A solid (or 3D shape) has three dimensions: length, width and height. Boxes, tins, dice, balls and ice-cream cones are all solids. In this lesson you will describe solids by their faces, edges and vertices, sort them into groups, and learn about nets – the flat patterns that fold up to make a solid.
A face is a surface of the solid. It can be flat (like the side of a box) or curved (like the side of a tin).
An edge is a line where two faces meet.
A vertex (plural vertices) is a corner where edges meet. The pointed top of a cone or pyramid is called its apex.
Figure 1 – A cube, a square-based pyramid, a cone and a cuboid. Image: MamtaRani, CC BY 4.0, via Wikimedia Commons.
Solid
Faces
Edges
Vertices
What the faces are
Cube
6
12
8
6 equal squares
Cuboid
6
12
8
6 rectangles (opposite faces equal)
Triangular prism
5
9
6
2 triangles, 3 rectangles
Square-based pyramid
5
8
5
1 square, 4 triangles
Triangular pyramid (tetrahedron)
4
6
4
4 triangles
Cylinder
3
2
0
2 flat circles, 1 curved surface
Cone
2
1
1
1 flat circle, 1 curved surface
Sphere
1
0
0
1 curved surface
Table 1 – Properties of common solids. (Some textbooks say a cylinder has 3 “surfaces” and do not count curved edges; follow your teacher’s convention.)
Classifying solids
Polyhedra have only flat faces, which are polygons (cubes, cuboids, prisms, pyramids). Solids with curved surfaces (cylinders, cones, spheres) are not polyhedra.
A prism has the same shape all the way through (its cross-section), with two identical end faces joined by rectangles. It is named after its end face: triangular prism, hexagonal prism. A cuboid is a rectangular prism.
A pyramid has a polygon base and triangular faces that meet at an apex. It is named after its base: square-based pyramid, triangular pyramid.
For any prism with an n-sided end face: faces = n + 2, edges = 3n, vertices = 2n. For any pyramid with an n-sided base: faces = n + 1, edges = 2n, vertices = n + 1.
Euler’s rule: for every polyhedron, faces + vertices − edges = 2. Check the cube: 6 + 8 − 12 = 2 ✔. It is a great way to check your counting.
Nets of solids
A net is a flat (2D) shape that can be cut out and folded to make a solid. Open out a cereal box carefully and you will see its net. Each face of the solid appears once in the net.
Solid
Its net is made of
Cube
6 equal squares
Cuboid
6 rectangles in 3 matching pairs
Triangular prism
2 triangles and 3 rectangles
Square-based pyramid
1 square with 4 triangles around it
Cylinder
2 circles and 1 rectangle (the rectangle wraps around the circles exactly)
Cone
1 circle and a sector (a “fan” shape cut from a larger circle)
Figure 2 – The 11 different nets of a cube. Any other arrangement of six squares will not fold into a cube. Image: BergerGießen, CC BY-SA 4.0, via Wikimedia Commons.
Not every arrangement of six squares makes a cube. A straight row of six squares does not work, and neither does a 2 × 3 block of squares – when you fold them, faces overlap and some sides are left open. A quick test: imagine folding, and make sure no two squares land on the same face.
How to make a solid from its net
Draw the net accurately on card with a ruler (for a cube, use squares of side 5 cm).
Add small tabs (flaps) on alternate edges so the faces can be glued together.
Cut out the net, including the tabs.
Score the fold lines lightly with a ruler and the back of scissors, then fold along every line.
Glue or tape each tab inside the next face and hold it until it sticks.
Check your solid: count its faces, edges and vertices and test Euler’s rule.
Worked examples
Example 1: Counting a hexagonal prism
Question: Find the number of faces, edges and vertices of a hexagonal prism, and check with Euler’s rule.
The end faces are hexagons, so n = 6.
Faces = 2 hexagons + 6 rectangles = 8. Edges = 6 + 6 + 6 = 18 (top, bottom and the sides joining them). Vertices = 6 on top + 6 on the bottom = 12.
Euler’s rule: 8 + 12 − 18 = 2 ✔
Answer: 8 faces, 18 edges and 12 vertices.
Example 2: A pyramid with an octagonal base
Question: How many faces, edges and vertices does an octagonal pyramid have?
Base: 1 octagon. Sides: 8 triangles. Faces = 9.
Edges: 8 around the base + 8 going up to the apex = 16.
Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 1, Strand 1.4 Geometry, Topic 1.4.1 Solids and Plane Shapes – Learning Outcomes 1.4.1.1 (classifying solids by their properties), 1.4.1.2 (drawing the net of a solid), 1.4.1.3 (creating a solid from its net) and 1.4.1.6 (solving problems involving solids and plane shapes).
A plane shape is a flat, two-dimensional (2D) shape. A polygon is a plane shape that is closed and made only of straight sides. Triangles, squares and hexagons are polygons; a circle is not (it is curved), and an open shape with a gap is not either. In this lesson you will name and classify polygons by their properties, and then use them to make tessellations – patterns that cover a surface with no gaps and no overlaps, like floor tiles or a honeycomb.
Naming polygons
Polygons are named by their number of sides. The number of vertices (corners) is always the same as the number of sides.
Sides
Name
Everyday example
3
triangle
a slice of pizza, a warning sign
4
quadrilateral
a page, a door, a kite
5
pentagon
the outline of a simple house shape
6
hexagon
a cell of a honeycomb, a nut for a bolt
7
heptagon
8
octagon
a STOP sign
9
nonagon
10
decagon
12
dodecagon
the outline of the UK £1 coin
Classifying polygons by their properties
A regular polygon has all sides equal and all angles equal (for example a square or an equilateral triangle). An irregular polygon does not.
Count the sides and vertices, look for equal sides (often marked with small dashes), right angles (marked with a small square) and parallel sides (marked with arrows).
A diagonal joins two vertices that are not next to each other. A polygon with n sides has n(n − 3) ÷ 2 diagonals.
A convex polygon has every corner pointing outwards. A concave polygon has at least one corner pointing inwards, like an arrowhead.
You will study triangles and quadrilaterals in much more detail in Term 2.
Tessellations
A tessellation (or tiling) is a pattern of shapes that fit together to cover a flat surface completely, with no gaps and no overlaps. You can see tessellations in floor and wall tiles, brick walls, paving stones, honeycombs, patchwork quilts, fabric and gift-wrap patterns, and in the famous tilings of the Alhambra palace in Spain.
Figure 1 – Bees build honeycomb from hexagons. Hexagons tessellate with no wasted space between the cells. Image: Merdal at Turkish Wikipedia, CC BY-SA 3.0, via Wikimedia Commons.
