Home / Teaching / 4t ESO / Challenges / 01 · Challenge 1
4t ESO · Maths challenges Notes 14 Sep 2026

Challenge 1: the Àgora mural

A mural for the neighbourhood day centre: edges, scale, areas, paint cans and percentages. Each question has its solution hidden underneath.

Download the problem sheet (PDF, in Catalan)

The problem

Today my team has been given a new challenge we are really excited about: the neighbourhood community association has asked us whether we can paint a mural on the wall of the new day centre for the elderly. The coordinator has sent us the design and asked us how much paint of each colour we need. The budget is tight, so she wants us to be as accurate as possible.

To be honest, we need help.

ÀGORA2 cm2 cm1,5 cm4,5 cm11122334556
Tap a colour or a red number to see which piece or edge it is.
The design the coordinator sent us. The red numbers are the edges in question 1.

Notes

  1. The planned scale is that $1$ cm on the drawing is $75$ cm in real life.
  2. The real wall is $6$ m long and $5$ m high.
  3. The drawing must be perfectly centred.
  4. Each spray can holds $200$ ml of paint and covers about $1{,}5\ \text{m}^2$. Each purple can costs €$4{,}58$, each pink can €$4{,}00$, each light blue can €$4{,}25$ and each dark blue can €$4{,}40$.
  5. The letters do not count: we already have black paint.

Guidelines. We want complete calculations, using the appropriate combined operations. Any decimals are rounded to two decimal places. It matters that what we are doing can be understood.

Which concepts we will need

All this looks complicated, so we have made an outline of the steps we will follow. We will need to revise Pythagoras' theorem, similarity, some areas and perhaps percentages. We will see how it goes; at the end we will fill in the table to make it clearer.

About rounding. All results are rounded to two decimal places, and each step uses the rounded result of the previous one, as the guidelines ask. If you only round at the end, some values change in the second decimal place: for example, purple comes out as $2{,}99\ \text{m}^2$ instead of $3{,}01\ \text{m}^2$.

1. The missing edges

Work out every edge missing from the design. The measurements the coordinator gave us are not real ones: we need to calculate.

See the solution
Edge 1

It is the side of the left square, which is given. It is also the diameter of both semicircles and the height of the lower right rectangle.

Edge 1: $2$ cm

Edge 3

The top vertex of the dark blue triangle is at the midpoint of the top side of the square, which splits it in two halves:

$$\dfrac{2}{2} = 1 \text{ cm}$$

Edge 3: $1$ cm

Edge 2

It is the hypotenuse of a right triangle with legs $2$ cm (the height of the square) and $1$ cm (edge 3). Pythagoras:

$$x^2 = 2^2 + 1^2 = 5 \;\Longrightarrow\; x = \sqrt{5} \approx 2{,}24 \text{ cm}$$

Edge 2: $2{,}24$ cm

Edge 4

It is the diagonal of the right-hand rectangle, the hypotenuse of a right triangle with legs $1{,}5$ cm and $2$ cm:

$$y^2 = 1{,}5^2 + 2^2 = 2{,}25 + 4 = 6{,}25 \;\Longrightarrow\; y = \sqrt{6{,}25} = 2{,}5 \text{ cm}$$

Edge 4: $2{,}5$ cm

Edge 5

It is the width of the right-hand column, which is given.

Edge 5: $1{,}5$ cm

Edge 6

The right-hand column is $4{,}5$ cm high and the rectangle underneath is $2$ cm:

$$4{,}5 - 2 = 2{,}5 \text{ cm}$$

Edge 6: $2{,}5$ cm

2. The real measurements

To keep track, convert the drawing measurements into real ones and make a sketch.

See the solution

Each centimetre on the drawing is $75$ cm on the wall: we multiply each edge by $75$ and convert to metres.

