Numerical Reasoning and Data Interpretation Flashcards

A police division's overtime budget increased by 15% in 2024, then decreased by 15% in 2025. A commander claims the budget is back to its original level. Which of the following is correct?

Answer: The commander is wrong; the budget is now 2.25% below the original amount.

Percentage changes apply to different bases, so they do not cancel. Starting at $100,000: a 15% increase yields $115,000. A 15% decrease from $115,000 is $17,250, leaving $97,750 — which is 2.25% below the original. The key trap is assuming a +15% and −15% net to zero, which ignores that the second percentage is applied to the already-inflated figure.

A city's reported violent crimes fell from 840 to 756 in one year. Over the same period, the city's population grew from 280,000 to 315,000. By what percentage did the violent crime RATE (per 100,000 residents) change?

Answer: Decreased by 20%

Rate Year 1: (840 ÷ 280,000) × 100,000 = 300 per 100k. Rate Year 2: (756 ÷ 315,000) × 100,000 = 240 per 100k. Percentage change: (240 − 300) ÷ 300 × 100 = −20%. The raw incident count dropped by only 10%, but because the population grew significantly, the rate dropped twice as much. Choosing 10% is the classic trap of using absolute counts instead of population-adjusted rates.

During a special operation, an officer worked: 3 shifts at $42.00/hr, 5 shifts at $48.00/hr, and 2 shifts at $54.00/hr. All shifts were 8 hours long. What was the officer's weighted average hourly rate across all 10 shifts?

Answer: $47.40

Weighted average = (3 × $42 + 5 × $48 + 2 × $54) ÷ 10 = ($126 + $240 + $108) ÷ 10 = $474 ÷ 10 = $47.40. The most common error is the simple (unweighted) average: ($42 + $48 + $54) ÷ 3 = $48.00, which gives equal weight to each pay rate regardless of how many shifts were worked at that rate.

A precinct logs the following Priority 1 response times (in minutes) for one week: 7, 4, 9, 3, 14, 6, 8, 5, 11, 3. Department policy requires the median response time to be under 7 minutes. Which statement correctly evaluates compliance?

Answer: Compliant; the median is 6.5 minutes.

Ordered values: 3, 3, 4, 5, 6, 7, 8, 9, 11, 14. With 10 values (even count), the median is the average of the 5th and 6th values: (6 + 7) ÷ 2 = 6.5. Since 6.5 < 7, the precinct is compliant. The trap in choice B is treating the 5th value (6) alone as the median — a rule that only applies to odd-numbered data sets. The mean is actually (3+3+4+5+6+7+8+9+11+14) ÷ 10 = 70 ÷ 10 = 7.0, but policy references the median, not the mean.

In 2024, a police service deployed its 360 officers across three divisions in the ratio 5:4:3. In 2025, the service expanded to 408 officers and redistributed them in the ratio 4:3:5. How many more officers does the THIRD division have in 2025 compared to 2024?

Answer: 80 more officers

2024: Total ratio parts = 12. Each part = 360 ÷ 12 = 30 officers. Third division: 3 × 30 = 90. 2025: Total ratio parts = 12. Each part = 408 ÷ 12 = 34 officers. Third division: 5 × 34 = 170. Difference: 170 − 90 = 80. A common error is failing to notice that the third division's share jumps from 3 parts to 5 parts in addition to the overall workforce growing — both factors must be accounted for.

A precinct's annual budget is $4,800,000. Personnel costs consume 65% of the total budget. Equipment costs represent 18% of the remaining (non-personnel) budget. If equipment costs increase by 12% next year, how much additional funding is required?

Answer: $36,288

Step 1 — Non-personnel budget: 35% × $4,800,000 = $1,680,000. Step 2 — Equipment costs: 18% × $1,680,000 = $302,400. Step 3 — 12% increase: $302,400 × 0.12 = $36,288. Choice B ($103,680) results from incorrectly applying 18% to the full $4,800,000 budget rather than the non-personnel remainder. Choice C ($40,320) uses 20% instead of 18% for equipment. The phrase 'of the remaining budget' is the critical qualifier that narrows the base.