Free Irish Numerical Reasoning Exam Flashcards

• Order of Merit (OOM): A ranked numerical list of successful candidates at a particular selection stage (e.g., Stage 1 psychometric testing). It dictates the order and batches in which candidates are invited to subsequent assessment stages. • Appointment Panel: Established only after candidates pass all final-stage selection criteria (e.g., competency interviews). Candidates are drawn strictly in merit order from the panel as departmental vacancies arise. Panel placement is not a direct job offer and does not guarantee appointment if vacancies do not reach your ranking prior to panel expiration.

• Formula: Average Pacing Benchmark (seconds/item) = (Total Time in Minutes × 60) ÷ Total Number of Questions. • Micro-example: An assessment allocates 20 minutes for 24 items: (20 × 60) ÷ 24 = 1,200 ÷ 24 = 50 seconds per item. • Assessment Application: Use this benchmark as an average reference rather than a rigid limit. Fast single-step data lookups (20–30s) bank surplus time for multi-stage calculations (70–90s).

• Expected Value Formula: EV = P(correct) × (+1) + P(incorrect) × 0. • Micro-example: Eliminating 2 distractor options leaves 3 possibilities. P(correct) = 1/3. EV = (1/3 × 1) + (2/3 × 0) = +0.33 raw points. • Strategic Rule: When an assessment does not impose negative marking penalties, never leave questions blank before the timer expires. Eliminating even one option creates a positive expected value.

• Core Rule: Select 'Cannot Say' when a requested comparison or conclusion requires baseline figures, population counts, or definitions not contained in the provided table, chart, or accompanying text. • Micro-example: A table shows Department A processed 1,400 passport renewals and Department B processed 950 renewals. If asked 'Which department achieved a higher percentage completion rate?', the answer is 'Cannot Say' unless total submitted applications for each department are provided. • Trap: Do not extrapolate trends, assume unstated denominators, or import external knowledge.

• Official Rule: Permitted equipment is competition-specific and defined in the Candidate Information Booklet and assessment familiarisation material. • Calculators: Certain competitions permit an external handheld calculator or provide an integrated on-screen calculator; others evaluate mental arithmetic and prohibit calculators entirely. • Rough Work Paper: When permitted, use clean blank paper strictly for tracking formulas, units, and intermediate figures. Pre-written notes or formula reference sheets are prohibited.

1. Identify the Target Quantity & Units: Pinpoint exactly what is being asked (e.g., absolute change vs percentage change, thousands vs millions, calendar year vs fiscal year). 2. Locate Relevant Source Cells: Read table titles, row/column headers, footnotes, and currency notations to isolate the exact required values. 3. Formulate the Operation: Decide whether the solution requires a direct lookup, a multi-step chain, or order-of-magnitude estimation before entering numbers into a calculator.

• Formula: Percentage Change = [(New Value - Original Value) ÷ Original Value] × 100%. • Micro-example: Annual department ICT expenditure expands from €450,000 to €540,000: Change = [(540,000 - 450,000) ÷ 450,000] × 100% = (90,000 ÷ 450,000) × 100% = +20%. • Critical Trap: Always divide by the Original (initial baseline) Value. Dividing the difference by the new value (€90k ÷ €540k = 16.67%) is a common distractor error.

• Formula: Original Value = Final Value ÷ (1 ± r), where r is the rate expressed as a decimal. • Micro-example: A total public procurement invoice of €3,690 includes 23% VAT. Original Pre-VAT Amount = €3,690 ÷ (1 + 0.23) = €3,690 ÷ 1.23 = €3,000. • Fatal Trap: Never subtract 23% directly from the gross final amount (€3,690 × 0.23 = €848.70; €3,690 - €848.70 = €2,841.30, which is incorrect). VAT is charged on the net base, not deducted from the gross.

• Percentage Points (pp): The absolute arithmetic difference between two percentage figures. Difference = New Rate (%) - Old Rate (%). • Percentage Change (%): The relative proportional increase or decrease of the rate itself. Formula = [(New Rate - Old Rate) ÷ Old Rate] × 100%. • Micro-example: Staff turnover in a regional agency increases from 5% to 6%: • Rise in percentage points = 6% - 5% = 1.0 percentage point. • Relative percentage increase = [(6 - 5) ÷ 5] × 100% = 1/5 × 100% = +20%.

