Mathematical Modelling

🎯 Learning Objectives

Formulate mathematical models from word problemsRumuskan model matematik daripada masalah berayat

Solve problems using appropriate techniquesSelesaikan masalah menggunakan teknik yang sesuai

Interpret and verify solutions in contextTafsir dan sahkan penyelesaian dalam konteks

📖 Key Concepts

The Modelling Process (Form 5)Proses Pemodelan (Tingkatan 5)

Turn real-world problems into math problems, solve, then check the answer makes sense.Ubah masalah dunia sebenar kepada masalah matematik, selesaikan, kemudian semak jawapan munasabah.

Step 1: Read carefully — identify what you need to find
Step 2: Define variables (Let x = ...)
Step 3: Form equation(s) from the information
Step 4: Solve using appropriate math techniques
Step 5: Verify — does the answer make sense?

Common Model TypesJenis Model Biasa

Identify which type of model fits the problem.Kenal pasti jenis model yang sesuai untuk masalah.

Linear: y = mx + c (constant rate of change)
Quadratic: y = ax² + bx + c (accelerating/decelerating)
Ratio: a:b = c:d → ad = bc
Direct proportion: y = kx
Inverse proportion: y = k/x

SPM Must-Know Application QuestionsJenis Soalan SPM Kerap Muncul

Most common modelling questions in SPM.Soalan pemodelan yang paling kerap dalam SPM.

Age problems: Let x = younger person's age
Number problems: Let x = the unknown number
Geometry: Perimeter, area, volume formulas
Sharing/Ratio: Find one part, multiply
Distance-Speed-Time: D = S × T
Percentage: Original × (1 ± r)^n

✏️ Worked Examples

Ali is 3 years older than Bakar. Their total age is 27. Find both ages.Ali 3 tahun lebih tua dari Bakar. Jumlah umur mereka 27. Cari umur kedua-dua.

Step 1: Let Bakar's age = x, then Ali's age = x + 3

Step 2: Form equation: x + (x + 3) = 27

Step 3: Solve: 2x + 3 = 27 → 2x = 24 → x = 12

Step 4: Bakar = 12 years old, Ali = 15 years old

Step 5: Verify: 12 + 15 = 27 ✓ | 15 is 3 more than 12 ✓

Bakar = 12 years, Ali = 15 yearsBakar = 12 tahun, Ali = 15 tahun

RM500 is shared between X and Y in the ratio 3:2. Find each person's share.RM500 dikongsi antara X dan Y dalam nisbah 3:2. Cari setiap bahagian.

Step 1: Total parts = 3 + 2 = 5 parts

Step 2: Value of one part = 500 / 5 = RM100

Step 3: X = 3 × 100 = RM300, Y = 2 × 100 = RM200

Verify: 300 + 200 = 500 ✓ | Ratio 300:200 = 3:2 ✓

X = RM300, Y = RM200X = RM300, Y = RM200

A rectangle has perimeter 36 cm. Its length is 12 cm. Find its width and area.Segi empat tepat mempunyai perimeter 36 cm. Panjangnya 12 cm. Cari lebar dan luasnya.

Step 1: P = 2(l + w) → 36 = 2(12 + w)

Step 2: 36/2 = 12 + w → 18 = 12 + w

Step 3: w = 18 − 12 = 6 cm

Step 4: Area = l × w = 12 × 6 = 72 cm²

Verify: 2(12+6) = 36 ✓

Width = 6 cm, Area = 72 cm²Lebar = 6 cm, Luas = 72 cm²

⚠️ Common Mistakes

❌ Not defining variablesTidak takrifkan pemboleh ubah

Always write "Let x = ..." before forming equations. Examiners need to see what x represents! Without this, you lose method marks.Sentiasa tulis "Misalkan x = ..." sebelum membentuk persamaan. Pemeriksa perlu lihat apa x wakili! Tanpa ini, anda hilang markah kaedah.

❌ Not checking answersTidak semak jawapan

If you get age = −5 or speed = 3000 km/h, something is wrong! Always plug your answer back into the original problem to verify.Jika dapat umur = −5 atau laju = 3000 km/j, sesuatu adalah salah! Sentiasa masukkan jawapan balik ke masalah asal untuk sahkan.

❌ Wrong ratio setupNisbah salah

For ratio 3:2 — X gets 3 parts, Y gets 2 parts. Total = 5 parts. Divide total by 5, then multiply by 3 and 2. NOT divide by 3 or 2!Untuk nisbah 3:2 — X dapat 3 bahagian, Y dapat 2 bahagian. Jumlah = 5 bahagian. Bahagi jumlah dengan 5, bukan dengan 3 atau 2!

📝 SPM Practice

x : y = 2 : 3 and x + y = 60. Find x.x : y = 2 : 3 dan x + y = 60. Cari x.

A rectangle has perimeter 40 cm and length 14 cm. Width = ?Segi empat tepat perímetro 40 cm dan panjang 14 cm. Lebar = ?

6 cm

12 cm

26 cm

14 cm

5 books cost RM75. What is the cost of 8 books?5 buku berharga RM75. Berapakah kos 8 buku?

RM120

RM100

RM15

RM80

A number increased by 7 equals 25. Find the number.Satu nombor ditambah 7 menjadi 25. Cari nombor itu.

A father is 4 times older than his son. Their total age is 50. Son's age = ?Bapa 4 kali lebih tua dari anaknya. Jumlah umur mereka 50. Umur anak = ?

12.5

PAPER 2 — 6 MARKS

(a) A sum of money is divided between A, B and C in the ratio 2 : 3 : 5. If C receives RM200, find the total sum of money. [3 marks]
(b) The length of a rectangle is 5 cm more than its width. The area of the rectangle is 84 cm². Find the dimensions of the rectangle. [3 marks](a) Satu jumlah wang dibahagi antara A, B dan C dalam nisbah 2 : 3 : 5. Jika C menerima RM200, cari jumlah wang. [3 markah]
(b) Panjang segi empat tepat adalah 5 cm lebih dari lebarnya. Luas segi empat tepat ialah 84 cm². Cari dimensi segi empat tepat. [3 markah]

Solution (a)Penyelesaian (a)

Total parts = 2 + 3 + 5 = 10 parts

5 parts = 200 → 1 part = 200 / 5 = 40

Total sum = 10 × 40 = RM400

Answer: RM400Jawapan: RM400

Solution (b)Penyelesaian (b)

Let width = w cm, then length = (w + 5) cm

Area = w(w + 5) = 84

w² + 5w − 84 = 0

(w + 12)(w − 7) = 0

w = 7 cm (reject w = −12, width cannot be negative)

Length = 7 + 5 = 12 cm

Width = 7 cm, Length = 12 cmLebar = 7 cm, Panjang = 12 cm

📋 Formula Summary

The Modelling Process:
Read → Define variables → Form equation → Solve → Verify

Linear equation: ax + b = c → solve for x
Quadratic: ax² + bx + c = 0 → factorise or use formula
Ratio: a:b = c:d → ad = bc
Perimeter (rectangle): P = 2(l + w)
Area (rectangle): A = lw
Distance: D = S × T
Percentage change: New = Original × (1 ± r)^n

🧮 Mathematical ModellingPemodelan Matematik