7.2 Fractions, Decimals & Percentages in Fire Operations

7.2 Fractions, Decimals & Percentages in Fire Operations

Fractions in Fire Service Calculations

Fractions appear throughout fire service operations, governing equipment specifications, hose diameters (e.g., 1 3/4-inch, 2 1/2-inch, 4 1/2-inch), nozzle tip orifices (e.g., 7/8-inch, 15/16-inch, 1 1/8-inch), tank levels, and hydraulic friction loss formulas. Under civil service test conditions without a calculator, candidates must manipulate proper fractions, improper fractions, and mixed numbers quickly and accurately.

Finding the Lowest Common Denominator (LCD)

Fractions cannot be added or subtracted directly unless they share a common denominator. The Lowest Common Denominator (LCD) is the Least Common Multiple (LCM) of the denominators.

To find the LCD:

1. List the multiples of each denominator until you identify the smallest number that appears in both lists, or
2. Determine if the larger denominator is a multiple of the smaller denominator.

Example: Add 3/8 and 5/16.

• Multiples of 8: 8, 16, 24...
• Multiples of 16: 16, 32...
• The LCD is 16. Convert 3/8 into sixteenths by multiplying numerator and denominator by 2: (3 x 2) / (8 x 2) = 6/16.
• Now add numerators: 6/16 + 5/16 = 11/16.

Adding and Subtracting Mixed Numbers

A mixed number consists of a whole number combined with a fraction (e.g., 2 1/2). Two methods exist for adding and subtracting mixed numbers:

• Method 1 (Separate Whole Numbers and Fractions): Add or subtract the whole numbers and fractional parts independently. If subtracting and the upper fraction is smaller than the lower fraction, borrow 1 from the whole number and convert it into equivalent fractional units (e.g., convert 5 1/4 into 4 5/4).
• Method 2 (Convert to Improper Fractions): Convert mixed numbers to improper fractions, perform the operation over a common denominator, and convert the result back to a mixed number.

Example: A structural cribbing stack requires timber blocks of different thicknesses: one block is 3 3/8 inches thick, and a second block is 2 3/4 inches thick. What is the total height of the stacked cribbing?

1. Find the LCD for eighths and fourths: LCD is 8.
2. Convert 2 3/4 into eighths: 3/4 = (3 x 2) / (4 x 2) = 6/8. So 2 3/4 = 2 6/8.
3. Add whole numbers: 3 + 2 = 5.
4. Add fractions: 3/8 + 6/8 = 9/8.
5. Simplify improper fraction: 9/8 = 1 1/8.
6. Combine with whole number total: 5 + 1 1/8 = 6 1/8 inches.

Multiplying and Dividing Fractions

• Multiplying Fractions: Multiply the numerators together, multiply the denominators together, and simplify the product. When multiplying mixed numbers, you must convert them to improper fractions first.

(2/3) x (5/8) = (2 x 5) / (3 x 8) = 10 / 24 = 5 / 12

• Dividing Fractions (The Reciprocal Rule): To divide by a fraction, invert the divisor (flip the numerator and denominator to form its reciprocal) and multiply:

(3/4) / (1/8) = (3/4) x (8/1) = (3 x 8) / (4 x 1) = 24 / 4 = 6

Fireground Context: If a 3/4-mile supply line is laid using 1/8-mile hose rolls, how many rolls are required? (3/4) / (1/8) = 6 rolls.

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Fireground Benchmark Conversion Equivalents

Rapid mental conversion between fractions, decimals, and percentages is essential for civil service exam success. The following benchmark equivalents should be committed to memory:

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Decimal Arithmetic Operations

Addition and Subtraction: Decimal Point Alignment

The fundamental rule of decimal addition and subtraction is strict vertical alignment of the decimal points. Empty decimal places to the right should be padded with trailing zeros to maintain column accuracy.

Example: An engine's diesel fuel log records three consecutive refills: 18.4 gallons, 6.75 gallons, and 12.08 gallons. What is the total volume of fuel added?

      1 8 . 4 0   <-- Padded with zero
        6 . 7 5
   +  1 2 . 0 8
   ------------
      3 7 . 2 3   gallons

Decimal Multiplication: Counting Decimal Places

When multiplying decimals:

1. Multiply the numbers as if they were whole integers, completely ignoring the decimal points during multiplication.
2. Count the total number of decimal places to the right of the decimal point in both original factors.
3. In the final product, start at the far right and count leftward that exact same number of places to insert the decimal point.

Example: Water weighs 8.34 pounds per gallon. How much does 16.5 gallons of water weigh?

          8 . 3 4   (2 decimal places)
       x  1 6 . 5   (1 decimal place) --> Total = 3 decimal places
       ----------
          4 1 7 0   (834 x 5)
        5 0 0 4 0   (834 x 60)
      + 8 3 4 0 0   (834 x 100)
      -----------
      1 3 7 . 6 1 0 pounds

The product of 834 x 165 is 137,610. Shifting the decimal 3 places left yields 137.61 pounds.