• Method: Multiply the initial amount by the product of individual period multipliers: Final Value = Initial Value × (1 + r₁) × (1 + r₂). • Micro-example: A public agency budget grows by 15% in Year 1 and then contracts by 10% in Year 2. Combined Multiplier = 1.15 × 0.90 = 1.035. Net Two-Year Change = (1.035 - 1) × 100% = +3.5% net increase. • Core Rule: Percentages cannot be added directly (15% - 10% ≠ 5%) because the second percentage applies to the updated base.

• Formula: Index Number = (Value in Current Period ÷ Value in Base Period) × 100. • The base period value is standardized to exactly 100.0. • Micro-example: Base Year capital spending is €50m (Index = 100.0). Current Year spending is €58.5m. Current Year Index = (58.5 ÷ 50.0) × 100 = 1.17 × 100 = 117.0. • Quick Interpretation: An index of 117.0 indicates expenditure is 17.0% above the base period level (117 - 100 = +17%).

• Formula: Percentage Change = [(Index in Period 2 - Index in Period 1) ÷ Index in Period 1] × 100%. • Micro-example: The Consumer Price Index (CPI) increases from 125.0 in 2023 to 135.0 in 2025. Change = [(135.0 - 125.0) ÷ 125.0] × 100% = (10.0 ÷ 125.0) × 100% = +8.0%. • Critical Trap: Do not subtract 135 - 125 and call it 10%. The index rose by 10 index points, but the actual percentage increase from the 2023 baseline is 8.0%.

• Formula: New Index Value = (Original Index Value ÷ Original Index of New Base Year) × 100. • Micro-example: A series has Year 1 = 100, Year 2 = 120, Year 3 = 144. To rebase the series so Year 2 becomes the new base (100.0): • Year 1 New Index = (100 ÷ 120) × 100 = 83.33 • Year 2 New Index = (120 ÷ 120) × 100 = 100.0 • Year 3 New Index = (144 ÷ 120) × 100 = 120.0 • Check: Year 3 remains 20% higher than Year 2 (144 ÷ 120 = 1.20).

• Decimal Partitioning Steps: 1. 10%: Move decimal point one position to the left. 2. 5%: Halve the 10% value. 3. 15%: Sum the 10% value and 5% value. 4. 2.5%: Halve the 5% value (or divide 10% by 4). • Micro-example: Calculate 15% and 2.5% of €72,000: 10% = €7,200; 5% = €3,600. 15% = €7,200 + €3,600 = €10,800. 2.5% = €3,600 ÷ 2 = €1,800.

1. Sum the Parts: Total Parts = a + b + c. 2. Find One Part's Value: Value per Part = Total Sum ÷ Total Parts. 3. Multiply Out: Share of a = a × (Value per Part); Share of b = b × (Value per Part); Share of c = c × (Value per Part). • Micro-example: Divide €240,000 between three public projects in ratio 3 : 4 : 5. Total parts = 3 + 4 + 5 = 12 parts. 1 part = €240,000 ÷ 12 = €20,000. Allocations: Project A = €60,000; Project B = €80,000; Project C = €100,000.

• Part-to-Part Ratio: Compares one subset directly to another subset (e.g., Male to Female applicants = 3 : 2). • Part-to-Whole Fraction: Compares a subset to the combined total group. Fraction of group a = a ÷ (a + b). • Micro-example: An office has 4 Executive Officers (EO) and 16 Clerical Officers (CO). Ratio EO : CO is 4 : 16 = 1 : 4. The fraction of EOs in the entire team is 4 ÷ (4 + 16) = 4/20 = 1/5 = 20% (NOT 1/4 or 25%).