Which shapes tessellate?
Where the corners of the shapes meet at a point, the angles must add up to exactly 360° (a full turn). The corner angle of a regular triangle is 60°, of a square 90°, of a regular pentagon 108° and of a regular hexagon 120°.
Equilateral triangles: 6 × 60° = 360° ✔
Squares: 4 × 90° = 360° ✔
Regular hexagons: 3 × 120° = 360° ✔
Regular pentagons: 3 × 108° = 324°, which leaves a 36° gap, and 4 × 108° = 432° overlaps ✘
So only three regular polygons tessellate on their own: the equilateral triangle, the square and the regular hexagon. These are called regular tessellations. Two or more different regular polygons can also tessellate together – for example regular octagons and squares (135° + 135° + 90° = 360°), a pattern you often see on bathroom floors. And any triangle or any quadrilateral, even an irregular one, will tessellate if you rotate copies of it.
TRY ITRegular polygon and tessellation explorer
Move the slider to change the number of sides. Then press “Try to tessellate” to see whether copies of the shape fit together with no gaps.
Figure 2 – A tessellation in the Alhambra, Granada, Spain. The artist M. C. Escher sketched this tiling in 1936 and it inspired his own famous interlocking designs. Image: gruban, CC BY-SA 2.0, via Wikimedia Commons.
Make your own tessellating tile
Start with a square of card (about 6 cm by 6 cm).
Cut a shape (a “nibble”) out of the top edge.
Slide the piece straight down – do not turn it – and tape it to the bottom edge.
Do the same from the left edge to the right edge if you like.
Trace around your new tile over and over, fitting each copy against the last. Colour the pattern – maybe the tiles look like fish, birds or hibiscus flowers!
This works because whatever was removed from one side is added to the opposite side, so the tiles still fit together exactly, just like the original squares.
Worked examples
Example 1: Classifying a polygon
Question: A shape has 8 equal sides and 8 equal angles. Name it and state its number of vertices and diagonals.
Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 1, Strand 1.4 Geometry, Topic 1.4.1 Solids and Plane Shapes – Learning Outcomes 1.4.1.4 (classifying polygons by their properties), 1.4.1.5 (creating tessellating patterns with plane shapes) and 1.4.1.6 (solving problems involving solids and plane shapes).
Perimeter is the total distance around the outside of a shape – the length of fence needed to go around a garden, or of ribbon to go around a picture frame. It is a length, so it is measured in units such as cm, m or km. Area is the amount of surface a shape covers – how much grass, paint or tiles you need. It is measured in square units such as cm2 (square centimetres) and m2 (square metres). In this lesson you will find the perimeter and area of squares, rectangles and triangles, and the perimeter of compound (L-shaped) figures.
To find the perimeter of any shape with straight sides, add the lengths of all the sides. For common shapes there are shortcuts:
Shape
Perimeter
Why
Square with side s
P = 4s
Four equal sides.
Rectangle with length l and width w
P = 2l + 2w = 2(l + w)
Two lengths and two widths.
Triangle with sides a, b, c
P = a + b + c
Add the three sides.
Any polygon
Add every side
Mark each side as you add it so you do not miss one.
Area
Area tells you how many unit squares fit inside a shape. A square that is 1 cm by 1 cm has an area of 1 cm2. On a grid you can find area by counting squares. For a shape with curved or uneven edges, estimate: count the whole squares, then count each square that is half or more covered as one, and ignore squares that are less than half covered.
Units: mm2 for tiny areas (a stamp), cm2 for a page, m2 for a room or a garden, and km2 for an island. (Trinidad and Tobago covers roughly 5,100 km2.) Large plots of land are often measured in hectares: 1 hectare = 100 m × 100 m = 10,000 m2.
Rectangle:A = l × w. A 6 cm by 4 cm rectangle holds 6 squares in each of 4 rows, so 24 cm2.
Square:A = s × s = s2.
Triangle:A = ½ × base × height, where the height is measured at a right angle (perpendicular) to the base. Any triangle is exactly half of a rectangle with the same base and height.
TRY ITPerimeter and area on a grid
Drag the sliders (or use the arrow keys) to change the rectangle. Watch how the perimeter and the area change – they are not the same thing!
Worked examples
Example 1: A rectangle
Question: Find the perimeter and area of a rectangle 9 cm long and 5 cm wide.
P = 2(l + w) = 2 × (9 + 5) = 2 × 14 = 28 cm.
A = l × w = 9 × 5 = 45 cm2.
Answer: Perimeter 28 cm, area 45 cm2.
Example 2: A square
Question: Find the perimeter and area of a square of side 7 m.
P = 4 × 7 = 28 m.
A = 7 × 7 = 49 m2.
Answer: Perimeter 28 m, area 49 m2. Notice: this square and the rectangle in Example 1 have the same perimeter number (28) but different areas.
Example 3: A right-angled triangle
Question: A triangle has sides 6 cm, 8 cm and 10 cm. The 6 cm and 8 cm sides meet at a right angle. Find its perimeter and area.
P = 6 + 8 + 10 = 24 cm.
The two sides that meet at the right angle are the base and the perpendicular height: base 8 cm, height 6 cm.
A = ½ × 8 × 6 = ½ × 48 = 24 cm2.
Answer: Perimeter 24 cm, area 24 cm2.
Example 4: Area of any triangle
Question: A triangle has a base of 12 cm and a perpendicular height of 5 cm. Find its area.
A = ½ × base × height = ½ × 12 × 5.
½ × 60 = 30.
Answer: 30 cm2. (The slanted sides are not needed for the area.)
Example 5: An L-shaped (compound) figure
Question: A garden is a 10 m by 8 m rectangle with a 4 m by 3 m rectangle cut out of one corner. Find its perimeter and area.
Work out the missing sides: going around, the sides are 10 m (bottom), 5 m (right, 8 − 3), 4 m (step in), 3 m (up), 6 m (top, 10 − 4) and 8 m (left).
Perimeter = 10 + 5 + 4 + 3 + 6 + 8 = 36 m. (Shortcut: cutting a corner out of a rectangle does not change its perimeter – it is still 2 × (10 + 8) = 36 m.)
Area method 2 (split): a 10 × 5 strip plus a 6 × 3 piece = 50 + 18 = 68 m2 ✔
Answer: Perimeter 36 m, area 68 m2.
Example 6: A real-life problem
Question: A rectangular garden is 12 m long and 7.5 m wide. Fencing costs $45 per metre, and one bag of grass seed covers 20 m2. Find the cost of the fence and the number of bags of seed needed.
