Shapes can be one-, two- or three-dimensional. This chapter helps you identify 3D shapes like cube, cuboid, cylinder, cone, sphere, prism and pyramid, and count their faces, edges and vertices.
Key points
A line has only length: one-dimensional (1D).
Flat shapes like a square or the surface of a table have length and breadth: two-dimensional (2D).
Solid objects like a brick, drum or ball have length, breadth and height: three-dimensional (3D).
A polygon is a closed figure with many straight sides.
In 2D shapes we talk of sides and vertices; in 3D shapes, of faces, edges and vertices.
A face is a flat or curved surface; an edge is where two faces meet; a vertex is a corner.
Point, line and plane are the basic ideas of geometry. This chapter explains lines, line segments, rays, collinear points, angles, how to name them and the types of angles.
Key points
A point shows a location and is named by a capital letter (point P).
A line extends endlessly on both sides and has infinitely many points. It is named by two points (line AB) or a small letter (line m).
Points on the same line are collinear points; points not on one line are non-collinear.
A plane is a flat surface extending endlessly in all directions (plane E).
A line segment is a part of a line with two endpoints (seg PQ). It is the shortest path between two points.
A ray starts at a point and goes on endlessly in one direction (ray AB, starting point A). Sun rays and torch light are examples.
An angle is formed by two rays with a common starting point. The rays are its arms; the common point is its vertex.
An angle is named with the vertex in the middle: ∠AOB or ∠BOA.
Angles are measured in degrees with a protractor.
Types: acute (less than 90°), right (90°), obtuse (between 90° and 180°), straight (180°).
Important terms
Line segment (रेषाखंड) – part of a line with two endpoints.
Ray (किरण) – part of a line with one starting point.
Collinear points – points on the same line.
Vertex – common starting point of the arms of an angle.
Formulas
Angle
Measure
Acute
0° < angle < 90°
Right
90°
Obtuse
90° < angle < 180°
Straight
180°
5 important questions
Q1. What is the difference between a line, a ray and a line segment?
A line has no endpoints, a ray has one starting point, and a line segment has two endpoints.
Q2. What are collinear points?
Three or more points that lie on one line.
Q3. Name the vertex and arms of ∠PQR.
Vertex Q; arms ray QP and ray QR.
Q4. Classify: 30°, 90°, 150°, 180°.
30° acute, 90° right, 150° obtuse, 180° straight.
Q5. How many angles are formed by three rays BA, BC, BD from point B?
Three: ∠ABC, ∠CBD and ∠ABD.
Exam tip
Always write the vertex letter in the middle while naming an angle.
This chapter revises large numbers in the Indian system and introduces whole numbers and integers (positive and negative numbers) with the number line and rules for adding and subtracting integers.
Numbers and shapes often follow patterns. This chapter shows patterns in counting, even, odd, square, triangular and cube numbers, prime number facts, and fractals — shapes whose small parts look like the whole.
Key points
Even numbers: 2, 4, 6, … Odd numbers: 1, 3, 5, …
Square numbers: 1, 4, 9, 16, 25, … (n × n). The sum of the first n odd numbers is n² (1 + 3 + 5 = 9).
2 is the smallest prime and the only even prime; 3 is the smallest odd prime.
Twin primes differ by 2 (3 and 5, 11 and 13). There are 8 pairs of twin primes between 1 and 100.
Co-prime numbers have only 1 as a common divisor.
Fractals: shapes where every small part looks like the whole (fern leaf, cauliflower). The word was coined by Benoit Mandelbrot in 1975; it comes from Latin meaning 'broken'.
Important terms
Pattern – an arrangement that follows a rule.
Square number – a number multiplied by itself.
Triangular number – number of dots that make a triangle.
Twin primes – primes with difference 2.
Fractal – a self-similar shape.
Formulas
nth square number = n²
nth triangular number = n × (n + 1) ÷ 2
1 + 3 + 5 + … (n odd numbers) = n²
5 important questions
Q1. Write the next three numbers: 1, 8, 27, 64, 125, …
These are cubes: 216, 343, 512.