• Core Property: The product of the two variables remains constant: x₁ × y₁ = x₂ × y₂. Formula: y₂ = (x₁ × y₁) ÷ x₂. • Micro-example: 8 civil service caseworkers clear an audit batch in 15 working days. How many days will 12 caseworkers take at the same processing speed? Total Effort = 8 caseworkers × 15 days = 120 caseworker-days. Days for 12 caseworkers = 120 ÷ 12 = 10 working days. • Trap: Do not cross-multiply as in direct proportion; additional workers reduce duration.

• Method: Harmonize the shared term (B) by finding the Least Common Multiple (LCM) of its values (3 and 4; LCM = 12). • Scale Ratio 1 by 4: A : B = (2 × 4) : (3 × 4) = 8 : 12. • Scale Ratio 2 by 3: B : C = (4 × 3) : (5 × 3) = 12 : 15. • Unified Ratio: A : B : C = 8 : 12 : 15. • Micro-example: If C represents €45,000 (15 parts), 1 part = €3,000. Therefore, A (8 parts) = €24,000.

Divide the two percentage growth rates when both are meaningful and the denominator is nonzero. A grows from €2m to €3m: 50%. B grows from €10m to €12m: 20%. The rate ratio is 50/20 = 2.5, so A’s percentage growth rate is 2.5 times B’s. In euros, B’s €2m increase is twice A’s €1m increase. Avoid the ambiguous phrase “2.5 times faster.”

Weight each office’s average by the number of cases included in that average, not automatically by staff headcount. If 90 completed cases average 12 days and 10 completed cases average 22 days, the combined case average is (90×12 + 10×22)/(90+10) = 13 days. Staff counts alone do not establish how many cases contributed to each mean.

• Rule: For two ratios a / b and c / d with b and d nonzero, direct equality holds if and only if (a × d) = (b × c). • Micro-example: Are 21 / 56 and 15 / 40 equal in proportion? Cross-product 1: 21 × 40 = 840. Cross-product 2: 56 × 15 = 840. Because both cross-products equal 840, the ratios are strictly proportional (both simplify to 3/8 = 0.375).

• Critical Header & Footnote Traps: 1. Unit Modifiers: Headers like '(in €000s)' or '(headcount in 100s)' mean a cell reading '85' represents €85,000 or 8,500 people. 2. Asterisks & Scope Notes: Footnotes often state 'Excludes decentralized units', 'Provisional figures', or 'Includes capital grants'. 3. Timeframes: Distinguish between calendar years (Jan–Dec), government fiscal years, and rolling quarters. • Rule: Missing a unit modifier shifts answers by factors of 10, 100, or 1,000—common multiple-choice traps.

• Clustered Bar Chart: Displays independent bars side-by-side for each subcategory; each bar's top edge corresponds directly to its value on the numerical axis. • Stacked Bar Chart: Combines subcategories into a single vertical bar showing the cumulative total. • Reading Stacked Segments: To find the value of a middle or top segment: Segment Value = Upper Boundary Level - Lower Boundary Level of that color band (NOT the height on the y-axis).

• Formula: Sector Value = (Central Angle in Degrees ÷ 360°) × Total Dataset Value. • Percentage Equivalent: Sector % = (Central Angle ÷ 360°) × 100%. • Micro-example: A department annual budget is €108 million. Facilities management is depicted by a 60° sector. Budget Share = (60° ÷ 360°) × €108m = (1/6) × €108m = €18 million. Percentage Share = (60 ÷ 360) × 100% = 16.67%.

• Dual-Axis Trap: Two different metrics are plotted over time against separate vertical scales (e.g., Left Axis = Application Volume in thousands; Right Axis = Unit Processing Cost in €). • Danger: The physical intersection of the two lines on the graph does NOT mean volume equals unit cost. They simply share a pixel coordinate. • Strategy: Trace each curve strictly to its corresponding designated axis scale before evaluating numerical magnitudes or trends.

If bar heights are frequencies, sum them directly. If the vertical axis is frequency density, each class frequency is density × class width, even when all widths are equal. Example: bins [0,5), [5,10), [10,15) have frequencies 15,65,20. The total is 100 and 80/100 = 80% lie below 10. The equivalent density heights are 3,13,4; summing those heights gives 20, not the sample size.

