — Linear Equations & Systems

Inequality Sign Flip: When multiplying or dividing both sides of an inequality by a negative number, you MUST reverse the inequality symbol: $-3x < 12 \implies x > -4$ .
Infinite vs. No Solution: If solving leads to a true statement like $0 = 0$ , the system has infinitely many solutions (coincident lines). If it leads to a contradiction like $0 = 7$ , there is no solution (parallel lines).
Slope from Standard Form: For $Ax + By = C$ , the slope is $m = -A/B$ and the $y$ -intercept is $C/B$ . Do not forget the negative sign on $A/B$ !
Absolute Value Splitting: The equation $|2x - 5| = 9$ splits into two distinct equations: $2x - 5 = 9$ OR $2x - 5 = -9$ . If an absolute value equals a negative number (e.g. $|x + 3| = -4$ ), there is immediately NO solution.

Solve the system:

3x + 2y = 16

and

x - y = 2

From second equation:

x = y+2

. Substitute:

3(y+2)+2y=16 \Rightarrow 5y=10 \Rightarrow y=2, x=4

A school theater sold 250 tickets for a play, raising a total of $1,750. Adult tickets cost $9 each and student tickets cost $5 each. How many adult tickets were sold?

Set up system of equations: Let $a$ be the number of adult tickets and $s$ be student tickets: $\begin{cases} a + s = 250 \\ 9a + 5s = 1750 \end{cases}$

Elimination method: Multiply the first equation by 5: $5a + 5s = 1250.$ Subtract from the second equation: $(9a + 5s) - (5a + 5s) = 1750 - 1250 \implies 4a = 500 \implies a = 125.$

Exactly 125 adult tickets were sold (and $s = 250 - 125 = 125$ student tickets).

📝 Chapter Quiz

Questions Level

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.

▲ Reference — Quadratic forms

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.

⊙ Explorer · Parabola in vertex forminteractive

a: 10

h: 1

k: -2

vertex(1, -2)

roots-0.41, 2.41

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.

Quadratic formula

\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}

Vertex

x=-\dfrac{b}{2a}

Difference of squares

a^2-b^2=(a+b)(a-b)

Perfect square

(a+b)^2=a^2+2ab+b^2

Worked example 2.A

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)

Worked example — Discriminant & nature of roots

Find the values of $k$ for which $x^2 + kx + 9 = 0$ has two equal roots.

Equal roots when $\Delta = 0$ : $k^2 - 4(1)(9) = 0 \Rightarrow k^2 = 36 \Rightarrow k = \pm 6$ .

Parabola

y = x^2 - 4x + 3

opening upward. The axis of symmetry is

x = 2

, vertex is

(2, -1)

, and roots are at

x = 1

and

x = 3

▲ EST I Math Traps & Pitfalls — Quadratics & Polynomials

Vertex Coordinate Signs: In vertex form $y = a(x - h)^2 + k$ , the vertex is $(h, k)$ . Notice the minus sign on $h$ : $y = 2(x + 3)^2 - 5$ has vertex at $(-3, -5)$ , NOT $(3, -5)$ . From standard form, $h = -b/(2a)$ .
Cancelling Variables Loses Roots: In an equation like $x^2 = 6x$ , NEVER divide both sides by $x$ ! Doing so loses the root $x = 0$ . Instead, set to zero: $x^2 - 6x = 0 \implies x(x - 6) = 0$ , yielding both $x = 0$ and $x = 6$ .
Discriminant Root Classifications: $\Delta = b^2 - 4ac$ . If $\Delta > 0$ , two distinct real solutions (parabola intersects $x$ -axis twice); if $\Delta = 0$ , exactly one real solution (parabola touches $x$ -axis at its vertex); if $\Delta < 0$ , no real solutions (parabola lies entirely above or below $x$ -axis).
Sum and Product of Roots (Vieta's Formulas): For $ax^2 + bx + c = 0$ , root sum is $r_1 + r_2 = -b/a$ and root product is $r_1 r_2 = c/a$ . This avoids having to solve the equation when the question only asks for the sum or product of solutions.

