Percentages, Averages & Work/Time Word Problems
Key Takeaways
- Percentage calculations enable rapid parsing of call volume statistics, false alarm rates, and resource utilization
- Arithmetic mean provides daily average emergency response rates for staffing and resource planning
- Speed-distance-time formula S = D / T requires converting minutes to hours (e.g. 45 min = 0.75 hours) prior to calculation
- Combined work-rate formula T_combined = (A * B) / (A + B) solves simultaneous cistern filling and basement pumping tasks
- Clear step-by-step algebraic modeling prevents errors on complex multi-part FOE word problems
Last updated: July 2026
Percentages, Averages & Work/Time Word Problems
Numerical word problems on the Fire Officer Examination evaluate your capacity to extract key operational data, apply correct algebraic formulas, and calculate exact figures for administrative reports, resource planning, and emergency responses. This section covers percentage calculations, arithmetic averages, speed-distance-time rates, and combined work-rate/pipe-filling problems.
Percentages in Fire Incident Statistics
Percentage calculations are required for analyzing incident reports, equipment readiness rates, and emergency response statistics.
ight)$$ --- ### Worked Example: Incident Breakdown Calculation **Problem**: Over the course of a month, a municipal fire station logged **240 total emergency calls**. Statistical analysis reveals that **25%** of these calls were false alarms triggered by system malfunctions. How many calls were genuine emergency responses? **Step-by-Step Solution**: 1. Identify total calls: $N = 240$. 2. Calculate number of false alarms: $$ ext{False Alarms} = 240 imes 0.25 = 60 ext{ calls}$$ 3. Subtract false alarms from total calls to find genuine emergencies: $$ ext{Genuine Emergencies} = 240 - 60 = 180 ext{ calls}$$ 4. Direct calculation alternative: $240 imes (1 - 0.25) = 240 imes 0.75 = 180 ext{ calls}$. --- ## Arithmetic Mean and Weighted Averages The **arithmetic mean (average)** represents the central value of a set of numerical data: $$ar{X} = rac{\sum X}{n} = rac{ ext{Sum of all values}}{ ext{Total number of items}}$$ --- ### Worked Example: Daily Response Call Average **Problem**: A district fire station recorded the following emergency response call volumes across three consecutive days: Day 1 = **8 calls**, Day 2 = **12 calls**, Day 3 = **10 calls**. What is the average number of calls per day? **Solution**: 1. Sum the call volumes: $\sum X = 8 + 12 + 10 = 30 ext{ calls}$. 2. Count the number of days: $n = 3$. 3. Compute the arithmetic mean: $$ar{X} = rac{30}{3} = 10 ext{ calls per day}$$ --- ## Speed, Distance, and Time Rates Emergency response time analysis relies on the fundamental motion formula: $$ ext{Speed } (S) = rac{ ext{Distance } (D)}{ ext{Time } (T)} \quad \implies \quad D = S imes T \quad \implies \quad T = rac{D}{S}$$ *Important Rule*: Units must be uniform before performing calculations (e.g., converting minutes to hours). --- ### Worked Example: Mutual-Aid Engine Travel Speed **Problem**: A fire engine is dispatched on a mutual-aid call to a neighboring municipality located **60 kilometers** away. The engine arrives at the scene exactly **45 minutes** after departure. What was the average speed of the fire engine in kilometers per hour (km/h)? **Solution**: 1. Identify given values: Distance $D = 60 ext{ km}$, Time $T = 45 ext{ minutes}$. 2. Convert time from minutes to hours: $$T = rac{45 ext{ minutes}}{60 ext{ minutes/hour}} = 0.75 ext{ hours}$$ 3. Apply the speed formula: $$S = rac{D}{T} = rac{60 ext{ km}}{0.75 ext{ hours}} = 80 ext{ km/h}$$ 4. Answer: The average travel speed was **80 km/h**. --- ## Work-Rate and Pipe/Fill-Rate Problems Work-rate problems calculate how long individuals, pumps, or water tankers take to complete a task working independently versus working together. ### The Combined Work-Rate Formula If Tanker A completes a fill task in time $A$, and Tanker B completes the same fill task in time $B$, their individual rates are $ rac{1}{A}$ and $ rac{1}{B}$ task per unit time. When working together, their combined rate $R_{ ext{total}}$ is: $$ rac{1}{T_{ ext{combined}}} = rac{1}{A} + rac{1}{B} \quad \implies \quad T_{ ext{combined}} = rac{A imes B}{A + B}$$ --- ### Worked Example: Combined Cistern Filling **Problem**: Water Tanker A can fill a **10,000-liter** emergency storage cistern in **20 minutes** operating alone. Water Tanker B can fill the same 10,000-liter cistern in **30 minutes** operating alone. If both tankers discharge simultaneously into the cistern, how many minutes will it take to fill the cistern completely? | Supply Unit | Individual Time ($t$) | Work Rate ($ ext{cistern/min}$) | | :--- | :--- | :--- | | **Tanker A** | 20 minutes | $ rac{1}{20}$ of cistern per minute | | **Tanker B** | 30 minutes | $ rac{1}{30}$ of cistern per minute | | **Combined (A + B)** | $T_{ ext{combined}}$ | $ rac{1}{20} + rac{1}{30} = rac{5}{60} = rac{1}{12}$ per minute | **Calculation**: 1. Combined rate: $ rac{3}{60} + rac{2}{60} = rac{5}{60} = rac{1}{12}$. 2. Invert combined rate to find combined time: $$T_{ ext{combined}} = rac{1}{ rac{1}{12}} = 12 ext{ minutes}$$ 3. Using the Product-Over-Sum shortcut formula: $$T_{ ext{combined}} = rac{20 imes 30}{20 + 30} = rac{600}{50} = 12 ext{ minutes}$$Test Your Knowledge
In a quarterly safety report, a fire station recorded 320 total structural inspections. If 15% of the inspected buildings received non-compliance notices, how many buildings successfully passed inspection without any non-compliance notices?
A
B
C
D
Test Your Knowledge
A ladder truck responds to an incident 45 kilometers away and completes the journey in 36 minutes. What was the ladder truck's average speed in kilometers per hour (km/h)?
A
B
C
D
Test Your Knowledge
Pump A can drain a flooded basement in 15 hours. Pump B can drain the same basement in 10 hours. If both pumps operate together simultaneously, how many hours will it take to drain the basement completely?
A
B
C
D