You cannot buy half a bag, and 4 bags would not be enough, so round up.
Answer: The fence costs $1,755 and 5 bags of seed are needed.
Example 7: Finding a missing side
Question: A rectangle has an area of 54 cm2 and a length of 9 cm. Find its width.
A = l × w, so 54 = 9 × w.
w = 54 ÷ 9 = 6.
Answer: 6 cm.
Common mistakes
Wrong units. Perimeter is a length (cm); area is in square units (cm2). Writing “45 cm” for an area loses marks.
Adding only two sides. The perimeter of a 9 cm by 5 cm rectangle is 28 cm, not 14 cm.
Forgetting the ½ for a triangle, or using a slanted side instead of the perpendicular height.
Mixing units. Change everything to the same unit before calculating (for example 2 m by 50 cm → 200 cm by 50 cm).
Thinking a bigger perimeter always means a bigger area. A 9 × 1 rectangle has perimeter 20 but area only 9; a 5 × 5 square also has perimeter 20 but area 25.
Practice questions
Try these yourself first, then open each answer to check your working.
1. Find the perimeter and area of a rectangle 15 cm by 4 cm.
Show answer
Perimeter = 2 × (15 + 4) = 38 cm. Area = 15 × 4 = 60 cm2.
2. A square has a perimeter of 36 cm. Find its side and its area.
Show answer
Side = 36 ÷ 4 = 9 cm. Area = 9 × 9 = 81 cm2.
3. A triangle has a base of 10 m and a perpendicular height of 7 m. Find its area.
Show answer
½ × 10 × 7 = 35 m2.
4. Find the area of a rectangle 2 m long and 50 cm wide, in cm2.
Show answer
2 m = 200 cm, so 200 × 50 = 10,000 cm2. (That is 1 m2 – which shows that 1 m2 = 10,000 cm2, not 100!)
5. A floor is 5 m by 4 m. How many square tiles of side 50 cm are needed to cover it?
Show answer
Floor area = 20 m2; each tile = 0.5 × 0.5 = 0.25 m2; 20 ÷ 0.25 = 80 tiles. (Or: 10 tiles along the 5 m side × 8 along the 4 m side = 80.)
6. A 12 m by 9 m rectangle has a 5 m by 4 m rectangle cut from one corner. Find the perimeter and area of the shape that is left.
Show answer
Perimeter = 2 × (12 + 9) = 42 m (cutting a corner does not change it). Area = 108 − 20 = 88 m2.
7. Draw (or describe) three different rectangles with whole-number sides and a perimeter of 20 cm. Which has the biggest area?
Show answer
For example 9 × 1 (area 9 cm2), 6 × 4 (24 cm2) and 5 × 5 (25 cm2). The square has the biggest area.
8. A hectare is the area of a 100 m by 100 m square. How many square metres is that?
Show answer
100 × 100 = 10,000 m2.
Quick check
1. What is the difference between perimeter and area?
Show answer
Perimeter is the distance around a shape (a length); area is the surface inside it (square units).
2. Write the formula for the area of a triangle.
Show answer
Area = ½ × base × perpendicular height.
3. Which unit would you use for the area of a classroom floor?
Show answer
Square metres (m2).
4. A shape on a grid covers 9 whole squares and 4 squares that are more than half covered. Estimate its area in square units.
Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 1, Strand 1.5 Measurement, Topic 1.5.3 Perimeter – Learning Outcomes 1.5.3.1 (identifying the perimeter of plane shapes), 1.5.3.2 (calculating perimeter) and 1.5.3.3 (solving perimeter problems to a stated degree of accuracy); and Topic 1.5.4 Area – Learning Outcomes 1.5.4.1 (the concept of area), 1.5.4.2 (units for area), 1.5.4.3 (measuring and estimating surface area) and 1.5.4.4 (the area of triangles, squares and rectangles).
To simplify an algebraic expression means to write it in a shorter, neater form that has exactly the same value for every number you substitute. 3x + 5x and 8x are equal for every value of x, but 8x is simpler. In this lesson you will learn to spot like terms, collect them, multiply and divide terms, and use the distributive law to expand brackets.
Like terms have exactly the same letters, with the same powers. Only the numbers in front (the coefficients) may differ.
Like terms
Unlike terms
3x and 7x
3x and 3y (different letters)
2ab and 5ba (ab = ba)
x and x2 (different powers)
y2 and 4y2
4a and 4 (one has a letter, one does not)
Think of it like money: you can add 3 five-dollar notes and 7 five-dollar notes to get 10 five-dollar notes, but 3 five-dollar notes plus 3 ten-dollar notes is not “6 fifteen-dollar notes”.
Adding and subtracting: collect like terms
Add or subtract the coefficients of like terms; the letters stay the same: 3x + 5x = 8x and 7y − 2y = 5y.
Remember an invisible 1: x means 1x, so 5x − x = 4x.
Group like terms together, keeping each term’s sign with it: 4a + 3b + 2a − b = (4a + 2a) + (3b − b) = 6a + 2b.
Unlike terms cannot be combined: 6a + 2b is already as simple as it gets.
Multiplying and dividing terms
Unlike adding, you can multiply or divide any terms. Multiply (or divide) the numbers, then deal with the letters.
3a × 4b = 3 × 4 × a × b = 12ab
2x × 5x = 2 × 5 × x × x = 10x2
12ab ÷ 3a = 12ab3a = 4b (12 ÷ 3 = 4, and the a on top cancels with the a below)
10x ÷ 5 = 2x
The distributive law: expanding brackets
The distributive law says that a number outside a bracket multiplies every term inside it: a(b + c) = ab + ac. You already use it in mental arithmetic: 3 × 14 = 3 × (10 + 4) = 30 + 12 = 42.
Figure 1 – The area model: a rectangle 3 units tall and (x + 4) units long has area 3 × x plus 3 × 4.
Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 1, Strand 1.6 Algebra, Topic 1.6.1 Introducing Algebra – Learning Outcomes 1.6.1.7 (identifying like and unlike terms), 1.6.1.8 (simplifying expressions involving the four operations) and 1.6.1.9 (simplifying expressions using the distributive law).
Updates
2026/10/25
– Put each lettered part of the questions and answers on its own line.
Algebra is a way of doing maths with letters that stand for numbers. It lets us write rules that work for any number, solve puzzles where a number is unknown, and describe patterns in science, business and computing. The big idea is simple: a letter such as n is just a number we either do not know yet or that can change. In this lesson you will learn the language of algebra – variables, constants, terms and expressions – how to turn words into algebra, and how to substitute numbers into an expression.