Solve

2x^2 - 7x + 3 = 0

using the quadratic formula.

x = \dfrac{7 \pm \sqrt{49-24}}{4} = \dfrac{7 \pm 5}{4}

. So

x = 3

x = \dfrac{1}{2}

The height

h

in meters of a projectile launched upward from an initial height of 25 meters after

t

seconds is modeled by

h(t) = -5t^2 + 20t + 25

. (a) At what time

t

does the projectile reach its maximum height, and what is this maximum height? (b) After how many seconds does the projectile strike the ground?

(a) Maximum height at vertex: The time at which maximum occurs is the vertex: $t = -\dfrac{b}{2a} = -\dfrac{20}{2(-5)} = \dfrac{20}{10} = 2\text{ seconds}.$ Substitute $t = 2$ to find maximum height: $h(2) = -5(2^2) + 20(2) + 25 = -20 + 40 + 25 = 45\text{ meters}.$

(b) Time to strike the ground: Set $h(t) = 0$ : $-5t^2 + 20t + 25 = 0 \implies -5(t^2 - 4t - 5) = 0 \implies (t - 5)(t + 1) = 0.$ Since time cannot be negative, $t = 5$ seconds.

📝 Chapter Quiz

Questions Level

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.

▲ Reference — Functions

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.

Composition

(f\circ g)(x)=f(g(x))

Inverse

swap

x,y

then solve

Shift up k

f(x)+k

Shift right h

f(x-h)

Worked example 3.A

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.

Worked example — Inverse function

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 $f^{-1}(x) = \dfrac{2x+1}{3}$ .

Transformation of the parent function

f(x) = x^2

(blue) into

g(x) = (x - 2)^2 - 3

(coral), representing a horizontal shift of 2 units right and vertical shift of 3 units down to vertex

(2, -3)

▲ EST I Math Traps & Pitfalls — Functions & Graphs

Horizontal Shift Reversal: The transformation $f(x - h)$ shifts the graph to the RIGHT by $h$ , while $f(x + h)$ shifts the graph to the LEFT by $h$ . The sign inside the parentheses always behaves counter-intuitively.
Composition Evaluation Order: For $(f \circ g)(x) = f(g(x))$ , always calculate the inner function $g(x)$ first, then feed the result into the outer function $f$ . In general, $f(g(x)) \neq g(f(x))$ .
Finding Inverses: To find $f^{-1}(x)$ , replace $f(x)$ with $y$ , swap $x$ and $y$ , and solve for $y$ . The domain of $f$ becomes the range of $f^{-1}$ , and the range of $f$ becomes the domain of $f^{-1}$ .
Function Identification (Vertical Line Test): A relation is a function if and only if each input $x$ has at most one output $y$ . If a vertical line intersects the curve more than once (like a circle or sideways parabola), it is NOT a function.

The graph of

y = f(x)

is shifted 3 units right and reflected in the

x

-axis. Write the new equation in terms of

f

y = -f(x-3)

Let

f(x) = 3x - 4

and

g(x) = x^2 + 1

. (a) Evaluate

(g \circ f)(2)

. (b) Determine the value of

x

such that

f^{-1}(x) = 5

(a) Composition $g(f(2))$ : First evaluate the inner function: $f(2) = 3(2) - 4 = 6 - 4 = 2.$ Now substitute into $g$ : $g(2) = (2)^2 + 1 = 4 + 1 = 5.$

(b) Inverse evaluation: By definition, $f^{-1}(x) = 5 \iff f(5) = x$ . Compute $f(5)$ : $x = f(5) = 3(5) - 4 = 15 - 4 = 11.$ Thus $x = 11$ .

📝 Chapter Quiz

Questions Level

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.

Ratio

\dfrac{a}{b}

Proportion

\dfrac{a}{b}=\dfrac{c}{d}

Percentage

\dfrac{\text{part}}{\text{whole}}\times 100

Rate

\dfrac{\text{distance}}{\text{time}}

Worked example 4.A

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.

● Ratios, Rates & Percentages

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: $\bar{x} = \dfrac{\sum w_i x_i}{\sum w_i}$ .

Worked example — Percentage increase and reverse

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.

Bar chart showing student frequency across four grade categories (total 72 students). Section B represents the mode with 28 students (38.9%).