Q2. Write the first five triangular numbers. Find the tenth.
1, 3, 6, 10, 15. Tenth = 10 × 11 ÷ 2 = 55.
Q3. Find 1 + 3 + 5 + 7 + 9 + 11 using a pattern.
Six odd numbers, so sum = 6² = 36.
Q4. Write the next three: 2, 9, 28, 65, …
Rule: n³ + 1. Next: 126, 217, 344.
Q5. Write any three pairs of twin primes.
(3, 5), (5, 7), (11, 13).
Exam tip
Find the rule first (add, multiply, square, cube), then continue the series.
This chapter explains divisors (factors), prime factors and divisibility, and how to find the GCD (greatest common divisor) and LCM (least common multiple) of numbers, with word problems.
Key points
A number that divides a given number exactly is its divisor (factor).
Factors that are prime numbers are prime factors. 12 = 2 × 2 × 3.
A number is divisible by another if it is a multiple of it.
Divisibility tests: by 2 – last digit even; by 3 – sum of digits divisible by 3; by 5 – last digit 0 or 5; by 9 – sum of digits divisible by 9; by 10 – last digit 0.
GCD is the greatest number that divides all the given numbers. It is the product of their common prime factors.
LCM is the smallest number divisible by all the given numbers. It is the product of common and uncommon prime factors.
GCD is used for 'greatest equal parts' problems; LCM for 'when will they meet again' problems.
Important terms
Divisor / Factor (विभाजक) – divides a number exactly.
Prime factor – a factor that is prime.
GCD (मसावि) – greatest common divisor (also called HCF).
This chapter is about comparing, adding and subtracting fractions, including mixed fractions, by first making denominators equal using the LCM.
Key points
In a/b, a is the numerator and b is the denominator.
Proper fraction: numerator smaller than denominator (3/5). Improper: numerator bigger or equal (7/4).
Mixed number: a whole number with a fraction (2 1/3 = 7/3).
To compare fractions, make the denominators the same and compare numerators.
To add or subtract fractions: find the LCM of the denominators, write equivalent fractions with the LCM as denominator, then add or subtract the numerators.
Write the answer in its simplest form (divide numerator and denominator by their GCD).
To add mixed numbers, change them to improper fractions first.
Important terms
Equivalent fractions – fractions of equal value (1/2 = 3/6).
Like fractions – same denominators.
Simplest form – numerator and denominator have no common factor except 1.
Formulas
a/c + b/c = (a + b)/c
a/b + c/d = (a × d + b × c)/(b × d), or use the LCM of b and d.
Mixed to improper: p q/r = (p × r + q)/r
5 important questions
Q1. Add 1/9 + 2/9.
(1 + 2)/9 = 3/9 = 1/3.
Q2. Add 6/10 + 3/5.
LCM of 10 and 5 = 10. 3/5 = 6/10. 6/10 + 6/10 = 12/10 = 6/5 = 1 1/5.
A ratio compares two quantities of the same kind by division. This chapter explains ratio, equal ratios, proportion and the unitary method for finding the price of one or many items.
Key points
The ratio of a to b is written a : b and equals a/b. a is the first term, b the second term.
To find a ratio, both quantities must be in the same unit.
A ratio has no unit.
Order matters: 2 : 3 is not the same as 3 : 2.
Ratios can be simplified like fractions: 12 : 18 = 2 : 3.
Two equal ratios form a proportion: 2 : 3 = 4 : 6.
Unitary method: find the value of one item by dividing, then the value of many items by multiplying.
Important terms
Ratio (गुणोत्तर) – comparison of two quantities by division.
Terms of a ratio – the two numbers in it.
Proportion (प्रमाण) – equality of two ratios.
Unitary method – finding the value of one unit first.
Formulas
a : b = a/b (same units, no unit in answer)
If a : b = c : d, then a × d = b × c.