Varying quantities, variables and constants
Many quantities in everyday life vary (change): the temperature during the day, the price of a taxi ride as the distance increases, your height as you grow, the credit left on a phone as you use it. Other quantities stay the same: there are always 7 days in a week and 60 minutes in an hour.
A variable is a quantity that can change or is unknown. We represent it with a letter: t for temperature, d for distance, n for “a number”.
A constant is a quantity that does not change. In the expression 5 + 3k, the number 5 is a constant and k is a variable.
Any letter can be used, but choosing a letter that reminds you of the quantity (h for height, c for cost) makes the algebra easier to read. A letter always stands for a number – a means “the number of apples”, not “apples”.
How algebra is written
In arithmetic language
In algebra
Note
3 × n
3n
Leave out the × sign and write the number first.
a × b
ab
Letters are usually written in alphabetical order.
1 × x
x
We do not write the 1.
y × y
y2
“y squared”.
n ÷ 4
n4 or n/4
Division is usually written as a fraction.
Terms, expressions and equations
A term is a single number, a single letter, or numbers and letters multiplied (or divided) together: 7, x, 5y, 3ab, p2.
The number in front of a letter is its coefficient: in 5y the coefficient is 5.
An expression is one or more terms joined by + or − signs: 4a − 2b + 7 is an expression with three terms. The sign in front of a term belongs to that term.
An equation contains an equals sign: 3x + 5 = 11. An expression has no equals sign. (You will solve equations in Term 3.)
Turning words into expressions
Words
Expression
5 more than n
n + 5
5 less than n
n − 5 (not 5 − n!)
n subtracted from 20
20 − n
twice n / double n
2n
the product of a and b
ab
n shared equally among 4
n4
3 times the sum of x and 2
3(x + 2)
the square of y
y2
Tip: “more than” and “less than” describe what happens ton. “5 less than n” starts with n and takes 5 away. Test it with a number: 5 less than 12 is 7, and 12 − 5 = 7 ✔
Substitution
To substitute means to replace each letter with a given number and then work out the value. Put the number in brackets if it helps, and follow the order of operations: Brackets, Orders (powers), Division and Multiplication, Addition and Subtraction (BODMAS, also called BIDMAS or PEMDAS).
Worked examples
Example 1: Substituting into expressions
Question: If a = 3, b = 5 and c = 2, find the value of (a) 4a + b (b) 2ab (c) 5b − 3c (d) a(b + c) (e) a + bc (f) a2 + b
(e) (a + b) ÷ c = (3 + 5) ÷ 2 = 8 ÷ 2 = 4 (the fraction line acts like brackets)
(f) a2 + b = 3 × 3 + 5 = 9 + 5 = 14
Answer: 17, 30, 19, 21, 4, 14.
Example 2: An expression from a real situation
Question: A taxi charges a $5 pick-up fee plus $3 for every kilometre. Write an expression for the fare for k km, and find the fare for a 7 km ride.
Cost of the kilometres: 3 × k = 3k dollars. Add the fixed fee: 5 + 3k.
Here 5 is a constant and k is the variable.
Substitute k = 7: 5 + 3 × 7 = 5 + 21 = 26.
Answer: Fare = $(5 + 3k); a 7 km ride costs $26.
Example 3: Writing an expression
Question: Phone credit starts at $100 and $15 is used each week. Write an expression for the credit left after k weeks. How much is left after 4 weeks?
Credit used after k weeks: 15k. Credit left: 100 − 15k.
Substitute k = 4: 100 − 15 × 4 = 100 − 60 = 40.
Answer: 100 − 15k dollars; $40 after 4 weeks.
Common mistakes
Treating 3a as a two-digit number. If a = 4, then 3a = 3 × 4 = 12, not 34.
Reversing “less than”. “7 less than y” is y − 7, not 7 − y.
Ignoring the order of operations. With b = 5 and c = 2, 5b − 3c = 25 − 6 = 19. Working left to right without care gives wrong answers.
Calling an equation an expression. If there is an = sign, it is an equation.
Thinking a letter is an object. In 3a + 2b, the letters stand for numbers (such as the cost of an apple), not for “apples” and “bananas”.
Practice questions
Try these yourself first, then open each answer to check your working.
1. Constant or variable? (a) the number of hours in a day (b) your height (c) the price of a litre of gasoline
2. Write as expressions: (a) 7 less than y (b) the sum of p and q, divided by 2 (c) 4 times m, plus 9
Show answer
(a) y − 7 (b) p + q2 (c) 4m + 9
3. (a) Is 3x + 5 = 11 an expression or an equation? (b) How many terms are in 4a − 2b + 7? (c) What is the coefficient of b in 4a − 2b + 7?
Show answer
(a) An equation (it has an = sign). (b) Three terms: 4a, −2b and 7. (c) −2 (the sign belongs to the term; you will meet negative numbers properly in Term 2).
4. If x = 6 and y = 2, find (a) 3x − 4y (b) xy3 (c) 2(x + y) (d) x2 − y2
5. Pens cost $p each and a notebook costs $12. Write an expression for the cost of 6 pens and one notebook. Find the cost when p = 4.
Show answer
6p + 12. When p = 4: 24 + 12 = $36.
6. A rectangle has length l and width w. Write an expression for its perimeter and find it when l = 9 cm and w = 4 cm.
Show answer
2l + 2w (or 2(l + w)). 18 + 8 = 26 cm.
7. Ana is n years old. Her brother is 3 years older than she is, and her grandmother is five times her age. Write expressions for their ages, then find them when n = 12.
Show answer
Brother: n + 3 = 15. Grandmother: 5n = 60.
Quick check
1. What is a variable?
Show answer
A quantity that can change or is unknown, represented by a letter.
2. What is the difference between an expression and an equation?
Show answer
An equation has an equals sign; an expression does not.
Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 1, Strand 1.6 Algebra, Topic 1.6.1 Introducing Algebra – Learning Outcomes 1.6.1.1 (investigating varying quantities), 1.6.1.2 (constants and variables), 1.6.1.3 (using symbols for unknown quantities), 1.6.1.4 (translating word statements into expressions), 1.6.1.5 (identifying an expression) and 1.6.1.6 (substituting whole numbers for variables).
Updates
2026/10/25
– Put each lettered part of the questions and answers on its own line.
A set is a collection of objects, numbers or ideas that belong together. Each object in a set is called a member or an element. For example, the colours of the flag of Trinidad and Tobago form the set {red, white, black}. Sets help us sort and classify things, and they are the starting point for Venn diagrams, probability and logic. This lesson covers naming and describing sets, the main types of sets, subsets, and the ideas of union, intersection and complement shown on Venn diagrams.
