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Mathematics, eighth-grade exam

Topic revision, sample questions with explanations and tests. The questions are our own; official papers are on the CKE website.

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Date: 11 May 2027, 9:00
Working time: 125 min
CKE trial paper: 15 December 2026

Standard time; it can be longer with special arrangements. CKE: komunikat o harmonogramie egzaminów w 2027 r. (20 sierpnia 2026) · CKE: informacja dyrektora o arkuszach przed egzaminami w 2027 r. (30 września 2026). Data checked: 2026-11-04.

Topics and revision

Natural numbers and integers

Divisibility tests: by 2 – the last digit is even; by 3 – the digit sum is divisible by 3; by 4 – the number formed by the last two digits is divisible by 4; by 5 – the last digit is 0 or 5; by 9 – the digit sum is divisible by 9.
Find the GCD and LCM from prime factorisations: the GCD takes the common factors, the LCM takes every factor at its highest power. For two numbers: GCD · LCM = a · b.
Division with remainder: a = b · q + r, where 0 ≤ r < b. The remainder is always smaller than the divisor.
A prime has exactly two divisors. The number 1 is neither prime nor composite; 2 is the only even prime.
Roman numerals: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000. A smaller symbol before a larger one is subtracted: IV = 4, XL = 40, CM = 900.
Integers: adding a negative number means subtracting its absolute value, e.g. 3 + (−5) = −2; subtracting a negative number means adding, e.g. 3 − (−5) = 8.

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Fractions and decimals

To add or subtract fractions, first bring them to a common denominator; then add only the numerators.
Multiplication: a/b · c/d = (a · c)/(b · d). Division means multiplying by the reciprocal: a/b : c/d = a/b · d/c.
Before multiplying or dividing, change a mixed number into an improper fraction: 2 1/3 = 7/3.
To change a fraction into a decimal, divide the numerator by the denominator. The expansion may be finite (3/8 = 0.375) or infinite and repeating (1/3 = 0.333...).
Order of operations: brackets first, then powers and roots, then multiplication and division, and finally addition and subtraction.
Whole from a part: if 3/5 of a number is 60, the number is 60 : 3/5 = 100.

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Powers

Product of powers with the same base: aⁿ · aᵐ = aⁿ⁺ᵐ. Quotient: aⁿ : aᵐ = aⁿ⁻ᵐ (for a ≠ 0).
Power of a power: (aⁿ)ᵐ = aⁿ·ᵐ, e.g. (2³)⁴ = 2¹².
Product of powers with the same exponent: aⁿ · bⁿ = (a · b)ⁿ, e.g. 4⁵ · 25⁵ = 100⁵.
Scientific notation: a · 10ᵏ, where 1 ≤ a < 10 and k is an integer, e.g. 0.0032 = 3.2 · 10⁻³ and 54 000 = 5.4 · 10⁴.
A sum of equal powers is not a power of the sum: 2⁵ + 2⁵ = 2 · 2⁵ = 2⁶, not 2¹⁰ or 4¹⁰.

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Roots

√a is the non-negative number whose square equals a (a ≥ 0). ∛a is the number whose cube equals a.
√(a · b) = √a · √b and √(a : b) = √a : √b for a ≥ 0, b > 0.
Taking a factor out: √72 = √(36 · 2) = 6√2. Bringing a factor in: 3√5 = √(9 · 5) = √45.
Estimating: since 8² = 64 and 9² = 81, we have 8 < √70 < 9.
There is no rule for the root of a sum: √(9 + 16) = 5, while √9 + √16 = 7.

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Algebraic expressions

Like terms differ only in their numerical coefficient: 3x and −5x are like terms, but 3x and 3x² are not.
A minus in front of brackets changes the sign of every term inside: −(x − 4) = −x + 4.
Multiplying a sum by a monomial: a(b + c) = ab + ac.
Multiplying two binomials: (a + b)(c + d) = ac + ad + bc + bd, then collect like terms.
Value of an expression: substitute the number in brackets, especially a negative one: for x = −2 we get x² = (−2)² = 4.
A two-digit number with tens digit a and units digit b equals 10a + b.