Cost of many = cost of one × number of items.
5 important questions
Q1. Find the ratio of 40 cm to 2 m.
2 m = 200 cm. 40 : 200 = 1 : 5.
Q2. Simplify 45 : 60.
GCD is 15. 45 : 60 = 3 : 4.
Q3. 6 pens cost ₹90. Find the cost of 10 pens.
1 pen = 90 ÷ 6 = ₹15. 10 pens = 15 × 10 = ₹150.
Q4. Are 3 : 5 and 9 : 15 in proportion?
3 × 15 = 45 and 5 × 9 = 45. Equal, so yes.
Q5. In a class of 40 students, 18 are girls. Find the ratio of boys to girls.
Boys = 40 − 18 = 22. Boys : girls = 22 : 18 = 11 : 9.
Exam tip
Convert to the same unit before writing a ratio, and simplify it.
This chapter studies triangles and quadrilaterals (four-sided figures, including squares and rectangles): their parts and their types based on sides and angles.
Key points
A triangle has 3 sides, 3 vertices and 3 angles.
By sides: equilateral (all 3 sides equal), isosceles (2 sides equal), scalene (no sides equal).
By angles: acute-angled (all angles less than 90°), right-angled (one angle 90°), obtuse-angled (one angle more than 90°).
The sum of the three angles of a triangle is 180°.
A quadrilateral has 4 sides, 4 vertices, 4 angles and 2 diagonals.
In quadrilateral ABCD: opposite sides AB, DC and AD, BC (2 pairs); adjacent sides share a vertex (4 pairs); opposite angles ∠A, ∠C and ∠B, ∠D (2 pairs); adjacent angles (4 pairs).
A diagonal joins two opposite vertices (AC, BD).
Special quadrilaterals: square (all sides equal, all angles 90°), rectangle (opposite sides equal, all angles 90°), rhombus, parallelogram, trapezium.
Important terms
Equilateral / Isosceles / Scalene – triangle types by sides.
Quadrilateral (चौकोन) – a closed figure with four sides.
Diagonal (कर्ण) – line joining opposite vertices.
Adjacent – next to each other, sharing a common point.
Formulas
Sum of angles of a triangle = 180°
Sum of angles of a quadrilateral = 360°
5 important questions
Q1. A triangle has sides 5 cm, 5 cm and 7 cm. What type is it?
Two sides are equal, so it is isosceles.
Q2. A triangle has angles 90°, 40° and 50°. What type is it?
One angle is 90°, so it is a right-angled triangle.
Q3. Two angles of a triangle are 60° and 70°. Find the third.
180° − (60° + 70°) = 50°.
Q4. Name the pairs of opposite sides and diagonals of quadrilateral PQRS.
Opposite sides: PQ and SR, PS and QR. Diagonals: PR and QS.
Q5. How many pairs of adjacent angles does a quadrilateral have?
Four pairs.
Exam tip
Draw and label quadrilateral ABCD before listing pairs of sides and angles.
An equation is a statement that two expressions are equal. This chapter teaches the order of operations, how to form equations from word statements, and how to solve simple equations by doing the same operation on both sides.
Key points
Solve brackets first, then division, multiplication, addition and subtraction, in that order.
A letter like x, y, m that stands for an unknown number is a variable.
An equation has an '=' sign; the expressions on the left and right are equal, e.g. x + 5 = 12.
Adding, subtracting or multiplying the same number on both sides, or dividing both sides by the same non-zero number, keeps the equation true.
The value of the variable that makes the equation true is its solution.
Always check the solution by putting it back in the equation.
Important terms
Variable (चल) – a letter standing for an unknown number.
Expression – numbers and variables joined by operations (3x + 2).
Equation (समीकरण) – a statement that two expressions are equal.
Solution – the value of the variable that satisfies the equation.
Formulas
If a = b, then a + c = b + c, a − c = b − c, a × c = b × c, a ÷ c = b ÷ c (c ≠ 0).