Naming and describing a set
Name a set with a capital letter and put its members inside curly brackets { }, separated by commas: V = {a, e, i, o, u}.
A set must be well defined: you must be able to say for certain whether something belongs. “Even numbers” is well defined; “nice songs” is not.
You can define a set in two ways: by listing the members, V = {a, e, i, o, u}, or by describing them in words, V = {vowels in the English alphabet}.
The symbol ∈ means “is a member of” and ∉ means “is not a member of”: e ∈ V, but b ∉ V.
n(A) means the number of members in set A. For the vowels, n(V) = 5.
Each member is listed only once, and the order does not matter: {1, 2, 3} is the same set as {3, 1, 2}.
Types of sets
Type
Meaning
Example
Empty (null) set
Has no members. Written { } or ∅.
{months with 32 days} = { }
Finite set
You could count all its members.
{pupils in your class}
Infinite set
The members go on forever. Use “…” in a list.
{multiples of 5} = {5, 10, 15, …}
Equal sets
Exactly the same members (A = B).
{letters in LISTEN} = {letters in SILENT}
Equivalent sets
The same number of members, n(A) = n(B), even if the members differ.
{1, 2, 3} and {a, b, c}
Subsets
If every member of set A is also in set B, then A is a subset of B, written A ⊂ B (some books write A ⊆ B). For example, {2, 4} ⊂ {1, 2, 3, 4, 5}. Two useful facts: the empty set is a subset of every set, and every set is a subset of itself. When we classify the members of a set, we divide it into subsets: {1, 2, 3, 4, 5, 6} can be split into the subsets {odd numbers} = {1, 3, 5} and {even numbers} = {2, 4, 6}.
Universal set, complement, union and intersection
The universal set, U, contains every member being talked about in a question – for example U = {whole numbers from 1 to 12}.
The complement of A, written A′, is everything in U that is not in A.
The unionA ∪ B is everything in A or B or both (join the sets together, without repeats).
The intersectionA ∩ B is everything in both A and B (the overlap).
Sets that share at least one member are intersecting sets. Sets with no members in common are disjoint: A ∩ B = { }.
Venn diagrams
A Venn diagram shows sets as circles inside a rectangle that stands for U. Overlapping circles show intersecting sets; separate circles show disjoint sets; a circle drawn completely inside another shows a subset. Members outside every circle are still in U – do not forget to write them in the rectangle.
Figure 1 – A Venn diagram in a stained-glass window at Gonville and Caius College, Cambridge, honouring John Venn, who made these diagrams famous in 1880. Image: Schutz (photo); stained glass designed by Maria McClafferty, CC BY-SA 2.5, via Wikimedia Commons.
TRY ITVenn diagram builder
Choose a rule for set A and for set B. Every number from 1 to 20 drops into the correct region, and the sets are written out for you in set notation.
Universal set: U = {whole numbers from 1 to 20}
Worked examples
Example 1: Listing a set
Question: List the members of A = {odd numbers between 2 and 12} and find n(A).
“Between 2 and 12” means we do not include 2 or 12.
The odd numbers in that range are 3, 5, 7, 9, 11.
Answer: A = {3, 5, 7, 9, 11}, n(A) = 5.
Example 2: Describing a set in words
Question: Describe the set {2, 4, 6, 8} in words.
Look for what the members have in common: they are all even, start at 2 and stop at 8.
Answer: {even numbers from 2 to 8} (or {the first four even numbers}).
Example 3: Union, intersection and complement
Question: U = {whole numbers from 1 to 12}, A = {factors of 12} and B = {even numbers}. List A, B, A ∩ B, A ∪ B, A′ and (A ∪ B)′, and draw the Venn diagram.
A = {1, 2, 3, 4, 6, 12} and B = {2, 4, 6, 8, 10, 12}.
A ∩ B (in both) = {2, 4, 6, 12}. These go in the overlap.
Answer: Venn diagram: 1 and 3 in A only; 2, 4, 6, 12 in the overlap; 8 and 10 in B only; 5, 7, 9, 11 outside the circles. (Choose “Factors of 20”, “Even numbers” and other rules in the builder above to practise more of these.)
Example 4: Equal or equivalent?
Question: Are {letters in LISTEN} and {letters in SILENT} equal sets? Are {1, 2, 3} and {a, b, c} equal?
LISTEN gives {L, I, S, T, E, N} and SILENT gives {S, I, L, E, N, T} – the same six letters, so the sets are equal.
{1, 2, 3} and {a, b, c} have different members, but both have 3 members.
Answer: The first pair are equal sets. The second pair are equivalent but not equal.
Example 5: Listing all subsets
Question: List every subset of {p, q, r}.
No members: { }.
One member: {p}, {q}, {r}.
Two members: {p, q}, {p, r}, {q, r}.
Three members: {p, q, r}.
Answer: 8 subsets. (A set with n members has 2n subsets: 23 = 8.)
Common mistakes
Repeating members. The letters in “BANANA” form the set {B, A, N}, not {B, A, N, A, N, A}.
Mixing up ∪ and ∩. Think “∪ for union = everything joined”; “∩ looks like an n for intersection = only the overlap”.
Forgetting members outside the circles. Every member of U must appear somewhere in the rectangle.
Confusing ∅ with {0}. ∅ has no members; {0} has one member, the number 0.
Using ∈ for sets. 3 ∈ {1, 3, 5} (a member), but {3} ⊂ {1, 3, 5} (a subset).
Practice questions
Try these yourself first, then open each answer to check your working.
1. List the set C = {prime numbers less than 20}. What is n(C)?
Show answer
C = {2, 3, 5, 7, 11, 13, 17, 19}, n(C) = 8.
2. Which of these sets are infinite? (a) {multiples of 5} (b) {pupils in your school} (c) {even numbers less than 1,000,000}
Show answer
Only (a). Sets (b) and (c) are very large but could be counted, so they are finite.
3. U = {1, 2, 3, …, 10}, P = {1, 2, 3, 4, 5} and Q = {4, 5, 6, 7}. List P ∩ Q, P ∪ Q and Q′.
Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 1, Strand 1.2 Sets, Relations and Functions, Topic 1.2.1 Sets – Learning Outcomes 1.2.1.1 (classifying, describing and naming sets), 1.2.1.2 (defining sets by listing or describing), 1.2.1.3 (dividing a set into subsets), 1.2.1.4 (empty, equal, equivalent, finite and infinite sets), 1.2.1.5 (universal sets, complements, union, intersecting sets, subsets and disjoint sets) and 1.2.1.6 (Venn diagrams for two sets).
