3.2 Fractions, Decimals & Percentages

3.2 Fractions, Decimals & Percentages

Fractional, decimal, and percentage computations constitute the quantitative backbone of practical administrative and technical duties in the armed forces. Whether calculating field ration disbursements, vehicle fuel consumption rates, equipment depreciation, or medical supply allocations, recruits must translate effortlessly across numerical forms. Recruit-style aptitude word problems often integrate these concepts, requiring candidates to perform multi-step conversions, evaluate fractional remaining balances, and solve commercial arithmetic challenges such as profit, loss, and simple interest without calculation aids.

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Conversions Between Fractions, Decimals, and Percentages

Fractions, decimals, and percentages represent identical mathematical values in different notations:

• A fraction represents parts of a whole ($\frac{a}{b}$).
• A decimal represents parts of a whole based on powers of ten ($0.25 = \frac{25}{100}$).
• A percentage expresses a ratio as a fraction of one hundred ($25\% = \frac{25}{100}$).

Common Benchmark Equivalencies

Committing benchmark equivalencies to memory allows candidates to eliminate conversion steps during time-sensitive tests.

Systematic Conversion Rules

1. Fraction to Decimal: Divide numerator by denominator (e.g., $\frac{5}{8} = 5 \div 8 = 0.625$).
2. Decimal to Percentage: Shift decimal point two places right (e.g., $0.075 = 7.5\%$).
3. Percentage to Fraction: Place percentage value over $100$ and reduce (e.g., $45\% = \frac{45}{100} = \frac{9}{20}$).

Applied Decimal Operations in Field Calculations

Handling decimal quantities swiftly requires strict precision and place-value accounting:

• Addition & Subtraction: Line up decimal points vertically before adding or subtracting. Pad missing places with zeroes (e.g., $18.4 - 5.75 = 18.40 - 5.75 = 12.65$).
• Multiplication: Multiply as whole numbers, count total decimal places across all factors, and restore the point (e.g., $1.4 \times 0.03 \rightarrow 14 \times 3 = 42$, with three decimal digits giving $0.042$).
• Division: Shift the decimal point in the divisor to form an integer, shift the dividend by the same number of places, and proceed with standard division ($16.8 \div 0.4 = 168 \div 4 = 42$).

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Arithmetic Operations with Fractions

Addition and Subtraction

Fractions cannot be directly combined unless they share a common denominator:

1. Identify the Lowest Common Denominator (LCD) of all terms.
2. Convert each fraction into an equivalent fraction with this denominator.
3. Add or subtract numerators while keeping the common denominator unchanged.
4. Simplify to lowest terms.

$$\frac{3}{4} + \frac{2}{5} - \frac{1}{2} = \frac{15}{20} + \frac{8}{20} - \frac{10}{20} = \frac{13}{20}$$

Multiplication of Fractions

Multiply numerators together and denominators together. Simplify diagonally before multiplying:

$$\frac{14}{25} \times \frac{15}{28} = \frac{1}{5} \times \frac{3}{2} = \frac{3}{10}$$

Here, $14$ and $28$ reduce by $14$, while $15$ and $25$ reduce by $5$.

Division of Fractions: The Reciprocal Rule

Dividing by a fraction is equivalent to multiplying by its reciprocal (invert divisor and multiply):

$$\frac{7}{12} \div \frac{14}{15} = \frac{7}{12} \times \frac{15}{14} = \frac{1}{4} \times \frac{5}{2} = \frac{5}{8}$$

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Applied Percentage Calculations

Percentage of a Quantity

Multiply the quantity by the fractional or decimal percentage: $\text{Value} = \frac{P}{100} \times \text{Total}$. Mental shortcut: For $15\%$ of GH₵ $480$, take $10\%$ (GH₵ $48$) plus $5\%$ (GH₵ $24$), yielding GH₵ $72$.

Percentage Increase, Decrease, and Change

Use direct multiplier factors:

• Increase by $r\%$: $\text{New} = \text{Original} \times (1 + \frac{r}{100})$
• Decrease by $r\%$: $\text{New} = \text{Original} \times (1 - \frac{r}{100})$
• Percentage Change: $\frac{|\text{New} - \text{Original}|}{\text{Original}} \times 100\%$

Commercial Mathematics: Profit and Loss

• Profit: $\text{Selling Price (SP)} - \text{Cost Price (CP)}$ when $SP > CP$. $\text{Profit } \% = \frac{\text{Profit}}{CP} \times 100\%$.
• Loss: $CP - SP$ when $CP > SP$. $\text{Loss } \% = \frac{\text{Loss}}{CP} \times 100\%$.

Simple Interest Calculations

Simple interest reflects non-compounding returns or loan costs:

$$I = \frac{P \times R \times T}{100}$$

Where $P$ is principal, $R$ is annual rate, $T$ is time in years, and Total Amount $A = P + I$. When time is in months, convert: $T = \frac{\text{months}}{12}$.

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Worked Practical Word Problems

Example 1: Multi-Stage Ration Allocation

Problem: A garrison mess receives $480\text{ kg}$ of grain. In Week 1, the kitchen prepares $\frac{3}{8}$ of the initial stock. In Week 2, it uses $\frac{2}{5}$ of the remaining supply. How many kilograms remain in store after Week 2?

Step-by-step Solution:

1. Week 1 use: $\frac{3}{8} \times 480 = 180\text{ kg}$.
2. Remaining after Week 1: $480 - 180 = 300\text{ kg}$.
3. Week 2 use: $\frac{2}{5} \times 300 = 120\text{ kg}$.
4. Final balance: $300 - 120 = 180\text{ kg}$. Alternative fraction approach: Remaining fraction $= (1 - \frac{3}{8}) \times (1 - \frac{2}{5}) = \frac{5}{8} \times \frac{3}{5} = \frac{3}{8}$. Total remaining $= \frac{3}{8} \times 480 = 180\text{ kg}$. Final Answer: $180\text{ kg}$.

Example 2: Fuel Consumption and Percentage Reserve

Problem: A patrol craft begins operations with $2,400\text{ litres}$ of diesel. In transit, it consumes $35\%$ of its fuel. During patrol manoeuvres, it uses an additional $25\%$ of the original capacity. How many litres remain, and what fraction of full capacity does this represent?

Step-by-step Solution:

1. Total percentage consumed: $35\% + 25\% = 60\%$.
2. Percentage remaining: $100\% - 60\% = 40\%$.
3. Remaining volume: $0.40 \times 2,400 = 960\text{ litres}$.
4. Simplified fraction: $40\% = \frac{40}{100} = \frac{2}{5}$. Final Answer: $960\text{ litres}$ remain ($\frac{2}{5}$ of full capacity).