AO Numerical Reasoning: Three Worked Practice Examples
Practise AO numerical reasoning with worked examples on percentage change, weighted averages and budget decisions, then try free AO numerical questions.
Numerical questions become easier to organise when you separate the decision from the calculation. First establish what the question asks. Then choose the figures, write the relationship and calculate. A calculator cannot correct a comparison built on the wrong base.
The examples below use invented public-service data. Try each question before reading its solution. They teach methods used in PSP’s AO practice; use your own assessment instructions for the official subjects, format and timing.
Example 1: percentage change with two moving parts
A programme received 2,400 applications in Year 1 and approved 65% of them. In Year 2 it received 3,000 applications and approved 60%.
By what percentage did the number of approved applications increase?
The approval rate fell, but the question asks about the number approved. Calculate the counts first:
- Year 1: 2,400 × 0.65 = 1,560 approvals.
- Year 2: 3,000 × 0.60 = 1,800 approvals.
- Increase: 1,800 − 1,560 = 240 approvals.
- Percentage increase: 240 ÷ 1,560 × 100 = 15.38%, rounded to two decimal places.
The denominator is the original number of approvals. Dividing by 1,800 gives the increase as a proportion of the later total, which answers a different question. Subtracting 60% from 65% gives a five-percentage-point fall in the approval rate, not the change in approved applications.
Before calculating a percentage change, finish this sentence: “I am comparing the change with the original number of ___.”
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Example 2: a weighted average
Two teams completed the following casework in one month:
| Team | Cases completed | Mean handling time per case |
|---|---|---|
| North | 120 | 18 minutes |
| South | 80 | 28 minutes |
What was the mean handling time across all 200 cases?
The teams handled different numbers of cases, so each team average does not carry equal weight.
- North’s total handling time: 120 × 18 = 2,160 minutes.
- South’s total handling time: 80 × 28 = 2,240 minutes.
- Combined time: 2,160 + 2,240 = 4,400 minutes.
- Combined mean: 4,400 ÷ 200 = 22 minutes per case.
Simply averaging 18 and 28 gives 23. That would be appropriate if both teams had completed the same number of cases. Here the faster team handled more cases, pulling the combined mean towards 18.
A useful sense check is that the answer must lie between 18 and 28 minutes. That will not distinguish 22 from 23, but it can catch a misplaced decimal or a division error.
Example 3: a budget decision with constraints
A team can fund whole projects from the table. Each project can be funded once. The budget is €100,000, and at least one funded project must be outside Dublin. Benefits are additive, and there are no other dependencies.
| Project | Location | Cost | Expected annual benefit units |
|---|---|---|---|
| A | Dublin | €55,000 | 90 |
| B | Cork | €45,000 | 65 |
| C | Galway | €40,000 | 62 |
| D | Dublin | €30,000 | 42 |
Which combination gives the highest expected benefit within the rules?
Compare feasible combinations. A + B costs €100,000 and gives 155 benefit units. A + C costs €95,000 and gives 152. B + C gives 127; B + D gives 107; C + D gives 104. A + D fits the budget but fails the outside-Dublin condition.
No three-project combination fits: even the cheapest three, B + C + D, cost €115,000. Single projects cannot exceed the 155 units from A + B. The answer is therefore A + B.
The best individual benefit-to-cost ratio does not settle every whole-project decision. Check the permitted combinations and all stated constraints. In this example, choosing the slightly cheaper A + C leaves €5,000 unused and produces three fewer benefit units.
Review the method, then add time pressure
For every missed question, identify whether the error involved selecting data, choosing a method, arithmetic, units or rounding. Redo the calculation from the table rather than copying the solution. Keep intermediate values unrounded unless the question specifies otherwise.
Try the free AO numerical sample to practise answering before viewing the explanation. PSP Premium includes 228 AO numerical questions: 120 in the main reasoning bank and 108 in three secondary 36-question papers. The secondary papers use shared data sets, charts, five answer choices and a 30-minute practice timer, following the publicly linked AO self-assessment. Untimed practice is also available. This older self-assessment does not establish the vacancy 9005 specification. Follow your own invitation for question counts, response method and timing.
Explore AO Premium practice at €39/month, renewing until cancelled, or open AO practice if you are already a member. Use the AO study plan to organise your sessions and the verbal and judgement guide to work on evidence and decisions.