Pattern and relationship problems in Foundation Mathematics require you to combine number skills — operations, percentages, fractions, rates — with pattern recognition to solve multi-step problems in real contexts.
KEY TAKEAWAY: Solving pattern problems is a three-step process: identify the pattern, write the rule, then apply number skills to answer the specific question.
Most practical pattern problems involve:
- Arithmetic sequences with operations (add/subtract/multiply/divide each step)
- Rate problems where a quantity changes at a constant rate
- Cost models that combine a fixed component and a variable component
- Percentage growth/decline over repeated periods
Worked Example — Phone Plan:
A phone plan costs $\$25$ per month plus $\$0.20$ per SMS.
$$\text{Monthly cost} = 25 + 0.20 \times n \quad (n = \text{number of SMSs})$$
| SMSs | Monthly cost ($\$$) |
|---|---|
| 0 | 25.00 |
| 50 | 35.00 |
| 100 | 45.00 |
| 200 | 65.00 |
To find how many SMSs for a $\$55$ budget:
$\$25 + 0.20n = 55$$
$$0.20n = 30$$
$$n = 150 \text{ SMSs}$$
EXAM TIP: Read the question carefully — it may ask for the number of steps, the value at a specific step, or when a value is first exceeded. Each requires a different calculation.
Worked Example — Savings Pattern:
Jake saves $\$50$ in week 1. Each week he saves $\$15$ more than the previous week. How much has he saved in total after 8 weeks?
Step 1 — Sequence:
$\$50, 65, 80, 95, 110, 125, 140, 155$$
Step 2 — Sum (arithmetic series):
$$S = \frac{n}{2}(t_1 + t_n) = \frac{8}{2}(50 + 155) = 4 \times 205 = \$820$$
Worked Example — Commission:
A salesperson earns a base salary of $\$600$/week plus a $3\%$ commission on sales.
$$\text{Weekly earnings} = 600 + 0.03 \times \text{sales}$$
To earn $\$900$ in a week:
$\$600 + 0.03 \times S = 900$$
$$0.03S = 300$$
$$S = \frac{300}{0.03} = \$10000 \text{ in sales}$$
Creating a systematic table helps when a formula is unclear.
Worked Example — Tile Patterns:
A path is made from square tiles. The first section uses 5 tiles; each extra section adds 4 tiles.
| Sections | Tiles |
|---|---|
| 1 | 5 |
| 2 | 9 |
| 3 | 13 |
| 4 | 17 |
| $n$ | $4n + 1$ |
Number of tiles for 12 sections:
$$4(12) + 1 = 49 \text{ tiles}$$
Cost if each tile costs $\$4.80$:
$\$49 \times 4.80 = \$235.20$$
Worked Example — Two Tradespeople:
Plumber A charges $\$80$ call-out + $\$60$/hour. Plumber B charges $\$50$ call-out + $\$70$/hour. When is the total cost the same?
$$\text{Plumber A: } C_A = 80 + 60h$$
$$\text{Plumber B: } C_B = 50 + 70h$$
Set equal:
$\$80 + 60h = 50 + 70h$$
$\$30 = 10h$$
$$h = 3 \text{ hours}$$
At $3$ hours both cost:
$$C = 80 + 60(3) = 80 + 180 = \$260$$
Plumber A is cheaper for jobs over $3$ hours; Plumber B is cheaper for shorter jobs.
VCAA FOCUS: VCAA tasks often give two options (e.g. two payment plans, two transport routes) and ask which is better value. Set up both rules, find the break-even point, and give a clear written conclusion.