▲ EST I Math Traps & Pitfalls — Ratios, Rates & Percentages

Percent Change Base Value: Always divide by the ORIGINAL quantity: $\frac{\text{new} - \text{old}}{\text{old}} \times 100\%$ . For example, if a price drops from $80 to $60, the percent decrease is $\frac{20}{80} = 25\%$ , NOT $\frac{20}{60} = 33.3\%$ .
Successive Percentage Multipliers: Never add or subtract percentages directly! A 30% markup followed by a 20% discount is $1.30 \times 0.80 = 1.04$ (a net 4% increase), NOT $30\% - 20\% = 10\%$ .
Average Speed is NOT Average of Speeds: Average speed equals $\frac{\text{Total Distance}}{\text{Total Time}}$ . If you travel 60 miles at 30 mph (2 hours) and return 60 miles at 60 mph (1 hour), your average speed is $\frac{120}{3} = 40\text{ mph}$ , NOT $\frac{30+60}{2} = 45\text{ mph}$ .
Square and Cubic Unit Conversions: When converting area and volume units, square or cube the conversion factor! $1\text{ m} = 100\text{ cm} \implies 1\text{ m}^2 = (100)^2 = 10,000\text{ cm}^2$ and $1\text{ m}^3 = 1,000,000\text{ cm}^3$ .

A car travels 240 km in 3 hours, then 180 km in 2 hours. Find the average speed for the whole journey.

Total distance

= 420

km. Total time

= 5

h. Average speed

= 420/5 = 84

km/h.

A retailer purchases a watch from a wholesaler for $80. The retailer marks up the wholesale price by 40% for retail display, and later offers a 25% discount off the display price during a promotion. (a) What is the promotional selling price of the watch? (b) What is the retailer's net percent profit relative to the wholesale cost?

(a) Promotional selling price:

Marked-up display price: $80 \times (1 + 0.40) = 80 \times 1.40 = \$112.$
Promotional price after 25% discount: $112 \times (1 - 0.25) = 112 \times 0.75 = \$84.$

(b) Net profit percentage: Profit is $84 - 80 = \$4$ . The net percent profit relative to wholesale cost is: $\dfrac{4}{80} \times 100\% = 5\%.$ Notice that $1.40 \times 0.75 = 1.05$ directly gives a 5% net gain.

📝 Chapter Quiz

Questions Level

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.

Mean

\bar{x}=\dfrac{\sum x}{n}

Median

middle value when sorted

Range

max

-

min

Probability

P(E)=\dfrac{\text{favourable}}{\text{total}}

Worked example 5.A

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

B

P(A\cup B)=P(A)+P(B)-P(A\cap B)

● Statistics & Probability

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: $P(A \cap B)=0$ .

Worked example — Combined probability

A bag has 4 red and 6 blue balls. Two are drawn without replacement. Find $P(\text{both red})$ .

$P = \dfrac{4}{10} \times \dfrac{3}{9} = \dfrac{12}{90} = \dfrac{2}{15}$ .

Box plot illustrating the five-number summary: Min = 52, Q1 = 65, Median = 78, Q3 = 88, Max = 98. The interquartile range (IQR) spans

88 - 65 = 23

points.

▲ EST I Math Traps & Pitfalls — Statistics & Data Displays

Outlier Influence on Mean vs. Median: The mean is heavily sensitive to extreme high or low values (skewed data), whereas the median is resistant. If an extreme outlier is introduced to a data set, the mean shifts significantly toward the outlier while the median changes minimally.
Standard Deviation Transformations: Adding or subtracting a constant $k$ to every data point shifts the mean by $k$ , but leaves standard deviation and IQR completely unchanged! Multiplying every value by $c$ multiplies standard deviation by $|c|$ .
Two-Way Table Conditional Probability: Watch the wording of the condition! "Given that a student is female, find the probability she passed" uses the female total as the denominator. In contrast, "Find the probability that a student is female AND passed" uses the entire grand total as the denominator.
With vs. Without Replacement: Drawing items without replacement reduces both the numerator and the denominator on subsequent draws: $P(\text{both red}) = \frac{R}{N} \times \frac{R-1}{N-1}$ .