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Equations and formulas

A solution of an equation is a number which, when substituted, makes the equation true; you can always check it.
An equivalent equation is obtained by adding the same thing to both sides or multiplying both sides by the same number other than 0.
An equation with fractions is easiest after multiplying by a common denominator: multiply (x − 1)/2 + x/3 = 3 by 6.
Word problem: name the unknown, write the other quantities using it, form an equation, solve it, check against the text.
Rearranging formulas: treat the letter you want as the unknown. From s = v · t we get t = s : v.
Consecutive even or odd numbers can be written as x, x + 2, x + 4.

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Percentages

p% of a number b is (p/100) · b, e.g. 15% of 240 = 0.15 · 240 = 36.
What percentage of b is a: (a : b) · 100%.
The number b of which p% is a: b = a : (p/100). The price before a 20% discount, when it is now 80 zł: 80 : 0.8 = 100 zł.
An increase of p%: multiply by (1 + p/100). A decrease of p%: multiply by (1 − p/100).
Two successive changes are calculated from the new value. A 10% increase and a 10% decrease do not return to the starting price: 1.1 · 0.9 = 0.99.
Concentration = (mass of substance : mass of solution) · 100%. Adding water does not change the mass of the substance.

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Proportionality and practical calculations

Directly proportional quantities: y = k · x; when x grows some number of times, y grows the same number of times.
Sharing in the ratio 2 : 3 : 4 means 2 + 3 + 4 = 9 equal parts, of which we take 2, 3 or 4.
A scale of 1 : 25 000 means that 1 cm on the map is 25 000 cm = 250 m on the ground.
Distance, speed, time: s = v · t, v = s : t, t = s : v. Convert minutes to hours: 15 min = 0.25 h.
Converting speed: 1 m/s = 3.6 km/h, e.g. 54 km/h = 15 m/s.
Units: 1 km = 1000 m, 1 t = 1000 kg, 1 kg = 100 dag = 1000 g, 1 l = 1 dm³ = 1000 cm³ = 1000 ml.
Calendar calculations: 7 days later it is the same day of the week. February has 28 or 29 days.

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Angles, triangles, Pythagoras' theorem

Vertically opposite angles are equal; adjacent angles on a line add up to 180°. With parallel lines, corresponding and alternate angles are equal.
The interior angles of a triangle add up to 180°. In an isosceles triangle the base angles are equal.
Triangle inequality: each side is shorter than the sum of the other two. If a + b = c, there is no triangle, as the points lie on one line.
Congruence tests: SSS (three sides), SAS (two sides and the angle between them), ASA (a side and the two angles next to it).
Pythagoras' theorem: in a right-angled triangle a² + b² = c², where c is the hypotenuse.
Diagonal of a square: d = a√2. Height of an equilateral triangle: h = a√3/2.

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Polygons and their areas

Areas: rectangle A = a · b; square A = a²; parallelogram A = a · h; triangle A = a · h / 2.
Trapezium: A = (a + b) · h / 2. Rhombus: A = e · f / 2, where e and f are the diagonals.
The diagonals of a rhombus cross at a right angle and bisect each other, so the side comes from Pythagoras' theorem.
Every square is a rectangle and a rhombus. A rectangle need not be a square.
A regular polygon has all sides equal and all angles equal, e.g. an equilateral triangle and a square. A rhombus that is not a square is not regular.
The area of a triangle can be calculated from different base–height pairs; so the shortest height is the one onto the longest side.
The area of a shape can often be found by subtracting the areas of cut-off triangles from the area of a rectangle.