This fun chapter has maths games and puzzles: smart cards that guess a hidden number, the angles of a five-pointed star, pentomino puzzles and addition–subtraction grids with fractions.
Key points
Smart cards: each of five cards has 16 numbers from 1 to 31. A friend says which cards hold his number; adding the first (smallest) number on each of those cards gives his number. (The cards start with 1, 2, 4, 8, 16.)
Star: a five-pointed star has a regular pentagon inside it.
Useful facts: angles on a straight line add to 180°; the angles of a triangle add to 180°; all angles of a regular pentagon are equal.
The sum of the angles of a pentagon is 540°, so each angle of a regular pentagon is 108°.
Angle next to 108° on a straight line = 180° − 108° = 72°. Each point of the star = 180° − 72° − 72° = 36°.
A pentomino is a shape of 5 equal squares joined side by side. There are 12 pentominoes, named by letters (F, I, L, N, P, T, U, V, W, X, Y, Z).
The 12 pentominoes (area 60 squares) can exactly fill rectangles of 6 × 10, 5 × 12, 4 × 15 and 3 × 20 — no gaps, no overlaps (tessellation).
Addition–subtraction grids help practise fractions.
Important terms
Regular polygon – all sides and angles equal.
Pentomino – shape of five joined squares.
Tessellation – covering a surface with shapes without gaps or overlaps.
Formulas
Sum of angles of a polygon with n sides = (n − 2) × 180°
Pentagon: (5 − 2) × 180° = 540°
5 important questions
Q1. Find the sum of the angles of a pentagon.
(5 − 2) × 180° = 540°.
Q2. Find each angle of a regular pentagon.
540° ÷ 5 = 108°.
Q3. In the star, one base angle of a point triangle is 72°. Find the angle at the tip.
180° − 72° − 72° = 36°.
Q4. How many squares are in all 12 pentominoes together?
12 × 5 = 60 squares. That is why they fit a 6 × 10 rectangle.
Q5. Find the sum of the angles of a hexagon.
(6 − 2) × 180° = 720°.
Exam tip
Remember the polygon angle-sum formula; it solves many star and shape puzzles.
A fraction whose denominator is 10, 100, 1000 … is a decimal fraction. This chapter teaches reading, writing, comparing, adding and subtracting decimals and their uses in daily life.
Key points
7/10 = 0.7, 23/100 = 0.23, 125/1000 = 0.125.
Places after the decimal point: tenths, hundredths, thousandths.
To convert a fraction, make its denominator 10, 100 or 1000: 3/4 = 75/100 = 0.75.
To compare decimals, compare whole parts first, then tenths, then hundredths. Adding zeros at the end does not change the value (0.5 = 0.50).
To add or subtract, write the numbers so that decimal points are one below another; fill blanks with zeros.
Decimals are used in shops, money, measurement, calculators, science and engineering.
When a shopkeeper sells something for more than its cost, he makes a profit; if for less, he makes a loss. This chapter teaches cost price, selling price, profit, loss and total cost including other expenses.
Key points
Cost price (CP): the price at which an item is bought.
Selling price (SP): the price at which it is sold.
If SP > CP, there is a profit. If SP < CP, there is a loss. If SP = CP, no profit, no loss.
Other expenses like transport, repairs or labour are added to the cost price to get the total cost price.
Profit or loss is always calculated on the total cost price.
Important terms
Cost price (खरेदी किंमत) – buying price.
Selling price (विक्री किंमत) – selling price.
Profit (नफा) – gain when SP is more than CP.
Loss (तोटा) – when SP is less than CP.
Formulas
Profit = SP − CP
Loss = CP − SP
SP = CP + Profit; SP = CP − Loss
Total CP = Original CP + Other expenses
5 important questions
Q1. CP = ₹418, profit = ₹72. Find SP.
SP = 418 + 72 = ₹490.
Q2. SP = ₹1275, loss = ₹84. Find CP.
CP = 1275 + 84 = ₹1359.