EdgeDrawing (cm)CalculationReal
1$2$$2 \cdot 75 = 150 \text{ cm}$$1{,}5 \text{ m}$
2$2{,}24$$2{,}24 \cdot 75 = 168 \text{ cm}$$1{,}68 \text{ m}$
3$1$$1 \cdot 75 = 75 \text{ cm}$$0{,}75 \text{ m}$
4$2{,}5$$2{,}5 \cdot 75 = 187{,}5 \text{ cm}$$1{,}88 \text{ m}$
5$1{,}5$$1{,}5 \cdot 75 = 112{,}5 \text{ cm}$$1{,}13 \text{ m}$
6$2{,}5$$2{,}5 \cdot 75 = 187{,}5 \text{ cm}$$1{,}88 \text{ m}$
1,5 m1,5 m1,13 m1,5 m1,88 m3,38 m1,5 m0,750,751,681,88r = 0,75
Sketch of the mural with the real measurements.

The radius of the semicircles is half of edge 1: $0{,}75$ m.

3. The square metres of each colour

Work out the areas to know the square metres of each colour.

See the solution
Pink

In the left square, the two pink corners add up to half of the $1{,}5$ m square. In the right-hand rectangle there is a right triangle with legs $1{,}5$ m and $1{,}13$ m.

$$1{,}5 \cdot 0{,}75 + \dfrac{1{,}5 \cdot 1{,}13}{2} \approx 1{,}97 \text{ m}^2$$
$$1{,}97 \text{ m}^2$$
Purple

A semicircle of radius $0{,}75$ m and the top rectangle, $1{,}13$ m by $1{,}88$ m.

$$\dfrac{\pi \cdot 0{,}75^2}{2} + 1{,}13 \cdot 1{,}88 \approx 3{,}01 \text{ m}^2$$
$$3{,}01 \text{ m}^2$$
Dark blue

The triangle in the left square: base $1{,}5$ m and height $1{,}5$ m.

$$\dfrac{1{,}5 \cdot 1{,}5}{2} \approx 1{,}13 \text{ m}^2$$
$$1{,}13 \text{ m}^2$$
Light blue

The other semicircle, radius $0{,}75$ m, and the triangle below the diagonal, legs $1{,}5$ m and $1{,}13$ m.

$$\dfrac{\pi \cdot 0{,}75^2}{2} + \dfrac{1{,}5 \cdot 1{,}13}{2} \approx 1{,}73 \text{ m}^2$$
$$1{,}73 \text{ m}^2$$

Rounding only the final result. Working with the unrounded measurements ($1{,}125$ and $1{,}875$ m) and rounding only at the end:

$$\text{Pink: }1{,}5 \cdot 0{,}75 + \dfrac{1{,}5 \cdot 1{,}125}{2} = 1{,}96875 \approx 1{,}97 \text{ m}^2$$
$$\text{Purple: }\dfrac{\pi \cdot 0{,}75^2}{2} + 1{,}125 \cdot 1{,}875 = 2{,}9929\ldots \approx 2{,}99 \text{ m}^2$$
$$\text{Dark blue: }\dfrac{1{,}5 \cdot 1{,}5}{2} = 1{,}125 \approx 1{,}13 \text{ m}^2$$
$$\text{Light blue: }\dfrac{\pi \cdot 0{,}75^2}{2} + \dfrac{1{,}5 \cdot 1{,}125}{2} = 1{,}7273\ldots \approx 1{,}73 \text{ m}^2$$

Only purple changes.

4. Centring the mural on the wall

Work out exactly the measurements to centre the design on the wall. Remember it must be perfectly centred.

See the solution
Horizontal margin

The drawing is $2 + 2 + 1{,}5 = 5{,}5$ cm wide. Using the scale as a conversion factor:

$$5{,}5 \text{ cm} \cdot \dfrac{75 \text{ cm}}{1 \text{ cm}} = 412{,}5 \text{ cm} \approx 4{,}13 \text{ m}$$

The wall is $6$ m long: what is left over is shared between both sides.

$$\dfrac{6 - 4{,}13}{2} \approx 0{,}94 \text{ m}$$

$0{,}94$ m margin on the left and on the right

Vertical margin

The drawing is $4{,}5$ cm high:

$$4{,}5 \text{ cm} \cdot \dfrac{75 \text{ cm}}{1 \text{ cm}} = 337{,}5 \text{ cm} \approx 3{,}38 \text{ m}$$
$$\dfrac{5 - 3{,}38}{2} = 0{,}81 \text{ m}$$

$0{,}81$ m margin at the top and at the bottom

4,13 m × 3,38 m0,94 m0,94 m0,81 m0,81 m6 m5 m
The mural centred on the 6 m × 5 m wall.