You already know how to find the factors and multiples of a number. In this lesson you will use them to find two very useful numbers: the highest common factor (HCF) and the lowest common multiple (LCM) of two or more numbers. You will learn three methods – listing, prime factors and the ladder – and how to tell which one a word problem needs.
Common factors and the HCF
A common factor of two numbers is a number that is a factor of both of them.
The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The common factors of 12 and 18 are 1, 2, 3 and 6.
The highest common factor (HCF) is the biggest of the common factors. The HCF of 12 and 18 is 6. (Some books call it the greatest common factor, GCF.)
1 is a common factor of every pair of numbers. If the HCF is 1, the numbers have no other factor in common – for example 8 and 15.
Common multiples and the LCM
A common multiple of two numbers is a number that is in both of their times tables.
Multiples of 4: 4, 8, 12, 16, 20, 24, … Multiples of 6: 6, 12, 18, 24, … The common multiples of 4 and 6 are 12, 24, 36, …
The lowest common multiple (LCM) is the smallest of the common multiples (not counting 0). The LCM of 4 and 6 is 12.
The list of common multiples never ends, but each one is a multiple of the LCM: 12, 24, 36, 48, …
Tip: the HCF can never be bigger than the smaller number, and the LCM can never be smaller than the bigger number. Use this to check your answers.
Method 1: Listing
For the HCF: list all the factors of each number (find them in pairs), ring the ones that appear in both lists, and pick the biggest.
For the LCM: write out the multiples of the bigger number one at a time, and stop at the first one that the smaller number also divides into exactly.
Listing is quick for small numbers. For bigger numbers the lists get long, so use prime factors or the ladder instead.
Writing a number as a product of prime factors
Every whole number bigger than 1 can be written as prime numbers multiplied together. This is called its prime factorisation. A factor tree is an easy way to find it: split the number into any two factors, keep splitting until every branch ends in a prime, then multiply the primes at the ends of the branches.
Figure 1 – A factor tree for 60. The primes at the ends of the branches (in orange) give 60 = 2 × 2 × 3 × 5 = 22 × 3 × 5.
We write repeated primes with index notation: 2 × 2 = 22, so 60 = 22 × 3 × 5. It does not matter how you start the tree (60 = 4 × 15 or 60 = 2 × 30 work too) – you always end up with the same primes.
Method 2: Using prime factors
Write each number as a product of primes. For 24 and 36: 24 = 2 × 2 × 2 × 3 = 23 × 3 and 36 = 2 × 2 × 3 × 3 = 22 × 32.
HCF: take only the primes that are in both numbers, using the lower power of each. Both have 2s and 3s: the lower powers are 22 and 3. HCF = 22 × 3 = 4 × 3 = 12.
LCM: take every prime that appears in either number, using the higher power of each: 23 and 32. LCM = 23 × 32 = 8 × 9 = 72.
Prime
Power in 24
Power in 36
For the HCF (lower)
For the LCM (higher)
2
23
22
22
23
3
3
32
3
32
Method 3: The ladder (repeated division)
Write the two numbers side by side. Divide both by a prime that goes into both of them exactly, and write the answers underneath. Keep going until no prime (other than 1) divides both numbers.
Divide by
24
36
2
24
36
2
12
18
3
6
9
2
3
HCF = multiply the numbers down the left side: 2 × 2 × 3 = 12.
LCM = multiply the left side and the numbers left on the bottom row: 2 × 2 × 3 × 2 × 3 = 72.
A useful check for two numbers: HCF × LCM = the two numbers multiplied together. Here 12 × 72 = 864 and 24 × 36 = 864 ✔
HCF or LCM? Reading word problems
Use the HCF when you are sharing or cutting things into the largest equal groups or pieces, with nothing left over. Key words: largest, greatest, longest, share equally, as many identical bags as possible.
Use the LCM when things repeat and you want to know when they next happen together, or the smallest amount that fits both. Key words: next together, at the same time, smallest number, least.
Quick test: the HCF answer is smaller than the numbers in the question; the LCM answer is bigger.
TRY ITHCF and LCM finder
Choose two whole numbers from 2 to 200. The finder lists their factors and multiples, writes them as products of primes, works through the ladder and gives the HCF and LCM.
Worked examples
Example 1: HCF by listing
Question: Find the HCF of 16 and 40.
Factors of 16: 1, 2, 4, 8, 16.
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40.
Common factors: 1, 2, 4, 8. The biggest is 8.
Answer: The HCF of 16 and 40 is 8.
Example 2: LCM by listing
Question: Find the LCM of 6 and 8.
Multiples of the bigger number, 8: 8, 16, 24, …
Does 6 divide 8? No. 16? No. 24? Yes, 24 = 6 × 4.
Answer: The LCM of 6 and 8 is 24. (Notice 6 × 8 = 48 is a common multiple, but it is not the lowest.)
Example 3: A factor tree
Question: Write 90 as a product of prime factors.
Split 90 = 9 × 10.
Split 9 = 3 × 3 and 10 = 2 × 5. All the branches now end in primes.
Write the primes in order: 2 × 3 × 3 × 5.
Answer: 90 = 2 × 3 × 3 × 5 = 2 × 32 × 5.
Example 4: HCF and LCM using prime factors
Question: Find the HCF and the LCM of 36 and 60.
36 = 22 × 32 and 60 = 22 × 3 × 5.
HCF: the shared primes are 2 and 3. Lower powers: 22 and 3. HCF = 4 × 3 = 12.
LCM: every prime, higher power: 22, 32 and 5. LCM = 4 × 9 × 5 = 180.
Check: 12 × 180 = 2,160 and 36 × 60 = 2,160 ✔
Answer: HCF = 12, LCM = 180.
Example 5: The ladder method
Question: Use the ladder to find the HCF and LCM of 48 and 72.
Divide both by 2: 24 and 36. Again by 2: 12 and 18. Again by 2: 6 and 9. Then by 3: 2 and 3.
No prime divides both 2 and 3, so stop.
HCF = 2 × 2 × 2 × 3 = 24.
LCM = 2 × 2 × 2 × 3 × 2 × 3 = 144.
Answer: HCF = 24, LCM = 144.
Example 6: Buses leaving together (LCM)
Question: Two buses leave the same terminal in San Fernando at 6:00 a.m. One leaves every 12 minutes and the other every 18 minutes. When will they next leave together?
The buses repeat, and we want the next time they match, so we need the LCM.
Multiples of 18: 18, 36, 54, … 12 does not divide 18, but 36 = 12 × 3.
The LCM of 12 and 18 is 36, so they leave together every 36 minutes.
