Calculate and apply simple models
Turn rules into ordered calculations and respect thresholds, fixed charges and rounding instructions.
Original StudyForA scenarios and formative model answers; no Cambridge marks, predicted grades or certified mastery. Alternative reasoned answers may be valid. Full-length unseen papers, a wider mixed-problem bank and calibrated specialist teacher marking remain outstanding.
Before you begin
- Read tables and instructions in English
- Use arithmetic, ratios and whole-number constraints
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Before calculating, what should you identify?
What you will learn
- Choose and organise the required operations
- Apply a stated model without changing its rules
Write the rule before substituting
A simple model is an explicit rule linking inputs and outputs. Identify fixed amounts, amounts per unit, caps and thresholds. State the units of each term. Round only where the rule requires it, because a ceiling for started blocks is different from rounding to the nearest whole number.
A fictional hire costs 6 credits plus 2 credits for every started 30-minute block. A 70-minute hire has three blocks, so its charge is 6 + 2×3 = 12 credits. The fixed charge is paid once, not once per block.
Check the rule at its boundaries
At exactly 60 minutes the hire uses two blocks; at 61 minutes it uses three. Boundary checks reveal whether you have silently replaced ‘started block’ with proportional charging. Test a small input whose result you can calculate directly before trusting a long calculation.
A calculator can execute operations accurately but cannot decide which model applies. Record the substituted expression and a short explanation, not only the final display.
Sources for this lesson
Pause and explain
How many started 30-minute blocks are charged at exactly 60 minutes?
Worked example
Calculate the charge for 70 minutes under the hire rule.
Show the worked solution
- 70÷30 is between two and three.
- Three blocks have started.
- Add the one-off 6-credit charge to 2×3.
Answer The charge is 12 credits. Using 70/30 proportionally would contradict the rule that each started block is charged in full.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Guided: Compare the hire charges at 60 and 61 minutes.
Give me a hint
Find how many blocks have started.
Compare my reasoning
- 60 minutes uses two blocks.
- 61 minutes starts a third.
- Add the fixed charge once.
At 60 minutes the charge is 6 + 2×2 = 10 credits. At 61 minutes it is 6 + 2×3 = 12 credits. The step change follows the charging rule.
Look for these in your work
- Counts blocks
- Adds the fixed charge once
- Explains the discontinuity
Independent: A stall sells 18 items at 7 credits each. Materials cost 3 credits per item and the pitch costs 20 credits. Find profit.
Give me a hint
Separate revenue from variable and fixed costs.
Compare my reasoning
- Revenue is 18×7 = 126.
- Costs are 18×3 + 20 = 74.
- Subtract costs from revenue.
Profit is 126 − 74 = 52 credits. The pitch cost is one fixed charge, while the material cost is incurred for each of the eighteen items.
Look for these in your work
- Calculates revenue
- Includes both costs
- Subtracts correctly
Application: A game awards 4 points per win, 1 per draw and deducts 2 per penalty. A team has 6 wins, 3 draws and 2 penalties. Calculate its score and explain the penalty operation.
Give me a hint
A penalty reduces the score twice.
Compare my reasoning
- Wins contribute 24 points.
- Draws contribute 3.
- Two penalties remove 4.
The score is 6×4 + 3×1 − 2×2 = 23 points. Treating a deduction as addition, or subtracting only one penalty, would apply a different rule.
Look for these in your work
- Applies all terms
- Uses a negative penalty
- Shows working
Common mistakes
- Repeating a one-off charge
- Ignoring a threshold
- Rounding before the rule permits it
What can you explain now?
How many started 30-minute blocks are charged at exactly 60 minutes?
Compare with the explanation
Two blocks
A third block has not started at exactly 60 minutes.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
How many started 30-minute blocks are charged at exactly 60 minutes?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.