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Circumference and area of a circle

Circumference: L = 2πr = πd, where d = 2r.
Area of a circle: A = πr². Substitute the radius, not the diameter.
If we know the circumference, the radius is r = L : (2π). If we know the area, then r = √(A : π).
Unless the task asks for an approximation, leave π in the result, e.g. 16π cm².

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Number line and coordinate plane

Midpoint of a segment with ends A = (x₁, y₁), B = (x₂, y₂): S = ((x₁ + x₂)/2, (y₁ + y₂)/2).
If you know the end A and the midpoint S, the other end is B = (2·xₛ − x₁, 2·yₛ − y₁).
The length of a segment comes from Pythagoras' theorem: |AB|² = (x₂ − x₁)² + (y₂ − y₁)².
Lattice points on the line AB can be found by repeating the same shift: if going from A to B is 3 right and 6 up, it is also 1 right and 2 up.
On the number line x ≥ 1.5 is a filled dot and a ray to the right; x < −4/7 is an open dot and a ray to the left.

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Prisms and pyramids

Volume of a prism: V = B · H. Volume of a pyramid: V = (1/3) · B · H, where B is the base area.
Total surface area: S = 2 · B + L for a prism, S = B + L for a pyramid, where L is the lateral area.
A regular prism is right and has a regular polygon as its base. A regular pyramid has a regular polygon as its base, with the foot of the height at its centre.
In a regular square pyramid: h² = H² + (a/2)², where h is the height of a lateral face and H is the height of the solid.
An n-gonal prism has 3n edges, 2n vertices and n + 2 faces. An n-gonal pyramid has 2n edges, n + 1 vertices and n + 1 faces.
Units: 1 dm³ = 1 l = 1000 cm³, 1 m³ = 1000 dm³.

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Counting and probability

In the eighth-grade exam objects are counted by listing them in an orderly way, without the multiplication rule.
Probability = number of favourable outcomes : number of all possible outcomes.
A probability is always a number from 0 to 1.
Rolling a six-sided die has 6 equally likely outcomes; tossing a coin has 2.
The numbers from 1 to 30 are 30 outcomes; list and count the favourable ones, e.g. multiples of 6: 6, 12, 18, 24, 30.

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Data and the arithmetic mean

Arithmetic mean = sum of the values : number of values.
Sum of the values = mean · number of values. This helps when looking for a missing result.
In a pie chart the whole circle is 360° and 100%. A sector with angle α is α/360 of the whole.
When reading data, check the unit and the question: is it about a value, a change or a share of the whole?
A change in value: new − old. Saying that “the values rise” means each one is greater than the one before.

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Sample questions

Question 1 · Natural numbers and integers

The number MCMXLVII written in the decimal system is:

2167
1927
1967
1947

Answer: 1947

M = 1000, CM = 900, XL = 40, VII = 7, so the total is 1947. The result 2167 comes from simply adding all the symbols, but C before M and X before L are subtracted.

Question 2 · Fractions and decimals

The sum 3/4 + 2/5 equals:

6/20
5/9
1 3/20
1 1/4

Answer: 1 3/20

The common denominator is 20: 15/20 + 8/20 = 23/20 = 1 3/20. The result 5/9 is wrong: denominators are not added.

Question 3 · Powers

The product 2³ · 2⁴ equals:

2¹²
4⁷
4¹²
2⁷

Answer: 2⁷

When multiplying powers with the same base, add the exponents and keep the base: 3 + 4 = 7, so 2⁷.

Question 4 · Roots

The value of the expression √(49/64) equals:

7/32
7/64
7/8
49/8

Answer: 7/8

Root of a quotient: √49 : √64 = 7/8. Both the numerator and the denominator must be rooted.

Question 5 · Algebraic expressions

After simplifying, the expression 3x − 2(x − 4) has the form:

x − 8
x − 4
5x − 8
x + 8

Answer: x + 8

3x − 2x + 8 = x + 8. The result x − 8 comes from not changing the sign of the second term in the brackets.

Official CKE papers

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CKE trial papers appear on the CKE website on the day of publication.