Answer: At 6:36 a.m.
Example 7: Gift bags (HCF)
Question: Mrs Ramdial has 24 mangoes and 36 oranges. She wants to pack them into identical gift bags, with nothing left over. What is the largest number of bags she can make, and what goes in each bag?
Sharing into the largest number of equal groups means we need the HCF.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. The HCF is 12.
Each bag gets 24 ÷ 12 = 2 mangoes and 36 ÷ 12 = 3 oranges.
Answer: 12 bags, each with 2 mangoes and 3 oranges.
Common mistakes
Mixing up HCF and LCM. The HCF is a factor, so it is small (no bigger than the smaller number). The LCM is a multiple, so it is big (no smaller than the bigger number).
Just multiplying the numbers to get the LCM. 6 × 8 = 48 is a common multiple of 6 and 8, but the lowest is 24. Multiplying only gives the LCM when the HCF is 1, like 5 and 7.
Leaving a non-prime at the end of a factor tree. Keep splitting until every branch ends in a prime. 1 is not prime, so never write it in a factor tree.
Mixing up the powers. For the HCF use only the shared primes and the lower power; for the LCM use every prime and the higher power.
Practice questions
Try these yourself first, then open each answer to check your working.
1. (a) List the factors of 20 and of 30. (b) Write down their common factors. (c) What is the HCF of 20 and 30?
6. Ravi visits his grandmother in Arima every 6 days. His cousin visits her every 9 days. They were both there today. In how many days will they next both be there on the same day?
Show answer
LCM of 6 and 9 = 18, so in 18 days.
7. A tailor has two pieces of cloth, 84 cm and 126 cm long. She wants to cut both into pieces of the same length, as long as possible, with nothing left over. (a) How long should each piece be? (b) How many pieces will she get altogether?
Show answer
(a) HCF of 84 and 126 = 42 cm (b) 84 ÷ 42 = 2 and 126 ÷ 42 = 3, so 5 pieces.
8. For a class sports day, hot dogs come in packs of 10 and buns come in packs of 8. What is the smallest number of hot dogs and buns you can buy so that there are exactly the same number of each? How many packs of each is that?
Show answer
LCM of 10 and 8 = 40. That is 40 ÷ 10 = 4 packs of hot dogs and 40 ÷ 8 = 5 packs of buns.
9. The HCF of two numbers is 6 and their LCM is 72. One of the numbers is 18. Find the other number.
Show answer
HCF × LCM = product of the numbers, so 6 × 72 = 432 = 18 × ?. The other number is 432 ÷ 18 = 24. Check: the HCF of 18 and 24 is 6 and the LCM is 72 ✔
10. Find (a) the HCF (b) the LCM of 12, 18 and 30.
Whole numbers come in “families” with special properties. Knowing them makes multiplication, division, fractions and algebra much easier later on. In this lesson you will meet factors and multiples, odd and even numbers, prime and composite numbers, and three kinds of “shape” numbers – square, triangular and rectangular numbers – plus square roots.
Factors and multiples
A factor of a number divides into it exactly, with no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12.
A multiple of a number is the result of multiplying it by a whole number – the numbers in its times table. The multiples of 12 are 12, 24, 36, 48, …
They are two sides of the same fact: 3 × 4 = 12 means 3 and 4 are factors of 12, and 12 is a multiple of 3 and of 4.
A number has a limited (finite) list of factors, but its multiples go on forever.
Tip: find factors in pairs. Start at 1 and work upwards: 1 × 12, 2 × 6, 3 × 4. Stop when the pairs start to repeat.
Odd and even numbers
An even number divides exactly by 2 – it ends in 0, 2, 4, 6 or 8. An odd number leaves a remainder of 1 when divided by 2 – it ends in 1, 3, 5, 7 or 9. Only the last digit matters: 3,457 is odd and 10,000 is even. Useful patterns: even + even = even, odd + odd = even, odd + even = odd.
Prime and composite numbers
A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13.
A composite number has more than two factors. Examples: 4 (1, 2, 4), 9 (1, 3, 9), 12.
1 is neither prime nor composite, because it has only one factor.
2 is the only even prime. Every other even number has 2 as a factor, so it is composite.
Figure 1 – The Sieve of Eratosthenes. Cross out the multiples of 2, then 3, then 5, then 7 (but not the numbers themselves). The numbers left over are the primes. Image: SKopp at German Wikipedia, CC BY-SA 3.0, via Wikimedia Commons.
Testing whether a number is prime: try dividing it by the primes 2, 3, 5, 7, 11, … You only need to test primes whose square is not bigger than the number. If none divides exactly, the number is prime.
Square numbers and square roots
A square number is a whole number multiplied by itself. 4 × 4 = 16, so 16 is a square number. We write 4 × 4 as 42 (“four squared”). As dots, a square number makes a perfect square.
n
1
2
3
4
5
6
7
8
9
10
11
12
n2
1
4
9
16
25
36
49
64
81
100
121
144
The square root undoes squaring. Since 42 = 16, the square root of 16 is 4. We write √16 = 4. Learn the table above both ways round: it gives you every square root up to √144 = 12.
Triangular and rectangular numbers
A triangular number can be drawn as a triangle of dots with rows of 1, 2, 3, … dots. 1 + 2 + 3 + 4 = 10, so 10 is triangular. The first ten are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55 – each time, add the next counting number.
A rectangular number can be drawn as a rectangle of dots with at least 2 rows and at least 2 columns. 12 is rectangular because it makes a 2 × 6 or a 3 × 4 rectangle. Every composite number is rectangular; a prime number can only make a single line of dots. (Some books use “rectangular” or “oblong” number to mean only n × (n + 1), such as 2, 6, 12, 20 and 30 – check which meaning your teacher or textbook uses.)
Some numbers belong to several families. 36 is square (6 × 6), triangular (1 + 2 + … + 8) and rectangular (4 × 9).
TRY ITNumber explorer (1 to 100)
Choose a number. The explorer lists its factors, says whether it is odd or even, prime or composite, square, triangular or rectangular, and draws it as dots.
Dot picture:
Worked examples
Example 1: All the factors of 36
Question: List the factors of 36 and describe what kind of number 36 is.
Find factor pairs: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6. (5 does not divide 36. After 6 the pairs repeat.)
List them in order: 1, 2, 3, 4, 6, 9, 12, 18, 36.
36 has more than two factors, so it is composite. 6 × 6 = 36, so it is a square number. 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36, so it is triangular too.
Answer: Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36. 36 is even, composite, square, triangular and rectangular.
Example 2: Is 91 prime?
