2.2 Fraction Operations, Percentage Calculations & Discounts

2.2 Fraction Operations, Percentage Calculations & Discounts

Fractions and percentages are essential mathematical concepts used throughout police management, crime statistics evaluation, and resource logistics. In the Jamaica Constabulary Force, officers frequently encounter fractional shift allocations, percentage reductions in crime stats, equipment procurement discounts, and statutory penalty structures. Mastering conversions between fractions, decimals, and percentages enables officers to evaluate data accurately and communicate findings clearly.

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1. Fraction Simplification & Basic Operations

A fraction represents a part of a whole, written as (\frac{a}{b}), where (a) is the numerator (number of parts taken) and (b) is the denominator (total equal parts in the whole, (b \neq 0)).

Simplifying Fractions

To express a fraction in its simplest form (lowest terms), divide both the numerator and denominator by their Greatest Common Factor (GCF).

• Example: A police post has 16 officers on duty out of a total station complement of 24 officers. The fraction on duty is (\frac{16}{24}). The GCF of 16 and 24 is 8. [ \frac{16 \div 8}{24 \div 8} = \frac{2}{3} ] Thus, (\frac{2}{3}) of the station complement is on duty.

Adding and Subtracting Fractions with Unlike Denominators

To add or subtract fractions with different denominators, you must first convert them into equivalent fractions with a Least Common Denominator (LCD)—the least common multiple of the denominators.

Worked Example 2.1: Officer Duty Hours

During an operational shift, Constable Brown spent (3\frac{1}{2}) hours conducting highway traffic enforcement, (2\frac{3}{4}) hours attending court in Spanish Town, and (1\frac{1}{3}) hours preparing incident statements. What is the total number of hours Constable Brown logged for these duties?

Step-by-step Solution:

1. Express the mixed numbers as improper fractions or separate the whole numbers and fractional parts:
   • Whole numbers sum: (3 + 2 + 1 = 6) hours.
   • Fractional parts: (\frac{1}{2} + \frac{3}{4} + \frac{1}{3}).
2. Determine the LCD for denominators 2, 4, and 3. The least common multiple of 2, 4, and 3 is 12.
3. Convert each fraction to an equivalent fraction with denominator 12: [ \frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12}, \quad \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}, \quad \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} ]
4. Add the numerators: [ \frac{6 + 9 + 4}{12} = \frac{19}{12} = 1\frac{7}{12} \text{ hours} ]
5. Combine whole numbers and fractional sum: (6 + 1\frac{7}{12} = 7\frac{7}{12}) hours.
6. Total Time Logged = (7\frac{7}{12}) hours (or 7 hours and 35 minutes).

Multiplication and Division of Fractions

• Multiplication: Multiply numerator by numerator and denominator by denominator: (\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}).
• Division: Multiply the dividend by the reciprocal of the divisor (Keep-Change-Flip): (\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}).

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2. Converting Between Fractions, Decimals & Percentages

Percentages express amounts as parts per hundred (symbol %). Converting fluently between these three forms is vital for standard police assessments.

Conversion Procedures

Common JCF Reference Equivalents

• (\frac{1}{4} = 0.25 = 25%)
• (\frac{1}{3} \approx 0.333 = 33.33%)
• (\frac{1}{2} = 0.50 = 50%)
• (\frac{2}{3} \approx 0.667 = 66.67%)
• (\frac{3}{4} = 0.75 = 75%)
• (\frac{4}{5} = 0.80 = 80%)

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3. Percentage Calculations & Percentage Change

Finding a Percentage of a Total Amount

To find a percentage of a given quantity, convert the percentage to a decimal or fraction and multiply by the total amount: [ \text{Part} = \text{Percentage (as decimal)} \times \text{Whole} ]

Calculating Percentage Change (Increase or Decrease)

Percentage change measures relative growth or reduction between an original value and a new value. [ \text{Percentage Change} = \frac{|\text{New Value} - \text{Original Value}|}{\text{Original Value}} \times 100% ]

Worked Example 2.2: Divisional Crime Reduction

St. James Police Division recorded 160 reported incidents of commercial break-ins in Year 1. Following intensified patrol strategies, reported break-ins dropped to 116 incidents in Year 2. Calculate the percentage decrease in commercial break-ins.

Step-by-step Solution:

1. Identify the original value and new value: Original = 160, New = 116.
2. Calculate the absolute decrease (change): [ 160 - 116 = 44 \text{ incidents} ]
3. Divide the change by the original value: [ \frac{44}{160} = 0.275 ]
4. Multiply by 100% to convert to a percentage: [ 0.275 \times 100% = 27.5% ]
5. Result = 27.5% reduction in commercial break-ins.

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4. Equipment Discounts & Statutory Fine Adjustments

In police administration, procurement departments receive bulk discounts on gear, while court administration handles statutory discount or surcharge schemes on early fine payments.

Calculating Discounted Prices

• Discount Amount: (\text{Original Price} \times \text{Discount Rate})
• Final Price: (\text{Original Price} - \text{Discount Amount}) (or (\text{Original Price} \times (1 - \text{Discount Rate})))

Worked Example 2.3: Fleet Equipment Procurement

The Ministry of National Security places a bulk requisition order for 50 body-worn camera units at a retail price of JMD $84,000 per unit. The manufacturer offers a 15% bulk discount for public safety agencies. Calculate the total cost for the 50 camera units after applying the discount.

Step-by-step Solution:

1. Calculate the total retail price without discount: [ 50 \times 84,000 = \text{JMD } $4,200,000 ]
2. Calculate the remaining percentage factor: [ 100% - 15% = 85% = 0.85 ]
3. Multiply total retail price by 0.85: [ 4,200,000 \times 0.85 = \text{JMD } $3,570,000 ]
4. Total Discounted Procurement Cost = JMD $3,570,000.