A share uses the overall total as its denominator. If the total grows faster than a component, the component’s share can fall while its absolute count rises. Example: 3,000 of 10,000 cases is 30%; 4,000 of 20,000 is 20%. The share falls 10 percentage points, but component output rises 1,000 cases, or 33.33%. Compare both the numerator and the total before drawing a volume conclusion.

• Formula: CAGR = (End Value ÷ Start Value)^(1/n) - 1, where n = number of intervening years. • Micro-example: Public service online portal registrations grow from 40,000 to 48,400 over a 2-year period (n = 2): Ratio = 48,400 ÷ 40,000 = 1.21. Square Root: √(1.21) = 1.10. CAGR = 1.10 - 1 = +10% per annum. • Verification: 40,000 × 1.10 = 44,000 (Year 1); 44,000 × 1.10 = 48,400 (Year 2).

• Reconciliation Check: Confirm that secondary dimensions—specifically reporting year, geographic definitions, and inclusion criteria—are identical across both sources. • Micro-example: Table 1 lists 2024 caseloads by county; Table 2 lists 2025 staff allocations. Dividing Table 1 output by Table 2 staff creates an invalid metric due to the 1-year temporal mismatch. • Protocol: Confirm exact matching of column/row identifiers before carrying values into multi-table calculations.

• Quoted Rate: €1 = X Foreign Currency (FC). 1. EUR to Foreign Currency: Multiply by Rate: FC = EUR × X. 2. Foreign Currency to EUR: Divide by Rate: EUR = FC ÷ X. • Micro-example: An Irish departmental contract specifies payment of £68,000 GBP when €1 = £0.85 GBP. Cost in Euro = £68,000 ÷ 0.85 = €80,000. • Sanity Check: Because £1 is stronger than €1 (£0.85 buys €1.00), the euro figure must be numerically greater than the pound figure.

Identify which currency the headings refer to and whose viewpoint is used. If bank buy/sell refers to foreign currency, buying it from the bank uses its sell rate; selling it back uses its buy rate. Then check units: foreign-per-euro and euro-per-foreign quotes require opposite multiplication/division. Do not select a rate merely because its number is higher or lower.

• Formula: Total FTE = Full-Time Staff Count + Σ (Part-Time Weekly Hours ÷ Standard Full-Time Weekly Hours). • Micro-example: A section employs 12 full-time staff (37 hours/week) and 8 part-time staff working 18.5 hours/week. Part-time FTE = 8 × (18.5 ÷ 37) = 8 × 0.5 = 4.0 FTE. Total Staffing = 12 + 4.0 = 16.0 FTE. • Note: Physical headcount is 20 individuals, but operational capacity equals 16.0 FTE.

• Area Multipliers (Squared Dimensions): • 1 m² = (100 cm)² = 10,000 cm² • 1 km² = (1,000 m)² = 1,000,000 m²; 1 Hectare = 10,000 m² • Volume Multipliers (Cubed Dimensions): • 1 m³ = (100 cm)³ = 1,000,000 cm³ • 1 m³ = 1,000 Litres; 1 Litre = 1,000 cm³ (or 1,000 ml) • Trap: Converting 4 km² into square metres requires multiplying by 1,000,000 (= 4,000,000 m²), not 1,000.

• Numerical Exponents: • 1 Thousand (k) = 10³ = 1,000 • 1 Million (m) = 10⁶ = 1,000,000 • 1 Billion (b) = 10⁹ = 1,000,000,000 • Micro-example: An agency combines €3.5m in capital infrastructure with €450k in software licences. Convert to common base: €3,500,000 + €450,000 = €3,950,000 (€3.95m). • Distractor Trap: Adding 3.5 + 450 = 453.5 ignores the thousand-fold multiplier difference.

• Formula: Per Capita Expenditure = Total Aggregate Expenditure ÷ Total Population. • Standardization Rule: Express both numerator and denominator in absolute raw numbers or identical powers of 10 prior to division. • Micro-example: National training expenditure is €14 billion for a population of 5 million citizens. Convert to scientific notation: (1.4 × 10¹⁰) ÷ (5.0 × 10⁶) = (1.4 ÷ 5.0) × 10⁴ = 0.28 × 10,000 = €2,800 per person.

