Percentages, Averages & Work/Time Word Problems

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.

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Percentages in Fire Incident Statistics

Percentage calculations are required for analyzing incident reports, equipment readiness rates, and emergency response statistics.

$$\text{Percentage Amount} = \text{Base Total} \times \left( \frac{\text{Percentage Rate}}{100} \right)$$

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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: $$\text{False Alarms} = 240 \times 0.25 = 60 \text{ calls}$$
3. Subtract false alarms from total calls to find genuine emergencies: $$\text{Genuine Emergencies} = 240 - 60 = 180 \text{ calls}$$
4. Direct calculation alternative: $240 \times (1 - 0.25) = 240 \times 0.75 = 180 \text{ calls}$.

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Arithmetic Mean and Weighted Averages

The arithmetic mean (average) represents the central value of a set of numerical data:

$$\bar{X} = \frac{\sum X}{n} = \frac{\text{Sum of all values}}{\text{Total number of items}}$$

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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: $\sum X = 8 + 12 + 10 = 30 \text{ calls}$.
2. Count the number of days: $n = 3$.
3. Compute the arithmetic mean: $$\bar{X} = \frac{30}{3} = 10 \text{ calls per day}$$

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Speed, Distance, and Time Rates

Emergency response time analysis relies on the fundamental motion formula:

$$\text{Speed } (S) = \frac{\text{Distance } (D)}{\text{Time } (T)} \quad \implies \quad D = S \times T \quad \implies \quad T = \frac{D}{S}$$

Important Rule: Units must be uniform before performing calculations (e.g., converting minutes to hours).

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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 $D = 60 \text{ km}$, Time $T = 45 \text{ minutes}$.
2. Convert time from minutes to hours: $$T = \frac{45 \text{ minutes}}{60 \text{ minutes/hour}} = 0.75 \text{ hours}$$
3. Apply the speed formula: $$S = \frac{D}{T} = \frac{60 \text{ km}}{0.75 \text{ hours}} = 80 \text{ km/h}$$
4. Answer: The average travel speed was 80 km/h.

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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 $\frac{1}{A}$ and $\frac{1}{B}$ task per unit time. When working together, their combined rate $R_{\text{total}}$ is:

$$\frac{1}{T_{\text{combined}}} = \frac{1}{A} + \frac{1}{B} \quad \implies \quad T_{\text{combined}} = \frac{A \times B}{A + B}$$

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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?

Calculation:

1. Combined rate: $\frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12}$.
2. Invert combined rate to find combined time: $$T_{\text{combined}} = \frac{1}{\frac{1}{12}} = 12 \text{ minutes}$$
3. Using the Product-Over-Sum shortcut formula: $$T_{\text{combined}} = \frac{20 \times 30}{20 + 30} = \frac{600}{50} = 12 \text{ minutes}$$