Q3. A cycle bought for ₹3,500 was sold for ₹3,150. Find profit or loss.
SP < CP. Loss = 3500 − 3150 = ₹350.
Q4. A TV bought for ₹12,000; ₹500 spent on transport; sold for ₹13,000. Find profit.
A percentage is a fraction with denominator 100. This chapter teaches what percent means, converting between fractions, decimals and percentages, and finding a percentage of a number in daily-life problems.
Key points
Percent means 'per hundred'. Symbol: %.
A percentage is a fraction whose denominator is 100, or a ratio whose second term is 100: 25% = 25/100.
To change a fraction to a percentage, multiply by 100: 3/4 = 3/4 × 100 = 75%.
To change a percentage to a fraction, divide by 100 and simplify: 40% = 40/100 = 2/5.
To find a percentage of a number: (percentage ÷ 100) × number.
Percentages are used in marks, results, profit and loss, discounts, interest rates and inflation.
Important terms
Percent (शेकडा) – out of every hundred.
Percentage – a number expressed out of 100.
Formulas
x% = x/100
x% of N = (x × N) ÷ 100
Percentage = (part ÷ whole) × 100
5 important questions
Q1. Find 10% of 500.
(10 × 500) ÷ 100 = 50.
Q2. Write 3/5 as a percentage.
3/5 × 100 = 60%.
Q3. Write 35% as a fraction in simplest form.
35/100 = 7/20.
Q4. Ravi scored 42 marks out of 50. What is his percentage?
(42 ÷ 50) × 100 = 84%.
Q5. In a school of 800 students, 45% are girls. How many boys are there?
This chapter teaches how to use geometry instruments — ruler, divider, compass, protractor and set squares — to draw line segments, angles, perpendiculars and angle bisectors.
Key points
A line segment of a given length is drawn with a ruler; a divider can transfer the length accurately.
A protractor is used to measure an angle and to draw an angle of a given measure.
Lines that meet at a right angle are perpendicular to each other.
A perpendicular to a line can be drawn through a point using a set square or a compass.
The perpendicular bisector of a segment passes through its midpoint and is perpendicular to it.
A ray that divides an angle into two equal angles is its angle bisector.
To bisect an angle with a compass: with centre at the vertex draw an arc cutting both arms; from these two points draw arcs of equal radius that cut each other; join the vertex to the cutting point.
Always keep a sharp pencil and keep construction arcs visible.
Important terms
Divider – an instrument to measure and copy lengths.
Perpendicular (लंब) – meeting at 90°.
Angle bisector (कोनदुभाजक) – ray dividing an angle into two equal parts.
Perpendicular bisector – line perpendicular to a segment through its midpoint.
Formulas
If ray BD bisects ∠ABC, then ∠ABD = ∠DBC = ½ ∠ABC.
5 important questions
Q1. Which instrument draws an angle of a given measure?
A protractor.
Q2. What is an angle bisector?
A ray that divides an angle into two equal parts.
Q3. ∠RTG = 105° is bisected. Find each part.
105° ÷ 2 = 52.5°.
Q4. Write the steps to bisect ∠PQR with a compass.
With centre Q draw an arc cutting QP at A and QR at B. With centres A and B and equal radius draw arcs cutting at C. Draw ray QC; it bisects ∠PQR.
Q5. When are two lines perpendicular?
When they intersect at a right angle (90°).
Exam tip
Do not erase construction arcs; they show your method and earn marks.
Perimeter is the total length of the boundary of a figure, and area is the space it covers. This chapter covers perimeter of polygons and area of rectangles, squares and triangles.
Key points
Perimeter is the sum of the lengths of all sides of a closed figure. Its unit is the same as the unit of the sides (cm, m).
Area is the space occupied by a figure on a flat surface. Its unit is square units (cm², m²).
Rectangle: opposite sides equal. Square: all sides equal.
Area of a triangle is half the area of a rectangle with the same base and height.
In a right-angled triangle, the two sides forming the right angle act as base and height.