The ages of 7 students are: 14, 15, 15, 16, 16, 16, 18. A student is chosen at random. Find (a) the median age, (b)

P(\text{age} > 15)

(a) 4th value = 16. (b) Ages > 15: three 16s and one 18 = 4 students.

P = 4/7

A survey of 200 high school students asked whether they play a varsity sport. Of 90 seniors, 36 play a varsity sport. Of 110 juniors, 55 play a varsity sport. (a) What is the probability that a randomly chosen student plays a varsity sport? (b) Given that a student is a senior, what is the probability that the student does NOT play a varsity sport?

(a) Probability of playing a sport: Total athletes $= 36 + 55 = 91$ . $P(\text{Sport}) = \dfrac{91}{200} = 0.455\, (45.5\%).$

(b) Conditional probability for seniors: Condition on the 90 seniors (restricted denominator). The number of non-athlete seniors is $90 - 36 = 54$ : $P(\text{No Sport} \mid \text{Senior}) = \dfrac{54}{90} = \dfrac{3}{5} = 0.60\, (60\%).$

📝 Chapter Quiz

Questions Level

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.

Circle area

\pi r^2

Circumference

2\pi r

Pythagorean theorem

a^2+b^2=c^2

Distance

\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

Worked example 6.A

A right triangle has legs $6$ and $8$ . Find the hypotenuse. $c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10.$

⊙ Explorer · The Pythagorean theoreminteractive

leg a: 6

leg b: 8

a² + b²100

c² (hypotenuse²)100

c10.00

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)

\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

● Geometry — Key Formulas

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 $(n-2) \times 180°$ .

Worked example — Similar triangles

Triangles $ABC$ and $DEF$ are similar. $AB = 6$ , $DE = 9$ , $BC = 8$ . Find $EF$ .

Scale factor $k = 9/6 = 1.5$ . $EF = 8 \times 1.5 = 12$ .

Right triangle

ABC

with legs

AC = 3

and

AB = 4

. By the Pythagorean theorem, the hypotenuse is

BC = \sqrt{3^2 + 4^2} = 5

▲ EST I Math Traps & Pitfalls — Geometry & Measurement

Area vs. Circumference Formulas: Circle area is $A = \pi r^2$ ; circumference is $C = 2\pi r$ . If a question provides the diameter, divide by 2 to get the radius $r = d/2$ before squaring! Squaring the diameter is the most common geometry blunder on EST I.
Inscribed vs. Central Angles: An inscribed angle with vertex on the circle circle circumference measures exactly HALF the central angle intercepting the same arc: $\angle_{\text{inscribed}} = \frac{1}{2}\angle_{\text{central}}$ . Any inscribed angle intercepting a diameter measures $90^\circ$ .
Similar Solids Scaling: If the ratio of linear lengths between two similar figures is $k$ , their surface areas scale as $k^2$ , and their volumes scale as $k^3$ . Never use the linear scale factor to scale volume directly.
Pythagorean Triples Speed: Memorize the common integer right triangles: $3-4-5$ , $5-12-13$ , $8-15-17$ , $7-24-25$ , and the special right triangles $45^\circ-45^\circ-90^\circ$ ( $1 : 1 : \sqrt{2}$ ) and $30^\circ-60^\circ-90^\circ$ ( $1 : \sqrt{3} : 2$ ).

A rectangle has perimeter 48 cm and length twice its width. Find its area.

Let width

= w

2(2w+w)=48 \Rightarrow 6w=48 \Rightarrow w=8

. Length

=16

. Area

= 128

^2

A circle with center

O

has radius

r = 6

cm. A central angle

\angle AOB = 60^\circ

subtends minor arc

AB

. Find (a) the exact length of arc

AB

in terms of

\pi

, and (b) the exact area of sector

AOB

(a) Arc length: The angle fraction is $\dfrac{60^\circ}{360^\circ} = \dfrac{1}{6}$ . $s = \dfrac{1}{6}(2\pi r) = \dfrac{1}{6}(2\pi \cdot 6) = 2\pi\text{ cm}.$

(b) Sector area: Using the same fraction of total area: $A_{\text{sector}} = \dfrac{1}{6}(\pi r^2) = \dfrac{1}{6}(\pi \cdot 6^2) = \dfrac{36\pi}{6} = 6\pi\text{ cm}^2.$