• Formula: Decimal Hours = Hours + (Minutes ÷ 60) + (Seconds ÷ 3,600). • Standard Reference Conversions: • 12 mins = 0.20 hrs; 15 mins = 0.25 hrs; 30 mins = 0.50 hrs; 45 mins = 0.75 hrs; 48 mins = 0.80 hrs. • Micro-example: A specialist works 6 hours and 36 minutes at a recharge rate of €50/hour. Decimal Hours = 6 + (36 ÷ 60) = 6 + 0.60 = 6.60 hours. Cost = 6.60 × €50 = €330.00 (NOT 6.36 × €50 = €318.00).

• Formula: Budget Variance = Actual Expenditure - Budgeted Expenditure. • Classification for Expenditures: • Favourable (F): Actual < Budget (underspend under the stated cost-variance convention). • Adverse / Unfavourable (A): Actual > Budget (overspend; budget exceeded). • Micro-example: Public outreach budget was €250,000; actual spend was €232,000. Variance = €232,000 - €250,000 = -€18,000 Favourable (a 7.2% underspend). An underspend alone does not prove better service performance or authorize retaining unspent public funds; actual funding rules determine their treatment.

• Formulas: Total Cost = Fixed Overhead + (Variable Cost per Unit × Output Volume). Unit Cost = Total Cost ÷ Output Volume. • Micro-example: A service center has €400,000 fixed overhead and €15 variable cost per applicant. At 20,000 applicants: Total Cost = €400,000 + (20,000 × €15) = €700,000. Unit Cost = €700,000 ÷ 20,000 = €35.00. • Scale Effect: If volume increases to 40,000, Total Cost = €1,000,000; Unit Cost drops to €25.00.

• Adding VAT (Net to Gross): Gross = Net × 1.23; VAT Amount = Net × 0.23. • Extracting VAT (Gross to Net): Net = Gross ÷ 1.23; VAT Amount = Gross - Net = Gross × (0.23 ÷ 1.23) ≈ Gross × 0.18699. • Micro-example: A supplier submits an all-inclusive gross invoice for €14,760 including 23% VAT. Net Price = €14,760 ÷ 1.23 = €12,000. VAT Paid = €14,760 - €12,000 = €2,760 (€12,000 × 0.23 = €2,760).

• Formula: Real Value = (Nominal Value ÷ Current Price Index) × 100 (where Base Year Index = 100.0). • Micro-example: An infrastructure program had €150m budget in Year 1 (Index 100.0) and €176m in Year 3 (Index 110.0). Real Value in Year 3 = (€176m ÷ 110.0) × 100 = €160m. • Analysis: Nominal expenditure grew by €26m (+17.33%), but after accounting for 10% price inflation, real purchasing power expanded by €10m (+6.67%).

Apply the stated rates and bands, then subtract permitted non-refundable tax credits with a zero floor. Hypothetical inputs: salary €50,000; first €42,000 at 20%; remainder at 40%; credits €3,550. Tax before credits = 8,400+3,200 = €11,600; tax payable = max(0,11,600−3,550) = €8,050; income after this tax = €41,950. These are exercise inputs, not current Irish tax bands; actual payroll can also deduct USC, PRSI, pensions and other amounts.

Simple net ROI = (total financial savings−cost)/cost ×100%; BCR = present value of benefits / present value of costs. Example: present-value benefits €450,000 and present-value costs €300,000 give BCR 1.50 and a present-value net-benefit/cost ratio of 50%. If €450,000 instead means an undiscounted four-year savings total, its BCR cannot be determined without timing and the discount rate. Do not mix nominal cash totals with discounted measures.

When the exercise specifies equal monthly accrual, allocation = annual amount × eligible months/12. A €360,000 calendar-year allocation starting 1 May includes May–December, eight months: €360,000×8/12 = €240,000. For a daily convention, use the stated eligible-day count and year denominator (365 or 366 as applicable); confirm inclusion of start/end dates. Real grants follow their own terms rather than automatically using this exercise rule.