Question: Is 91 a prime number?
10 × 10 = 100 is bigger than 91, so we only need to test the primes 2, 3, 5 and 7.
91 is odd, so 2 does not divide it. 9 + 1 = 10, which is not a multiple of 3, so 3 does not divide it. It does not end in 0 or 5, so 5 does not divide it.
91 ÷ 7 = 13 exactly.
Answer: No. 91 = 7 × 13, so it is composite. (Many people guess 91 is prime – it is a famous trap!)
Example 3: Is 97 prime?
Question: Is 97 a prime number?
Again we test the primes 2, 3, 5 and 7 (because 10 × 10 = 100 > 97).
97 is odd; 9 + 7 = 16 is not a multiple of 3; it does not end in 0 or 5; 97 ÷ 7 = 13 remainder 6.
Answer: Yes, 97 is prime – no prime up to 7 divides it.
Example 4: Multiples
Question: Write the first five multiples of 7. Is 84 a multiple of 7?
Multiply 7 by 1, 2, 3, 4, 5: 7, 14, 21, 28, 35.
84 ÷ 7 = 12 with no remainder.
Answer: 7, 14, 21, 28, 35. Yes, 84 = 7 × 12 is a multiple of 7.
Example 5: A square root
Question: Find √225.
225 ends in 5, so try 5: 225 ÷ 5 = 45, and 45 = 5 × 9 = 5 × 3 × 3.
Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 1, Strand 1.1 Number Operations and Number Theory, Topic 1.1.1 Whole Numbers – Learning Outcome 1.1.1.6 (telling apart rectangular, triangular and square numbers; factors and multiples; odd and even numbers; prime and composite numbers; and square numbers and their square roots).
Rounding means replacing a number with a simpler, “rounder” number that is close to it. A newspaper might say a stadium held “about 27,000 people” when the exact count was 27,465. Rounded numbers are easier to remember, say and calculate with. Estimating means making a sensible guess at a quantity or an answer without counting or calculating everything exactly. In this lesson you will round whole numbers to the nearest 10, 100, 1,000 and up to the nearest million, and use benchmarks such as 100 to estimate large quantities.
Find the rounding place (for example the hundreds digit if you are rounding to the nearest 100).
Look at the digit immediately to its right – this is the “decider” digit.
If the decider is 5, 6, 7, 8 or 9, round up: add 1 to the rounding digit.
If the decider is 0, 1, 2, 3 or 4, round down: leave the rounding digit as it is.
Replace every digit to the right of the rounding place with 0. Digits to the left do not change (unless rounding up turns a 9 into 10 – then carry, just like in addition).
Why does 5 round up? A number ending in exactly 5 is halfway between the two round numbers, and the usual convention in school maths is to round halfway values up. On a number line you can see that numbers above the halfway point are nearer the higher round number.
TRY ITRounding on a number line
Type a number and choose the place to round to. The number line shows the two “round” numbers on either side, the halfway point, and which one is nearer. Use the step buttons to see exactly where the answer changes.
(steps move the number by one tenth of the rounding place)
Worked examples
Example 1: One number, three different roundings
Question: Round 3,746 to the nearest 10, the nearest 100 and the nearest 1,000.
Nearest 10: the tens digit is 4 and the decider (ones) is 6. 6 ≥ 5, so round up: 4 tens become 5 tens → 3,750.
Nearest 100: the hundreds digit is 7 and the decider (tens) is 4. 4 < 5, so round down → 3,700.
Nearest 1,000: the thousands digit is 3 and the decider (hundreds) is 7. Round up → 4,000.
Answer: 3,750; 3,700; 4,000. Notice that the answer depends on which place you round to.
Example 2: A decider of exactly 5
Question: Round 48,650 to the nearest 100 and to the nearest 1,000.
Question: Round 2,495,318 to the nearest 10,000 and to the nearest million.
Nearest 10,000: the ten-thousands digit is 9 and the decider (thousands) is 5, so round up. 9 ten-thousands + 1 = 10 ten-thousands, so carry 1 into the hundred-thousands place: 249 ten-thousands become 250 ten-thousands → 2,500,000.
Nearest million: the millions digit is 2 and the decider (hundred thousands) is 4. Round down → 2,000,000.
Answer: 2,500,000 and 2,000,000.
Example 4: Rounding up past a 9
Question: Round 69,960 to the nearest 100.
The hundreds digit is 9 and the decider (tens) is 6, so round up.
How many people are in a big crowd? How many beans are in a jar? Counting every single one would take far too long, so we estimate using a benchmark – a group whose size we know. A benchmark of 100 works well:
Count out 100 objects (or 100 people in one part of a photo) and notice how much space they take up.
Mentally group the rest of the objects into chunks of about the same size.
Count the chunks and multiply by 100.
Verify: if you can, count the real total and see how close you were.
Example 5: Estimating a crowd
Question: A photo of a crowd is divided into equal sections. One section contains about 100 people, and the crowd fills about 12 sections. Estimate the size of the crowd.
Benchmark: 1 section ≈ 100 people.
12 sections ≈ 12 × 100.
Answer: about 1,200 people.
Estimating answers and giving answers to a degree of accuracy
Before you calculate, round each number to make a quick estimate. If your exact answer is far from the estimate, check your working. For example, 412 × 29 ≈ 400 × 30 = 12,000. The exact answer is 11,948, which is close, so it is reasonable.
Sometimes a question asks you to give your answer “to the nearest thousand” or “correct to the nearest hundred dollars”. Work out the exact answer first, then round it once at the end. Example: a school fair raised $18,653. To the nearest hundred dollars this is $18,700 (the tens digit, 5, means round up).
Common mistakes
Rounding in a chain. To round 1,448 to the nearest 100, look only at the tens digit (4) → 1,400. Do not round to 1,450 first and then to 1,500.
Forgetting the zeros. 3,746 to the nearest 100 is 3,700, not 37.
Looking at the wrong digit. The decider is always the digit immediately to the right of the rounding place.
Changing digits on the left. When you round down, the digits to the left of the rounding place stay exactly the same.
Practice questions
Try these yourself first, then open each answer to check your working.
1. Round 8,362 to the nearest 10, the nearest 100 and the nearest 1,000.
Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 1, Strand 1.1 Number Operations and Number Theory, Topic 1.1.1 Whole Numbers – Learning Outcomes 1.1.1.4 (rounding to the nearest ten, hundred, thousand and up to millions), 1.1.1.5 (estimating a quantity using 100 as a benchmark and checking by counting) and 1.1.1.8 (solving whole-number problems and giving answers to a stated degree of accuracy).