📝 Chapter Quiz

Questions Level

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.

sin θ

\sin\theta=\dfrac{\text{opp}}{\text{hyp}}

cos θ

\cos\theta=\dfrac{\text{adj}}{\text{hyp}}

tan θ

\tan\theta=\dfrac{\text{opp}}{\text{adj}}

Radians

180^\circ=\pi\text{ rad}

Worked example 7.A

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. $\text{other leg}=\sqrt{13^2-5^2}=12,\quad \sin\theta=\frac{12}{13}.$

⊙ Explorer · SOH-CAH-TOAinteractive

angle θ: 30

opposite5.00

adjacent8.66

sin θ / cos θ / tan θ0.500 / 0.866 / 0.577

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

● Trigonometry & Complex Numbers

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: $i^2 = -1$ ; $|a+bi| = \sqrt{a^2+b^2}$ .

Worked example — Finding all sides of a right triangle

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 $= 13\cos 22.6° \approx 12$ .

Unit circle for

\theta = 30^\circ = \frac{\pi}{6}\text{ rad}

. The terminal point has coordinates

(\cos 30^\circ, \sin 30^\circ) = \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)

▲ EST I Math Traps & Pitfalls — Trigonometry & Additional Topics

Radian vs. Degree Mode: If evaluating trigonometric ratios on a calculator, always verify your active mode! Evaluating $\sin(30)$ in radian mode yields $-0.988$ instead of $0.5$ .
Complementary Angle Theorem: In any right triangle, the acute angles sum to $90^\circ$ , which implies $\sin(x) = \cos(90^\circ - x)$ and $\cos(x) = \sin(90^\circ - x)$ . If $\sin(A) = \cos(B)$ , then $A + B = 90^\circ$ .
Pythagorean Identity Variations: $\sin^2\theta + \cos^2\theta = 1$ rearranges to $\sin^2\theta = 1 - \cos^2\theta = (1 - \cos\theta)(1 + \cos\theta)$ . Recognizing this difference of squares simplifies many algebraic trig questions instantly.
Powers of $i$ Cyclicity: Powers of the imaginary unit cycle every 4 powers: $i^1 = i$ , $i^2 = -1$ , $i^3 = -i$ , $i^4 = 1$ . To evaluate $i^n$ , divide $n$ by 4 and use the remainder.

Simplify

\dfrac{\sin^2\theta}{1-\cos\theta}

Use

\sin^2\theta = 1-\cos^2\theta = (1-\cos\theta)(1+\cos\theta)

. So

\dfrac{(1-\cos\theta)(1+\cos\theta)}{1-\cos\theta} = 1+\cos\theta

In a right triangle with acute angles

A

and

B

, it is given that

\sin(3x + 10^\circ) = \cos(2x + 15^\circ)

. (a) Find the value of

x

. (b) Determine the value of

\tan(x + 5^\circ)

(a) Use complementary angle identity: Since $\sin A = \cos B$ for acute angles in a right triangle, $A + B = 90^\circ$ : $(3x + 10^\circ) + (2x + 15^\circ) = 90^\circ \implies 5x + 25^\circ = 90^\circ \implies 5x = 65^\circ \implies x = 13^\circ.$

(b) Evaluate tangent: Substitute $x = 13^\circ$ : $\tan(13^\circ + 5^\circ) = \tan(18^\circ) \approx 0.3249.$

📝 Chapter Quiz

Questions Level

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.

Exponent product

a^m\cdot a^n=a^{m+n}

Exponent quotient

\dfrac{a^m}{a^n}=a^{m-n}

Log definition

\log_b x = y \Leftrightarrow b^y=x

Log product

\log(ab)=\log a+\log b

Worked example 8.A

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).

● Exponentials & Logarithms

$\log_b x = y \Leftrightarrow b^y = x$ . 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: $\ln(e^x) = x$ ; $e^{\ln x} = x$ .

Worked example — Solving an exponential equation

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.