• Formula: Bidder's Price Score = (Lowest Compliant Tender Price ÷ Bidder's Tender Price) × Maximum Allocated Price Marks. • Micro-example: 50 marks available for commercial price. The lowest compliant bid is €200,000 (awards full 50 marks). Bidder Y quotes €250,000. Bidder Y Score = (€200,000 ÷ €250,000) × 50 = 0.80 × 50 = 40.0 marks. • Key Property: The formula is non-linear; higher prices lose marks in inverse proportion to the lowest bid.

• Formula: 1 / T_total = (1 / T₁) + (1 / T₂), which rearranges to: T_total = (T₁ × T₂) ÷ (T₁ + T₂). • Micro-example: Officer X can process a caseload file backlog alone in 4 hours. Officer Y can process the identical backlog alone in 6 hours. Combined Time = (4 × 6) ÷ (4 + 6) = 24 ÷ 10 = 2.4 hours (2 hours 24 minutes). • Reality Check: Combined duration (2.4h) must always be less than the faster individual's standalone duration (4.0h).

• 2-Step Staffing Formula: 1. Total Labor-Hours = Total Volume of Cases × Standard Hours per Case. 2. Staff Required = Total Labor-Hours ÷ (Available Working Days × Productive Daily Hours per Worker). • Micro-example: 3,600 complex applications must be audited within 15 working days. Each audit requires 0.75 hours. Staff work 6 productive hours/day. Total Labor-Hours = 3,600 × 0.75 = 2,700 hours. Available hours per worker = 15 × 6 = 90 hours. Staff Required = 2,700 ÷ 90 = 30 officers. If whole workers are required and the result is fractional, round up; FTE capacity and physical headcount are different quantities.

• Harmonic Mean Formula: Average Speed = (2 × v₁ × v₂) ÷ (v₁ + v₂), where v₁ and v₂ are outbound and return speeds. • Micro-example: A regional inspector drives 80 km/h outbound and encounters heavy traffic returning at 40 km/h over the same route. Average Speed = (2 × 80 × 40) ÷ (80 + 40) = 6,400 ÷ 120 = 53.33 km/h. • Fatal Trap: A simple arithmetic average (80 + 40) ÷ 2 = 60 km/h is false because the inspector spends twice as much time driving at 40 km/h as at 80 km/h.

• Daily Output per FTE = Total Completed Cases ÷ (FTE Staff Count × Operational Days). Productivity Variance (%) = [(Actual Daily Rate - Benchmark Rate) ÷ Benchmark Rate] × 100%. • Micro-example: A unit of 10 FTE officers finishes 1,200 cases across 20 operational days. Standard benchmark is 5.0 cases/FTE/day. Actual Rate = 1,200 ÷ (10 × 20) = 1,200 ÷ 200 = 6.0 cases/FTE/day. Variance = [(6.0 - 5.0) ÷ 5.0] × 100% = (+1.0 ÷ 5.0) × 100% = +20% above benchmark.

• Systematic Pattern Diagnosis: 1. First Differences: Calculate difference between successive terms. If constant, it is arithmetic. 2. Second Differences: If first differences increase at a constant rate (e.g., +2, +4, +6...). 3. Alternating Operations: Check for paired alternating rules (e.g., × 2, + 4, × 2, + 4...). • Micro-example: Find the next number: 3, 6, 10, 20, 24, 48, ? Pattern: 3 × 2 = 6; 6 + 4 = 10; 10 × 2 = 20; 20 + 4 = 24; 24 × 2 = 48; 48 + 4 = 52.

For 389×52, use 380 < 389 < 400, giving 19,760 < product < 20,800. Of options 18,228;20,228;20,226;24,228, only 20,228 and 20,226 remain. The product ends in 9×2 = 18, hence units digit 8, selecting 20,228. Merely rounding to 400×50 ≈20,000 would not safely exclude 18,228; use bounds narrow enough to distinguish the options.