Mathematical Modeling
How to Build and use a linear model
Use this to turn a real situation with a fixed part and a per-unit rate into a linear equation you can use.
Before you start
- Turning a worded situation into an expression
- Substituting a value into a formula
- Solving a simple linear equation
When to use it
Use this to turn a real situation with a fixed part and a per-unit rate into a linear equation you can use.
The steps
- Identify the fixed amount (the part that does not change) and the rate per unit.
- Write the model as total = fixed + rate × quantity.
- Substitute a known value to check the model fits the situation.
- Use the model to answer the question asked.
- Check the answer makes sense in the real situation.
Worked example
A plumber charges a fixed call-out fee of RM40 plus RM30 for each hour worked. Find the cost of a 3-hour job.
- Identify the fixed amount, RM40 (the call-out fee), and the rate, RM30 per hour.
- Write the model as total = fixed + rate × quantity: C = 40 + 30h, where h is hours.
- Check with a known value: a 1-hour job gives C = 40 + 30 × 1 = RM70, which fits the situation.
- Use the model for the question, h = 3: C = 40 + 30 × 3 = 40 + 90 = 130.
- RM130 for 3 hours is reasonable, so the answer makes sense.
A second example, with a twist
The rate is negative because the tank is draining, and you are given the total and must solve backwards for the quantity. A tank starts with 500 litres of water and drains at 20 litres per minute.
After how many minutes will 140 litres remain?
- Identify the fixed amount, 500 litres (the starting water), and the rate, −20 litres per minute because it is draining.
- Write the model as total = fixed + rate × quantity: V = 500 − 20t, where t is minutes.
- Check with a known value: after 1 minute V = 500 − 20 × 1 = 480 litres, which fits a draining tank.
- Use the model with V = 140: 140 = 500 − 20t, so 20t = 360 and t = 18.
- The tank would empty at 25 minutes, so 18 minutes to reach 140 litres makes sense.
Formula pages
- Average speed
- Circumference of circle
- Area of circle
- Arc length
- Area of sector
- Area of kite
- Area of trapezium
- Surface area of cylinder
- Surface area of cone
- Surface area of sphere
- Volume of prism
- Volume of cylinder
- Volume of cone
- Volume of sphere
- Volume of pyramid
Practise this in a KBAT problem
Frequently asked questions
How do I spot the fixed part and the rate in a word problem?
The fixed part is the amount that is there before any units are counted, a starting value or a one-off charge. The rate is the amount added or removed for each unit, so it usually comes with words like 'per hour' or 'each'.
Underline both before writing the model.
When is the rate negative?
The rate is negative whenever the total goes down as the quantity grows, like water draining, fuel being used, or a balance being spent. Write it with a minus sign, so the model becomes fixed − rate × quantity.
A positive rate is for things that build up over time.
Why check the model with a known value?
Substituting a value you already understand, like the cost at zero hours, or the amount after one minute, confirms the model matches the real situation before you trust it. If the check gives something impossible, you have set up the fixed part or the rate wrongly and can fix it early.