Graphs of exponential growth

y = 2^x

(blue) and exponential decay

y = (0.5)^x

(coral), reflecting across the

y

-axis with common

y

-intercept at

(0, 1)

▲ EST I Math Traps & Pitfalls — Exponentials & Radicals

Negative Exponents vs. Negative Values: A negative exponent denotes reciprocal, not a negative quantity: $x^{-n} = \frac{1}{x^n}$ . For instance, $3^{-2} = \frac{1}{9}$ , NOT $-9$ or $-6$ .
Fractional Exponent Definition: $x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$ . The denominator is the index of the radical root, and the numerator is the power: $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$ .
Adding Exponents vs. Multiplying Powers: Remember that $x^a \cdot x^b = x^{a+b}$ , but $(x^a)^b = x^{ab}$ . Do not confuse $2^3 \cdot 2^4 = 2^7$ with $(2^3)^4 = 2^{12}$ .
Extraneous Solutions in Radical Equations: Squaring both sides of an equation like $\sqrt{2x + 3} = x$ can introduce extraneous roots. You MUST substitute candidate roots back into the original unsquared equation to verify validity.

Solve

\log_3(x+1) + \log_3(x-1) = 3

\log_3[(x+1)(x-1)] = 3 \Rightarrow x^2-1 = 27 \Rightarrow x^2 = 28 \Rightarrow x = 2\sqrt{7}

(reject negative).

Solve the exponential equation

4^{x+1} = 8^{2x-1}

for

x

Express with common base 2: Rewrite $4 = 2^2$ and $8 = 2^3$ : $(2^2)^{x+1} = (2^3)^{2x-1} \implies 2^{2(x+1)} = 2^{3(2x-1)}.$

Equate exponents: Since bases match: $2x + 2 = 6x - 3 \implies 2 + 3 = 6x - 2x \implies 5 = 4x \implies x = \dfrac{5}{4} = 1.25.$

📝 Chapter Quiz

Questions Level

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 $3x - 7 = 14$ .

Basic

3x=21,\qquad x=7.

If $2x + y = 10$ and $x = 3$ , find $y$ .

Basic

6+y=10,\qquad y=4.

Find the slope of the line through $(2,\,5)$ and $(6,\,13)$ .

Basic

m=\dfrac{13-5}{6-2}=\dfrac{8}{4}=2.

Solve the system $x+y=8$ , $x-y=2$ .

Intermediate

Add the equations: $2x=10$ , so $x=5$ , $y=3$ .

x=5,\quad y=3.

Write the equation of the line with slope $-2$ through $(0,\,4)$ .

Intermediate

y=-2x+4.

Solve $\dfrac{x}{3}+2=5$ .

Intermediate

\dfrac{x}{3}=3,\qquad x=9.

Solve $|2x-3|=7$ .

Intermediate

$2x-3=7 \Rightarrow x=5$ or $2x-3=-7 \Rightarrow x=-2$ .

x=5\text{ or }x=-2.

Solve the system $2x+3y=12$ , $x-y=1$ .

Advanced

From 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?

Intermediate

m=-\dfrac{1}{3}.

Solve the inequality $4x - 5 > 11$ .

Intermediate

4x>16,\qquad x>4.

Solve $x^2 - 7x + 12 = 0$ by factoring.

Intermediate

(x-3)(x-4)=0 \Rightarrow x=3\text{ or }x=4.

Find 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 $x^2 + 2x - 8 = 0$ .

Intermediate

x=\dfrac{-2\pm\sqrt{4+32}}{2}=\dfrac{-2\pm 6}{2} \Rightarrow x=2\text{ or }x=-4.

How many real solutions does $x^2 + x + 1 = 0$ have?

Intermediate

Discriminant $\Delta=1-4=-3<0$ .

\text{No real solutions.}

A parabola has vertex $(3,-2)$ and passes through $(5,6)$ . Find $a$ in $y=a(x-3)^2-2$ .

Advanced

Substitute $(5,6)$ : $6=a(4)-2$ .

a=2.

Expand and simplify $(2x-3)^2$ .

Intermediate

4x^2-12x+9.

Find the roots of $2x^2 - 8 = 0$ .

Basic

x^2=4 \Rightarrow x=\pm 2.

Factor $3x^2 - 12$ .

Intermediate

3(x^2-4)=3(x-2)(x+2).

If $f(x) = 2x + 5$ , find $f(3)$ .

Basic

2(3)+5=11.

