2.3 Multi-Step Police Arithmetic

Multi-Step Police Arithmetic

The most challenging math questions on the NPOST do not merely ask you to add or multiply two numbers; they present a complex scenario requiring you to perform a sequence of calculations to arrive at the final answer. These are known as multi-step arithmetic problems.

Success in this area is less about knowing advanced math and more about reading comprehension, organization, and patience. You must break the problem down into distinct parts, solve each part sequentially, and use your intermediate answers to find the final solution. Writing down your work clearly is essential.

This section covers the most common categories of multi-step problems seen on the exam: travel/distance, vehicle fuel, and payroll calculations.

Travel Time, Distance, and Rate

Police officers constantly deal with travel times, whether rushing to an emergency, calculating a suspect's flight, or estimating accident reconstruction data. The foundational formula for all these problems is: Distance = Rate × Time (D = RT)

• Distance is how far you traveled (e.g., miles).
• Rate is the speed (e.g., miles per hour).
• Time is how long the travel took (e.g., hours).

You can rearrange this formula to solve for any missing variable:

• Time = Distance ÷ Rate
• Rate = Distance ÷ Time

Scenario 1: Intercepting a Suspect A suspect vehicle is fleeing a bank robbery heading North on the highway at a constant rate of 80 miles per hour. A highway patrol officer is dispatched from a station located 120 miles North of the robbery on the same highway, heading South to intercept. The officer travels at 100 miles per hour. If they leave at the exact same time, how long will it take for the two vehicles to meet?

Step-by-Step Solution:

1. Identify the goal: Find the time until they meet.
2. Determine relative speed: Because the vehicles are driving toward each other, you must add their speeds together to find the "closing rate." Closing Rate = 80 mph (suspect) + 100 mph (officer) = 180 mph.
3. Use the formula: Time = Distance ÷ Rate.
4. Calculate: The total distance between them is 120 miles. The closing rate is 180 mph. Time = 120 ÷ 180.
5. Simplify the fraction: 120/180 reduces to 12/18, which further reduces to 2/3 of an hour.
6. Convert to minutes: 2/3 of a 60-minute hour is (2 x 60) ÷ 3 = 120 ÷ 3 = 40 minutes. Answer: They will meet in 40 minutes.

Vehicle Fuel and Mileage Calculations

Fleet management and fuel economy are standard logistical issues in law enforcement. These problems typically require you to calculate fuel consumption over a specific distance and then determine the remaining range or cost.

Scenario 2: Fuel Consumption on Patrol Officer Jackson begins his 12-hour shift with a full 18-gallon tank of gas in his cruiser. His cruiser gets an average of 15 miles to the gallon. During his shift, he drives a total of 135 miles. At the end of his shift, gas costs $3.50 per gallon. How much will it cost to completely refill the cruiser's gas tank?

Step-by-Step Solution:

1. Identify the goal: Find the cost to replace the fuel used.
2. Calculate fuel used: To find out how many gallons were used, divide the total miles driven by the miles per gallon (MPG). Gallons Used = 135 miles ÷ 15 miles/gallon. 135 ÷ 15 = 9 gallons.
3. Verify tank status (optional but good practice): The tank holds 18 gallons, and he used 9, so he has 9 gallons left. He needs to replace the 9 gallons he used.
4. Calculate cost: Multiply the gallons needed by the price per gallon. Cost = 9 gallons x $3.50/gallon. 9 x 3 = 27 9 x 0.50 = 4.50 27 + 4.50 = 31.50. Answer: It will cost $31.50 to refill the tank.

Complex Payroll and Overtime

Law enforcement shifts are rarely standard 9-to-5 affairs. Officers frequently work overtime, night differentials, and holiday pay. You will often need to calculate regular pay separately from overtime pay and combine them.

Standard overtime is typically "time-and-a-half," meaning the overtime rate is 1.5 times the regular hourly rate.

Scenario 3: Calculating Weekly Payroll Officer Patel has a regular hourly wage of $24.00. Her standard workweek is 40 hours. Any hours worked over 40 are paid at time-and-a-half (1.5x regular pay). This week, she worked four 10-hour regular shifts, plus a special 6-hour weekend detail for a local parade. What is her total gross pay for the week?

Step-by-Step Solution:

1. Identify the goal: Calculate total pay (Regular Pay + Overtime Pay).
2. Calculate regular hours and pay: Regular Hours = four 10-hour shifts = 40 hours. Regular Pay = 40 hours x $24.00/hour. 40 x 24 = 960. (Regular pay = $960).
3. Calculate overtime rate: Overtime Rate = Regular Rate x 1.5. $24.00 x 1.5 = $36.00/hour.
4. Calculate overtime pay: Overtime Hours = 6 hours (the weekend detail). Overtime Pay = 6 hours x $36.00/hour. 6 x 30 = 180. 6 x 6 = 36. 180 + 36 = 216. (Overtime pay = $216).
5. Calculate total pay: Add regular pay and overtime pay. $960 + $216 = $1,176. Answer: Her total gross pay for the week is $1,176.

Multi-Step Budgeting

Budget problems require you to track a starting total, subtract various expenses, and often calculate how much of an item can be bought with the remainder.

Scenario 4: Purchasing Equipment A precinct has a discretionary fund of $5,000 for the quarter. The captain spends $1,200 on new office chairs and $850 on a coffee machine for the breakroom. The captain wants to use the entire remaining balance to purchase tactical flashlights for the officers. If each flashlight costs $45, how many flashlights can the captain buy, and how much money will be left over?

Step-by-Step Solution:

1. Calculate total expenses so far: $1,200 + $850 = $2,050.
2. Calculate the remaining budget: Starting budget ($5,000) - Expenses ($2,050). 5000 - 2050 = 2950. The remaining budget is $2,950.
3. Calculate how many items can be bought: Divide the remaining budget by the cost of one item. 2950 ÷ 45.
   • 45 goes into 295 six times (45 x 6 = 270).
   • 295 - 270 = 25. Bring down the 0 to make 250.
   • 45 goes into 250 five times (45 x 5 = 225).
   • 250 - 225 = 25 remainder.
   • The result is 65 with a remainder of 25.
4. Interpret the result: The captain can buy 65 complete flashlights. The remainder of 25 represents the leftover money ($25). Answer: The captain can buy 65 flashlights, with $25 left over.

By practicing these multi-step breakdowns, you will train your brain to calmly sequence the necessary math on exam day.