Use the same unit for all sides before calculating.
Important terms
Perimeter (परिमिती) – boundary length.
Area (क्षेत्रफळ) – surface covered.
Base and height – a side of a triangle and the perpendicular distance to it from the opposite vertex.
Formulas
Perimeter of rectangle = 2 × (l + b); of square = 4 × side
Area of rectangle = l × b; of square = side²
Area of triangle = ½ × base × height
Area of right-angled triangle = ½ × product of sides forming the right angle
5 important questions
Q1. Find the perimeter of a rectangle 15 m by 9 m.
2 × (15 + 9) = 48 m.
Q2. The perimeter of a square is 64 cm. Find its side and area.
Side = 64 ÷ 4 = 16 cm. Area = 16 × 16 = 256 cm².
Q3. Find the area of a triangle with base 12 cm and height 7 cm.
½ × 12 × 7 = 42 cm².
Q4. A right-angled triangle has sides 6 cm and 8 cm forming the right angle. Find its area.
½ × 6 × 8 = 24 cm².
Q5. Find the perimeter of a triangle with sides 5 cm, 7 cm and 9 cm.
5 + 7 + 9 = 21 cm.
Exam tip
Write units carefully: cm for perimeter, cm² for area.
In a right-angled triangle, the square of the longest side (hypotenuse) equals the sum of the squares of the other two sides. This rule was known to the ancient Indian mathematician Baudhayana and the Greek mathematician Pythagoras.
Key points
In a right-angled triangle, the side opposite the right angle is the hypotenuse. It is the longest side.
Theorem: (hypotenuse)² = (side 1)² + (side 2)².
Baudhayana stated this rule in the Shulbasutras for making altars, long ago.
Three whole numbers that satisfy this rule form a Baudhayana–Pythagoras triplet, e.g. (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (9, 40, 41).
Multiplying a triplet by the same number gives another triplet: (6, 8, 10).
If the sides of a triangle form such a triplet, the triangle is right-angled.
Important terms
Hypotenuse (कर्ण) – side opposite the right angle.
Right-angled triangle – triangle with one 90° angle.
Triplet – a set of three numbers.
Formulas
c² = a² + b² (c = hypotenuse)
a² = c² − b²
5 important questions
Q1. The sides forming the right angle are 6 cm and 8 cm. Find the hypotenuse.
c² = 36 + 64 = 100, so c = 10 cm.
Q2. Hypotenuse 13 cm, one side 5 cm. Find the other side.
b² = 169 − 25 = 144, so b = 12 cm.
Q3. Is (8, 15, 17) a triplet?
8² + 15² = 64 + 225 = 289 = 17². Yes.
Q4. Is a triangle with sides 4, 5, 6 cm right-angled?
4² + 5² = 41 ≠ 36. No.
Q5. Hypotenuse is 41 cm and one side is 9 cm. Find the other.
b² = 1681 − 81 = 1600, so b = 40 cm.
Exam tip
Learn the common triplets (3,4,5), (5,12,13), (8,15,17), (7,24,25); they save time.
Banks keep our money safe and give loans. This chapter explains bank accounts, principal, rate, period and simple interest, and how to calculate interest and total amount.
Key points
A bank accepts deposits and gives loans. It pays interest on deposits and charges interest on loans.
Principal (P): the amount deposited in or borrowed from the bank.
Period (N): the time for which money is deposited or borrowed, in years.
Rate of interest (R): the interest on ₹100 for one year, written as % per year.
Interest (I): extra money received on a deposit or paid on a loan.
Simple interest is calculated on the principal only.
Amount = Principal + Interest.
Banks offer savings accounts, fixed deposits and loans. Online and mobile banking are common now.
Important terms
Principal (मुद्दल) – money deposited or borrowed.
Rate of interest (व्याजाचा दर) – interest per ₹100 per year.
Simple interest (सरळ व्याज) – interest on principal only.
Amount (रास) – principal plus interest.
