9.2 Fractions, Decimals & Percent Calculations

9.2 Fractions, Decimals & Percent Calculations

Numerical reasoning on the IO Solutions NFSI exam heavily emphasizes working fluidly with fractions, decimals, and percentages. Firefighters encounter fractional measurements and percentage ratios constantly: measuring water levels in apparatus booster tanks ($1/4, 1/2, 3/4$ full), calculating remaining operational time on self-contained breathing apparatus (SCBA) cylinders, determining foam concentrate injection rates ($0.5%, 3%, 6%$), and calculating percentage pressure drops during fire flow testing. Because calculators are prohibited on the NFSI, candidate speed depends on instant recall of benchmark equivalents and efficient mental arithmetic techniques.

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Working with Fractions

A fraction represents a part of a whole, written as $\frac{\text{Numerator}}{\text{Denominator}}$, where the numerator indicates the number of active parts and the denominator indicates the total equal parts comprising the whole.

1. Equivalent Fractions & Reducing to Lowest Terms

To simplify fractions on scratch paper, divide both numerator and denominator by their Greatest Common Factor (GCF): $\text{Reduce } \frac{18}{24}: \quad \text{GCF} = 6 \rightarrow \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$

2. Converting Mixed Numbers and Improper Fractions

• Mixed Number to Improper Fraction: Multiply the whole number by the denominator, add the numerator, and place over the original denominator. $3 \frac{5}{8} = \frac{(3 \times 8) + 5}{8} = \frac{24 + 5}{8} = \frac{29}{8}$
• Improper Fraction to Mixed Number: Divide numerator by denominator; the quotient becomes the whole number and the remainder becomes the new numerator. $\frac{43}{6} = 43 \div 6 = 7 \text{ with a remainder of } 1 \rightarrow 7 \frac{1}{6}$

3. Adding and Subtracting Fractions with Unlike Denominators

To combine fractions with different denominators, find the Least Common Denominator (LCD), convert each fraction to an equivalent fraction with the LCD, and then add or subtract the numerators.

Worked Example: Summing Remaining SCBA Air Ratios

Three firefighters report their remaining SCBA cylinder air levels as $\frac{3}{4}$, $\frac{2}{3}$, and $\frac{1}{2}$. What is the total combined fractional air remaining across all three cylinders?

1. Find LCD: The LCD of 4, 3, and 2 is 12.
2. Convert to Equivalent Fractions: $\frac{3}{4} = \frac{9}{12}, \quad \frac{2}{3} = \frac{8}{12}, \quad \frac{1}{2} = \frac{6}{12}$
3. Sum Numerators: $\frac{9 + 8 + 6}{12} = \frac{23}{12} = 1 \frac{11}{12}\text{ full cylinder equivalents}$

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Fraction-Decimal-Percent Conversion Matrix

To perform rapid mental math on the NFSI without a calculator, candidates must memorize standard conversion benchmarks. Converting fractions to decimals involves dividing the numerator by the denominator. Converting decimals to percentages requires multiplying by 100 (shifting the decimal point two places to the right).

$\text{Fraction} \longrightarrow \text{Decimal} \quad (\text{Divide Numerator by Denominator})$ $\text{Decimal} \longrightarrow \text{Percent} \quad (\text{Shift Decimal 2 Places Right})$

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Mental Percentage Building Blocks

Calculating percentages mentally on the NFSI is effortless when using the 10% and 1% Building Block Method:

• Finding 10%: Move the decimal point one place to the left ($10% \text{ of } 450 = 45.0$).
• Finding 1%: Move the decimal point two places to the left ($1% \text{ of } 450 = 4.50$).
• Finding 5%: Take half of 10% ($5% \text{ of } 450 = \frac{45}{2} = 22.5$).
• Finding 15%: Add 10% and 5% ($15% \text{ of } 450 = 45 + 22.5 = 67.5$).
• Finding 20%: Double 10% ($20% \text{ of } 450 = 45 \times 2 = 90$).
• Finding 25%: Divide the number by 4, or add 20% and 5% ($25% \text{ of } 450 = 112.5$).

Percentage Increase & Decrease Formula

To determine the percentage change between an original value and a new value:

$\text{Percentage Change} = \left( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \right) \times 100\%$

Worked Example: Friction Loss Percentage Increase

When water flow through a 2-inch hose line increases from 100 GPM to 150 GPM, friction loss increases from 16 psi to 36 psi. What is the percentage increase in friction loss?

1. Identify Values: Original = 16 psi, New = 36 psi
2. Calculate Increase: $36 - 16 = 20\text{ psi}$
3. Divide by Original: $\frac{20}{16} = \frac{5}{4} = 1.25$
4. Multiply by 100%: $1.25 \times 100% = 125% \text{ increase}$

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Fireground Application: Foam Concentrate Proportioning

Firefighters use firefighting foam concentrates mixed with water to extinguish flammable liquid fires (Class B) and structure/wildland fires (Class A).

• Class A Foam: Typically proportioned from 0.1% to 1.0% (commonly 0.5%).
• Class B Foam: Proportioned at 3% for hydrocarbons (gasoline, diesel) or 6% for polar solvents (alcohol, acetone).

$\text{Gallons of Foam Concentrate} = \text{Total Finished Solution Gallons} \times \text{Concentration Percentage}$

$\text{Gallons of Water Required} = \text{Total Finished Solution Gallons} - \text{Gallons of Foam Concentrate}$

Worked Example: 3% Class B Foam Operation

An incident commander requests 1,000 gallons of finished 3% Class B foam solution to cover an aviation fuel spill. How many gallons of foam concentrate and how many gallons of water are required?

1. Concentrate Calculation: $1,000 \text{ gal} \times 3% = 1,000 \times 0.03 = 30\text{ gallons of foam concentrate}$
2. Water Calculation: $1,000 \text{ gal} - 30 \text{ gal} = 970\text{ gallons of water}$