Decimal Division: Shifting the Divisor

You cannot divide directly by a decimal divisor by hand. You must convert the divisor into a whole integer:

1. Move the decimal point of the divisor to the right until it is a whole number.
2. Move the decimal point of the dividend to the right by the exact same number of places (adding zeros if necessary).
3. Place the decimal point in the quotient directly above the new decimal point in the dividend, then divide as normal.

Example: Divide 45.6 by 1.2.

• Shift divisor 1.2 one place right to make 12.
• Shift dividend 45.6 one place right to make 456.
• Compute 456 / 12 = 38. Therefore, 45.6 / 1.2 = 38.

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Percentages in Fire Service Operations

A percentage represents a ratio out of 100 (per cent = per hundred). To convert a percentage to a decimal, shift the decimal point two places to the left (e.g., 3% = 0.03; 0.5% = 0.005; 33% = 0.33). To find the percentage of a quantity, multiply the total quantity by the decimal equivalent:

Part = Total x (Percentage / 100)

1. Foam Concentrate Proportioning

Firefighting foam is created by mixing water, foam concentrate, and air. The proportioning rate defines the percentage of foam concentrate mixed into the water stream before aeration:

• Class A Foam (Vegetation and Ordinary Combustibles): Proportioning rates range from 0.1% to 1.0%.
  • Overhaul / mop-up: commonly 0.2% to 0.5%.
  • Wildland direct attack or structural exposure protection: 1.0%.
  • Calculation: A 500-gallon water tank proportioned at 1.0% requires: 500 x 0.01 = 5 gallons of Class A foam concentrate.
• Class B Foam (Flammable Liquids):
  • Hydrocarbons (gasoline, diesel, jet fuel): proportioned at 3%.
  • Polar Solvents (alcohol, ethanol blends, acetone): proportioned at 6%.
  • Calculation: An incident requires flowing a 250 GPM foam handline for 15 minutes on a spilled gasoline tanker fire at 3%:
    • Total solution discharged = 250 GPM x 15 minutes = 3,750 gallons.
    • Foam concentrate required = 3,750 x 0.03 = 112.5 gallons of concentrate.
    • Because concentrate is supplied in 5-gallon pails, the crew needs: 112.5 / 5 = 22.5, rounded up to 23 pails.

2. SCBA Cylinder Pressure and Air Reserve Thresholds

Self-Contained Breathing Apparatus (SCBA) cylinders operate at high pressures (standard 4,500 psi cylinders). Firefighters track air consumption using critical percentage benchmarks:

• 100% Full Capacity: 4,500 psi on a 4,500-psi cylinder. Many departments set a minimum fill level, such as 90% (4,050 psi), for a cylinder to count as in service.
• 50% Mark: 2,250 psi on a 4,500-psi cylinder. Some departments use half a cylinder as a turn-around point. The NFPA air-management approach (NFPA 1404) aims to have firefighters out of the hazard area before the low-air alarm activates.
• 33% Low-Air Alarm Threshold: Standard SCBA End-of-Service-Time Indicators (EOSTI)—such as the Heads-Up Display (HUD) flashing red lights and mechanical vibralert tactile alarms—activate when 33% (one-third) of rated cylinder capacity remains:

4,500 psi x (1/3) = 1,500 psi

Air-management practice treats that remaining third as an emergency reserve for getting out, not as working air.

3. Water Tank Depletion and Reserve Percentages

Determining water reserve percentages on the fireground prevents running pumps dry, which destroys pump impellers and mechanical seals.

Calculation Formula:

Remaining Percentage = (Remaining Gallons / Total Tank Capacity) x 100%

Scenario: An engine with a 750-gallon booster tank has 180 gallons remaining. What percentage of the tank remains?

Remaining % = (180 / 750) x 100% = (18 / 75) x 100% = (6 / 25) x 100% = 6 x 4% = 24%

4. Percentage Increase and Decrease Calculations

Civil service word problems frequently ask for the percentage change between two operational figures (e.g., changes in water flow, staffing budgets, or incident call volumes):

Percentage Change = [(New Value - Original Value) / Original Value] x 100%

Example: A fire department's annual structure fire call volume increased from 400 incidents last year to 460 incidents this year. What is the percentage increase?

Change = 460 - 400 = 60 incidents
Percentage Increase = (60 / 400) x 100% = (6 / 40) x 100% = (3 / 20) x 100% = 15%

5. Sales Tax, Markups & Percent of a Total

FCTC's sample math items include a sales-tax total, a percent-of-a-fund question, and a percent decrease. Three patterns cover them:

• Total after tax or markup: Total = Price × (1 + rate). A $60 pair of boots with 7.5% tax costs 60 × 1.075 = $64.50. Computing only the tax ($4.50) is the classic wrong answer.
• Percent of a total: Percent = Part ÷ Whole × 100. If $9 of a $36 station fund goes to supplies, that is 9 ÷ 36 = 1/4 = 25%. Watch the answer format: choices may mix "25%" with the decimal ".25." Pick the form the question asks for.
• Percent decrease: Percent change = (New − Original) ÷ Original × 100. A price that drops from $250 to $200 changes by −50 ÷ 250 = −0.20, a 20% decrease. Divide by the original value, not the new one; 50 ÷ 200 = 25% is the trap.