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Mathematics, Classes VI-VIII (Paper II): key facts and practice questions

Revise Mathematics, Classes VI-VIII (Paper II) for CTET and state TETs: the facts and distinctions that keep coming up, a one-glance revision list, and 10 practice questions with answers and explanations.

Drafted with AI assistance and fact-checked on 2026-10-15; not yet read line by line by a human editor. Spotted a mistake? Tell us. Editorial standards & corrections

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Key concepts and facts

Knowing our numbers: Indian periods are ones, thousands, lakhs and crores, so 2,08,40,015 is two crore eight lakh forty thousand fifteen; international periods are ones, thousands and millions, so 1 million = 10 lakh. Place value is the digit times the value of its place (in 3,62,18,405 the 6 is worth 60,00,000). To estimate, round to the greatest place: 689 × 312 is about 700 × 300 = 2,10,000. In Roman numerals no symbol repeats more than three times, V, L and D never repeat, and only I, X and C are written before a larger symbol to subtract: 1984 = MCMLXXXIV.
Playing with numbers: A prime has exactly two factors; 1 is neither prime nor composite, and 2 is the only even prime. Divisibility: for 3 or 9, check the digit sum; for 4, the last two digits; for 8, the last three; for 11, the difference of the sums of alternate digits must be 0 or a multiple of 11. A number divisible by two co-prime numbers is divisible by their product, but one divisible by 4 and 6 need not be divisible by 24. HCF takes the common primes at their lowest powers, LCM all primes at their highest, and for two numbers HCF × LCM = their product. 'Greatest container' signals HCF; 'ring together again' signals LCM.
Whole numbers and integers: Whole numbers are closed under addition and multiplication, not under subtraction or division; 0 is the additive identity, 1 the multiplicative identity, and division by 0 is not defined. Integers model temperatures below zero, depths and losses. Subtracting an integer means adding its additive inverse: −12 − (−20) = 8. A product of non-zero integers is negative when the number of negative factors is odd.
Fractions, decimals and rational numbers: Make the denominators equal before adding or subtracting; to divide, multiply by the reciprocal: 4 1/2 ÷ 3/4 = 6. A product of decimals has as many decimal places as all the factors together: 0.3 × 0.02 = 0.006. A rational number is p/q (p, q integers, q ≠ 0), and between any two there are infinitely many: 9/24, 10/24 and 11/24 lie between 1/3 and 1/2. Exponents (a ≠ 0): aᵐ × aⁿ = aᵐ⁺ⁿ, a⁰ = 1, a⁻ᵐ = 1/aᵐ; standard form is k × 10ⁿ with 1 ≤ k < 10. A perfect square never ends in 2, 3, 7 or 8; square and cube roots come from pairing or tripling prime factors: √1,764 = 42.
Algebra: A variable stands for any number, as in the rule 3n + 1 for the matchsticks in a row of n squares. An equation is solved by doing the same operation on both sides. A ratio compares quantities of the same kind in the same unit (50 cm : 2 m = 1 : 4); a : b = c : d is a proportion when a × d = b × c, and the unitary method first finds the value of one unit. Profit and loss per cent are calculated on the cost price.
Geometry: Angles on a straight line add up to 180° and those around a point to 360°; the angles of a triangle add up to 180° and those of a quadrilateral to 360°. Every square is a rectangle and a rhombus, but not the other way round. Convex polyhedra satisfy F + V − E = 2; a cube has 6 faces, 12 edges and 8 vertices. A square has 4 lines of symmetry, a non-square rectangle 2 and an equilateral triangle 3. Ruler and compasses alone give perpendicular bisectors, angle bisectors and angles of 60°, 30°, 90° and 120°.
Mensuration and data handling: A rectangle has perimeter 2(l + b) and area l × b, a triangle has area 1/2 × base × height, and a circle has circumference 2πr and area πr². The volume of a cuboid is l × b × h, and 1 litre = 1,000 cm³. The mean is the sum divided by the number of values, the median the middle value in order (or the mean of the middle two), and the mode the most frequent value; a pie-chart sector is the same fraction of 360° as the item is of the total.
Common errors: adding numerators and denominators (1/2 + 1/3 is 5/6, not 2/5), treating −(−a) as −a, swapping HCF and LCM, counting the whole numbers between 20 and 50 as 30 instead of 29, writing 2³ = 6, and confusing area with perimeter. Check each answer by substitution, estimation or the inverse operation.

One-glance revision list

Indian and international systems: 4,07,30,500 = 40,730,500, read as four crore seven lakh thirty thousand five hundred, or forty million seven hundred thirty thousand five hundred.
Divisibility: 4,15,512 is divisible by 8 because 512 = 8 × 64, and by 9 because its digit sum 4 + 1 + 5 + 5 + 1 + 2 = 18 is a multiple of 9.
HCF and LCM: 36 = 2² × 3² and 48 = 2⁴ × 3, so HCF = 2² × 3 = 12 and LCM = 2⁴ × 3² = 144; check 12 × 144 = 36 × 48 = 1,728.
Integers: (−8) − (−13) + (−6) = −8 + 13 − 6 = −1, and (−3) × (−4) × (−5) = −60 because there are three negative factors.
Fractions and decimals: 2 1/2 ÷ 5/6 = 5/2 × 6/5 = 3, and 0.25 × 0.4 = 0.100 = 0.1.
Exponents: 3⁴ × 3⁻² = 3² = 9, (5/2)⁻² = 4/25, and 4,50,00,000 = 4.5 × 10⁷ in standard form.
Squares and cubes: 1,764 = 2² × 3² × 7², so √1,764 = 42; 2,744 = 2³ × 7³, so ∛2,744 = 14.
Algebra: if 3x + 7 = 25, then 3x = 18 and x = 6; check: 3 × 6 + 7 = 25.
Ratio and percentage: ₹1,200 shared in the ratio 3 : 5 gives ₹450 and ₹750; an article bought for ₹800 and sold for ₹920 gives a profit of 120/800 × 100 = 15%.
Geometry and mensuration: a triangle with angles 47° and 68° has third angle 180° − 115° = 65°; a rectangle 12 m by 5 m has perimeter 34 m and area 60 m².
Data handling: for 4, 7, 7, 9 and 13 the mean is 40 ÷ 5 = 8, the median is 7 and the mode is 7.