Formulas
I = (P × R × N) ÷ 100
Amount = P + I
5 important questions
Q1. Find the simple interest on ₹5,000 at 8% per year for 3 years.
I = 5000 × 8 × 3 ÷ 100 = ₹1,200.
Q2. Find the amount in Q1.
Amount = 5,000 + 1,200 = ₹6,200.
Q3. Raju borrowed ₹12,000 at 10% for 2 years. How much will he repay?
Data is information in numbers. This chapter explains frequency, frequency tables, bar graphs and infographics for showing data clearly.
Key points
Collected numerical information is data.
Frequency is how many times a value (score) occurs.
A frequency table lists each value with its frequency; tally marks help in counting.
A bar graph shows data with bars (columns) of equal width. The height of each bar shows its value.
A bar graph has a title, labelled axes, and a suitable scale (e.g. 1 cm = 10 students). Gaps between bars are equal.
Infographics show information artistically and attractively with pictures, icons and numbers. Scientists use them for planets and stars; ancient people drew pictures on walls.
Graphs make comparison easy at a glance.
Important terms
Data (सामग्री) – information in numbers.
Frequency (वारंवारता) – number of times a value occurs.
Bar graph (स्तंभालेख) – data shown with columns.
Scale – what one unit of length on the graph stands for.
Infographic – information shown with attractive pictures.
Formulas
Bar height = value ÷ scale
5 important questions
Q1. What is frequency?
The number of times a particular value occurs in data.
Q2. Marks of 10 students: 5, 7, 5, 8, 9, 7, 5, 8, 7, 5. Make a frequency table.
5 → 4, 7 → 3, 8 → 2, 9 → 1.
Q3. If the scale is 1 cm = 20 trees, what bar height shows 90 trees?
90 ÷ 20 = 4.5 cm.
Q4. What should every bar graph have?
A title, labelled axes, a scale, and bars of equal width with equal gaps.
Q5. What is an infographic?
Information presented in an artistic and attractive way using pictures and numbers.
Exam tip
Choose a scale that keeps the tallest bar within your page.
This chapter has interesting mathematics activities: the life of the great Indian mathematician Srinivasa Ramanujan, the Tower of Hanoi puzzle, a number-guessing trick, and finding areas of squares by cutting and arranging.
Key points
Srinivasa Ramanujan (22 December 1887 – 26 April 1920) was a world-famous Indian mathematician who worked on thousands of results (theorems).
22 December, his birthday, is celebrated as National Mathematics Day.
Tower of Hanoi: move discs from one peg to another, one at a time, never placing a bigger disc on a smaller one.
With 1, 2, 3, 4 discs the least moves are 1, 3, 7, 15. Each time, moves = 2 × previous + 1.
Number tricks work because of simple algebra hidden inside them.
Squares of side 6 units and 3 units have areas 36 and 9 square units; cutting and joining pieces helps find the area of a combined square.
Important terms
Theorem – a mathematical statement that is proved.
Tower of Hanoi – a disc-moving puzzle.
Number trick – a game where a hidden number is found by calculation.
Formulas
Least moves in Tower of Hanoi with n discs: t = 2ⁿ − 1
Area of square = side²
Important dates and people
Srinivasa Ramanujan: born 22-12-1887, died 26-04-1920.
National Mathematics Day: 22 December.
5 important questions
Q1. When is National Mathematics Day celebrated and why?
On 22 December, the birthday of Srinivasa Ramanujan.
Q2. What are the rules of the Tower of Hanoi?
Move one disc at a time, and never put a bigger disc on a smaller one.
Q3. How many moves are needed for 5 discs?
2⁵ − 1 = 32 − 1 = 31 moves.
Q4. Find the areas of squares with sides 6 units and 3 units.
6 × 6 = 36 sq units; 3 × 3 = 9 sq units.
Q5. What is the total area of these two squares?
36 + 9 = 45 sq units.
Exam tip
Remember Ramanujan's birth date and the Tower of Hanoi formula 2ⁿ − 1.