2.3 Ratio Simplification, Direct & Inverse Proportion, Unit Rates

2.3 Ratio Simplification, Direct & Inverse Proportion, Unit Rates

Police operations depend heavily on proportional reasoning. Whether analyzing officer-to-population ratios in urban divisions, allocating special duty stipends, projecting fuel requirements for highway patrols, or calculating crime scene processing times based on forensic team size, ratios, proportions, and unit rates provide the mathematical backbone for strategic planning in the Jamaica Constabulary Force.

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1. Ratio Simplification & Part-to-Part / Part-to-Whole Relationships

A ratio compares two or more quantities of the same kind, expressed using a colon ((:)), as a fraction, or with the word "to". Ratios must always be expressed in lowest terms by dividing each term by their Greatest Common Factor (GCF).

Types of Ratios

• Part-to-Part Ratio: Compares one subgroup to another subgroup (e.g., Corporals to Constables).
• Part-to-Whole Ratio: Compares one subgroup to the entire total population (e.g., Corporals to Total Station Officers).

Worked Example 3.1: Personnel Structure Simplification

A police divisional headquarters has 45 Constables, 30 Corporals, and 15 Sergeants assigned to tactical response units. Express the staffing structure as a simplified 3-part ratio of Constables to Corporals to Sergeants.

Step-by-step Solution:

1. Write the initial ratio: (45 : 30 : 15).
2. Identify the GCF of 45, 30, and 15, which is 15.
3. Divide each term by 15: [ (45 \div 15) : (30 \div 15) : (15 \div 15) = 3 : 2 : 1 ]
4. Simplified Staffing Ratio = 3 : 2 : 1. (For every 1 Sergeant, there are 2 Corporals and 3 Constables).

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2. Sharing Quantities in Given Ratios

Allocating financial stipends, overtime funds, or personnel across districts requires partitioning a whole quantity into proportional parts.

Step-by-Step Ratio Partitioning Method

1. Find Total Ratio Parts: Add all terms in the ratio together.
2. Find Value of One Part: Divide the total quantity by the total number of ratio parts.
3. Calculate Individual Allocations: Multiply the value of one part by each respective term in the ratio.
4. Verify: Ensure the sum of individual allocations equals the original total quantity.

Worked Example 3.2: Special Commendation Reward Allocation

A commendation reward grant of JMD $180,000 is to be shared among a Constable, a Corporal, and a Sergeant who jointly solved a major fraud case, in the ratio (5 : 3 : 1) based on operational field hours. Calculate each officer's reward share.

Step-by-step Solution:

1. Calculate total parts: (5 + 3 + 1 = 9 \text{ parts}).
2. Determine value per part: [ \text{Value per part} = \frac{\text{JMD } $180,000}{9} = \text{JMD } $20,000 ]
3. Calculate individual shares:
   • Constable Share (5 parts): (5 \times $20,000 = \text{JMD } $100,000)
   • Corporal Share (3 parts): (3 \times $20,000 = \text{JMD } $60,000)
   • Sergeant Share (1 part): (1 \times $20,000 = \text{JMD } $20,000)
4. Verification: ($100,000 + $60,000 + $20,000 = \text{JMD } $180,000).

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3. Direct Proportion vs. Inverse Proportion

Proportion is an equation stating that two ratios or rates are equal.

Proportion Comparison:
- Direct Proportion:  y1 / x1 = y2 / x2  ==>  As X increases, Y increases proportionally.
- Inverse Proportion: x1 * y1 = x2 * y2  ==>  As X increases, Y decreases proportionally.

Direct Proportion

In direct proportion, as one quantity increases, the other increases at a constant rate. The ratio (\frac{y}{x} = k) remains constant.

Worked Example 3.3: Tactical Gear Procurement

The procurement officer notes that 12 tactical flashlights cost JMD $45,600. How much will 28 identical flashlights cost for a new patrol team?

Step-by-step Solution:

1. Set up the direct proportion equation: [ \frac{\text{Cost}_1}{\text{Quantity}_1} = \frac{\text{Cost}_2}{\text{Quantity}_2} \implies \frac{45,600}{12} = \frac{C_2}{28} ]
2. Find unit cost: (45,600 \div 12 = \text{JMD } $3,800) per flashlight.
3. Multiply unit cost by new quantity: [ C_2 = 28 \times 3,800 = \text{JMD } $106,400 ]
4. Total Cost for 28 Flashlights = JMD $106,400.

Inverse Proportion

In inverse proportion, as one quantity increases, the other decreases proportionally such that their product remains constant: (x_1 \cdot y_1 = x_2 \cdot y_2 = k). This frequently applies to work-rate and time scenarios.

Worked Example 3.4: Evidence Processing Duration

A specialized team of 6 forensic officers can process a complex crime scene in 10 hours. If the division dispatches 15 forensic officers working at the exact same rate, how many hours will it take to complete the scene processing?

Step-by-step Solution:

1. Recognize inverse proportion: More officers result in fewer required processing hours.
2. Calculate total officer-hours required (constant work product (k)): [ k = 6 \text{ officers} \times 10 \text{ hours} = 60 \text{ officer-hours} ]
3. Set up the equation for 15 officers: [ 15 \text{ officers} \times T_2 = 60 \text{ officer-hours} ]
4. Solve for (T_2): [ T_2 = \frac{60}{15} = 4 \text{ hours} ]
5. Time Required for 15 Officers = 4 hours.

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4. Unit Rates & Operational Performance Metrics

A unit rate compares a quantity to one single unit of another measure (e.g., kilometers per hour, fuel consumption per kilometer, cost per unit).

Worked Example 3.5: Speed Calculation

A highway patrol interceptor travels 165 km along the PJ Patterson Highway in 1.5 hours to reach an emergency checkpoint. Calculate the cruiser's average speed in kilometers per hour (km/h).

Step-by-step Solution:

1. Apply speed formula: (\text{Speed} = \frac{165 \text{ km}}{1.5 \text{ hours}}).
2. Multiply numerator and denominator by 10 to clear decimal: (\frac{1650}{15}).
3. Divide: (1650 \div 15 = 110).
4. Average Speed = 110 km/h.