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.
| 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.
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?
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.
