Learning Objectives
- understand the main ideas in Real-Life Applications;
- use correct mathematical language and notation;
- apply the ideas to progressively harder problems;
- check and explain each solution.
Introduction
This lesson develops Real-Life Applications from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Visual Exploration
Use the diagrams, tables or examples in this lesson to identify what changes, what stays fixed and which relationship connects the values.
Variation in Real Life
Variation becomes useful when a real situation can be represented by a consistent mathematical relationship.
Application 1: Cost
The cost of data varies directly as the number of gigabytes. A 6 GB bundle costs US$9. Find the cost of 14 GB.
C = kg.
9 = 6k, so k = 1.5.
C = 1.5(14) = US$21.
Application 2: Fixed Work
Twelve machines produce a fixed order in 18 hours. How long would 27 identical machines take?
Machines and time vary inversely.
k = 12 × 18 = 216.
Time = 216/27 = 8 hours.
Application 3: Geometry
The area A of a triangle varies jointly as its base b and height h. If A = 24 cm² when b = 8 cm and h = 6 cm, find A when b = 10 cm and h = 9 cm.
A = kbh.
24 = k(8)(6), so k = 1/2.
A = ½(10)(9) = 45 cm².
Application 4: Fixed and Variable Charges
A service charges US$4 fixed plus US$2.50 per hour. Find the cost for 7 hours.
C = 2.5h + 4.
C = 2.5(7) + 4.
C = US$21.50.
⚠️ Assumptions Matter
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
Foundation- The cost of rice varies directly as mass. If 4 kg costs US$7.20, find the cost of 9 kg.
- Ten workers finish a fixed task in 24 days. Find the time for 16 equally efficient workers.
- The energy E used varies jointly as power P and time t. If 120 units are used when P = 6 and t = 5, find E when P = 8 and t = 7.
- A courier fee consists of a fixed amount and a charge per kilometre. A 5 km delivery costs US$8 and a 13 km delivery costs US$16. Find the formula and the cost of 20 km.
- The time t to complete a fixed task varies inversely as the product of the number of workers w and their efficiency e. If 8 workers of efficiency 3 complete the task in 15 hours, how long will 10 workers of efficiency 4 take?
✅ Check Your Work
Show practice answers
Foundation
-
US$16.20.
Unit price = 7.20/4 = 1.80.
Cost = 1.80(9) = 16.20. -
15 days.
workers × days = 10 × 24 = 240.
days = 240/16 = 15.
Developing
-
224 units.
E = kPt
120 = k(6)(5), so k = 4.
E = 4(8)(7) = 224. -
C = d + 3, and 20 km costs US$23.
8 = 5k + c
16 = 13k + c
8 = 8k, so k = 1.
c = 3.
Challenge
-
9 hours.
t = k/(we)
15 = k/(8×3), so k = 360.
t = 360/(10×4) = 9.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Identify the variation type before calculating.
- Use units to interpret k.
- Check assumptions in real-life models.
- Decide whether the final answer is sensible.