5. The paint cans

Now that you know the square metres, order exactly the paint cans we need. Show every calculation using combined operations that can be followed: when money is involved, the coordinator wants it clear.

See the solution

Each can covers $1{,}5\ \text{m}^2$. We use this as a conversion factor and always round up: you cannot buy part of a can, and if we run short the mural is left unfinished.

Pink
$$1{,}97 \text{ m}^2 \cdot \dfrac{1 \text{ can}}{1{,}5 \text{ m}^2} \approx 1{,}31 \text{ cans}$$

2 cans

Purple
$$3{,}01 \text{ m}^2 \cdot \dfrac{1 \text{ can}}{1{,}5 \text{ m}^2} \approx 2{,}01 \text{ cans}$$

3 cans

Dark blue
$$1{,}13 \text{ m}^2 \cdot \dfrac{1 \text{ can}}{1{,}5 \text{ m}^2} \approx 0{,}75 \text{ cans}$$

1 can

Light blue
$$1{,}73 \text{ m}^2 \cdot \dfrac{1 \text{ can}}{1{,}5 \text{ m}^2} \approx 1{,}15 \text{ cans}$$

2 cans

Purple shows why rounding matters. With $3{,}01\ \text{m}^2$ we get $2{,}01$ cans, and two are not enough, however small the shortfall. Rounding only at the end we get $2{,}99\ \text{m}^2$ and $1{,}995\ldots$ cans: two would be enough, but only just.

Total cost

With the prices in note 4, in a single combined operation:

$$2 \cdot 4{,}00 + 3 \cdot 4{,}58 + 1 \cdot 4{,}40 + 2 \cdot 4{,}25 = 8{,}00 + 13{,}74 + 4{,}40 + 8{,}50 = 34{,}64 \text{ euros}$$

€$34{,}64$ on paint

Rounding only the final result. With the unrounded areas, purple comes to $2{,}9929\ldots / 1{,}5 = 1{,}995\ldots$ cans: two cans are enough, by a very small margin. The other colours do not change.

$$2 \cdot 4{,}00 + 2 \cdot 4{,}58 + 1 \cdot 4{,}40 + 2 \cdot 4{,}25 = 30{,}06 \text{ euros}$$

6. Percentage of each colour

A question came up: what percentage of the whole wall does each colour cover?

See the solution

The wall is $5 \cdot 6 = 30\ \text{m}^2$. We divide the area of each colour by the total area and multiply by $100$.

ColourCalculation%
Pink$\dfrac{1{,}97}{30} \cdot 100$$6{,}57\ \%$
Purple$\dfrac{3{,}01}{30} \cdot 100$$10{,}03\ \%$
Dark blue$\dfrac{1{,}13}{30} \cdot 100$$3{,}77\ \%$
Light blue$\dfrac{1{,}73}{30} \cdot 100$$5{,}77\ \%$

Rounding only the final result. With the unrounded areas:

ColourCalculation%
Pink$\dfrac{1{,}96875}{30} \cdot 100$$6{,}56\ \%$
Purple$\dfrac{2{,}9929\ldots}{30} \cdot 100$$9{,}98\ \%$
Dark blue$\dfrac{1{,}125}{30} \cdot 100$$3{,}75\ \%$
Light blue$\dfrac{1{,}7273\ldots}{30} \cdot 100$$5{,}76\ \%$

7. The concepts in each question

State which maths concepts you used in each question.

See the solution
QuestionConcept
1Pythagoras, plane figures
2Scales, similarity
3Areas of plane figures
4Arithmetic
5Conversion factors
6Percentages