Lessons in depth

8 longer lessons on Mathematics, Classes VI-VIII (Paper II), one per sub-topic, each with its key facts and a quick revision.

Practice questions

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1. In the number 7,40,53,962, what is the difference between the place value of 4 and the place value of 5? (A) 3,95,000 (B) 39,95,000 (C) 39,50,000 (D) 3,50,000

Answer: (C) 39,50,000. The 4 is in the ten-lakhs place, so its place value is 40,00,000; the 5 is in the ten-thousands place, so its place value is 50,000. Difference = 40,00,000 − 50,000 = 39,50,000.

2. A teacher has more than 24 pieces of chalk. Whether she makes bundles of 12, of 16 or of 24, exactly 7 pieces are left over each time. What is the least number of pieces she can have? (A) 31 (B) 41 (C) 48 (D) 55

Answer: (D) 55. The number minus 7 must be a common multiple of 12, 16 and 24, and the least one is their LCM, 48. So the number is 48 + 7 = 55; check: 55 = 4 × 12 + 7 = 3 × 16 + 7 = 2 × 24 + 7.

3. A can holds 2 3/4 litres of milk. Then 1 5/6 litres more is poured in, and 2 1/3 litres is used to make tea. How much milk is left in the can? (A) 1 1/4 litres (B) 2 1/4 litres (C) 2 7/12 litres (D) 3 1/4 litres

Answer: (B) 2 1/4 litres. With the common denominator 12: 2 9/12 + 1 10/12 − 2 4/12 = 1 15/12 = 2 3/12 = 2 1/4 litres. Forgetting to carry the extra whole from 15/12 gives 1 1/4.

4. What is 0.000000307 in standard form? (A) 3.07 × 10⁷ (B) 3.07 × 10⁻⁶ (C) 307 × 10⁻⁹ (D) 3.07 × 10⁻⁷

Answer: (D) 3.07 × 10⁻⁷. Moving the decimal point 7 places to the right gives 3.07, so 0.000000307 = 3.07 × 10⁻⁷. 307 × 10⁻⁹ has the same value but is not in standard form, which needs a number from 1 up to (but less than) 10.

5. Each letter E made with matchsticks uses 5 matchsticks. How many matchsticks are needed to make n such letters E placed separately in a row? (A) n + 5 (B) 5n (C) 5n + 1 (D) n⁵

Answer: (B) 5n. Each letter needs 5 matchsticks and the letters do not share any stick, so n letters need 5 × n = 5n matchsticks (for example, 3 letters need 15).

6. Solve: (x − 2)/5 + 3 = (x + 4)/3. (A) 9.5 (B) −11.5 (C) 17.5 (D) 19

Answer: (A) 9.5. Multiply both sides by 15: 3(x − 2) + 45 = 5(x + 4), so 3x + 39 = 5x + 20, 2x = 19 and x = 9.5. Check: 7.5/5 + 3 = 4.5 and 13.5/3 = 4.5.

7. The complete factorisation of 3x² − 27 is: (A) 3(x − 3)² (B) 3(x + 9)(x − 9) (C) (3x + 3)(x − 9) (D) 3(x + 3)(x − 3)

Answer: (D) 3(x + 3)(x − 3). Take out the common factor 3 to get 3(x² − 9). Then x² − 9 = (x + 3)(x − 3), so 3x² − 27 = 3(x + 3)(x − 3).

8. At simple interest, a sum of ₹6,500 amounts to ₹8,190 in 4 years. What is the rate of interest per annum? (A) 5.2% (B) 6.5% (C) 13% (D) 26%

Answer: (B) 6.5%. SI = 8190 − 6500 = ₹1,690. Rate = (1690 × 100) ÷ (6500 × 4) = 6.5% per annum.

9. Seven points A, B, C, D, E, F and G are marked in that order on a straight line. How many different line segments have two of these points as their end points? (A) 6 segments (B) 7 segments (C) 20 segments (D) 21 segments

Answer: (D) 21 segments. Each segment is fixed by choosing 2 of the 7 points, and AB is the same segment as BA. A has 6 partners, B has 5 new ones, and so on: 6 + 5 + 4 + 3 + 2 + 1 = 21.

10. The supplement of an angle is five times its complement. What is the angle? (A) 22.5° (B) 45° (C) 67.5° (D) 112.5°

Answer: (C) 67.5°. 180° − x = 5(90° − x) gives 180° − x = 450° − 5x, so 4x = 270° and x = 67.5°. Check: the complement is 22.5° and the supplement is 112.5° = 5 × 22.5°.

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