If $f(x) = x^2$ and $g(x) = x + 1$ , find $f(g(2))$ .

Intermediate

g(2)=3,\quad f(3)=9.

Find the inverse of $f(x) = 3x - 6$ .

Intermediate

Swap $x$ 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.

Intermediate

f(x-3).

If $g(x) = \dfrac{1}{x-2}$ , what is the domain?

Advanced

x\neq 2,\quad\text{i.e. }(-\infty,2)\cup(2,\infty).

If $h(x)=-f(x)+2$ , describe the transformation.

Advanced

Reflect 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.

Basic

C=5+2(12)=29\text{ SAR}.

Divide 120 in the ratio $3:5$ . Find the larger part.

Intermediate

\text{Larger}=\dfrac{5}{8}\times120=75.

A price increased from 80 to 100 SAR. Find the percentage increase.

Intermediate

\dfrac{100-80}{80}\times100=25\%.

A car travels 240 km at 80 km/h. How long is the trip?

Intermediate

t=\dfrac{240}{80}=3\text{ hours}.

If $5$ workers finish a task in $12$ days, how many days for $3$ workers?

Advanced

Total 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.

Intermediate

200\times0.85=170\text{ SAR}.

Find the mean of $6,\,9,\,11,\,14,\,10$ .

Basic

\bar{x}=\dfrac{50}{5}=10.

Find the median of $3,\,7,\,8,\,12,\,15,\,20$ .

Intermediate

\text{Median}=\dfrac{8+12}{2}=10.

A bag has 4 red, 3 blue, 5 green balls. Find $P(\text{blue})$ .

Intermediate

P=\dfrac{3}{12}=\dfrac{1}{4}.

A data set has mean 12 and range 10. If all values are doubled, find the new mean.

Advanced

\text{New mean}=2\times12=24.

Find the mode of $2,\,5,\,7,\,5,\,3,\,5,\,8$ .

Basic

\text{Mode}=5.

Find the area of a triangle with base 10 and height 7.

Basic

A=\tfrac{1}{2}\times10\times7=35.

Find the hypotenuse of a right triangle with legs 7 and 24.

Intermediate

c=\sqrt{49+576}=\sqrt{625}=25.

Find the area of a circle with diameter 10.

Intermediate

A=\pi r^2=\pi(5)^2=25\pi.

Two angles of a triangle are $55^\circ$ and $75^\circ$ . Find the third.

Basic

180^\circ-55^\circ-75^\circ=50^\circ.

A rectangle has perimeter 36 and length 10. Find the width.

Intermediate

2(10+w)=36 \Rightarrow w=8.

Find $\sin\theta$ if the opposite side is 5 and the hypotenuse is 13.

Intermediate

\sin\theta=\dfrac{5}{13}.

A right triangle has $\tan\theta=\dfrac{3}{4}$ . Find $\sin\theta$ .

Advanced

Hypotenuse $=\sqrt{9+16}=5$ .

\sin\theta=\dfrac{3}{5}.

Simplify $2^3 \times 2^4$ .

Basic

2^{3+4}=2^7=128.

Solve $2^x = 64$ .

Intermediate

2^6=64 \Rightarrow x=6.

A population of 500 grows at 10% per year. Find it after 2 years.

Advanced

500\times(1.1)^2=500\times1.21=605.

Evaluate $\log_2 32$ .

Intermediate

2^5=32 \Rightarrow \log_2 32=5.

Solve $\log_3(x+1)=2$ .

Advanced

x+1=3^2=9 \Rightarrow x=8.

Simplify $\log 5 + \log 20$ .

Advanced

\log(5\times20)=\log 100=2.

Find the volume of a rectangular prism $4\times 3\times 5$ .

Basic

V=4\times3\times5=60.

A fair die is rolled twice. Find $P(\text{both show 6})$ .

Advanced

\dfrac{1}{6}\times\dfrac{1}{6}=\dfrac{1}{36}.

✎

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EST Math

Test format & strategy

Heart of Algebra

Linear inequalities

Quadratics & Polynomials

Polynomial operations

Functions & Graphs

Domain and range

Ratios, Rates & Data

Interpreting graphs

Statistics & Probability

Counting and probability

Geometry

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Growth and decay

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