Math Word Problems: Arithmetic, Percentages, Ratios, Time, Distance, and Speed in Police Context

Math Word Problems on the CAT

The NJ LEE Cognitive Ability Test (CAT) includes math word problems that test arithmetic, percentages, ratios, and time/distance/speed calculations in police contexts. No calculator is allowed — all work is mental or scratch-paper. The key is reading the scenario carefully, identifying what is being asked, and applying the right operation.

Core Arithmetic Operations

Most CAT math questions require only addition, subtraction, multiplication, and division. The trap is misreading the question, not the difficulty of the arithmetic. Always underline what the question asks before computing.

Worked example — patrol mileage budget: A patrol unit is allocated 1,200 miles per month. Officer Rivera has used 348 miles in the first week. Assuming even usage across four weeks, will she exceed the monthly allocation?

Step 1: Project total usage — 348 × 4 = 1,392 miles. Step 2: Compare to allocation — 1,392 > 1,200, so yes, she will exceed by 192 miles.

Percentages

Percentage questions on the CAT typically ask for percent of a total, percent increase/decrease, or reversing a percentage. Master three formulas:

• Percent of a total: (part ÷ whole) × 100
• Percent change: (new − old) ÷ old × 100
• Finding the whole from a part: part ÷ (percent ÷ 100)

Worked example — budget allocation: A department's annual budget is $4.8 million. Training accounts for 15% of the budget. How much is spent on training?

Step 1: Convert 15% to decimal — 0.15. Step 2: Multiply — $4,800,000 × 0.15 = $720,000.

Worked example — crime rate change: A township reported 240 burglaries in 2024 and 192 in 2025. What is the percent decrease?

Step 1: Find the decrease — 240 − 192 = 48. Step 2: Divide by the original — 48 ÷ 240 = 0.20. Step 3: Convert — 0.20 × 100 = 20% decrease.

Worked example — reversing a percentage: After a 10% raise, an officer's salary is $66,000. What was the original salary?

Step 1: The new salary is 110% of the original. Step 2: Divide — $66,000 ÷ 1.10 = $60,000.

Mental Math Shortcuts

• To find 10% of any number, move the decimal one place left: 10% of 4,800 = 480.
• To find 1%, move two places: 1% of 4,800 = 48.
• Build other percentages from these: 15% = 10% + 5% (half of 10%).
• For 12.5%, use 1/8: $3,600,000 ÷ 8 = $450,000.

Common CAT Math Traps

1. Unit mismatch — speed in mph, time in minutes. Always convert before computing.
2. Percent vs percentage points — "increased by 10%" is multiplicative; "increased by 10 points" is additive.
3. Asking for the wrong value — you compute distance but the question asks for time. Reread before answering.
4. Distractor numbers — the question includes a number you do not need. Identify relevant data before computing.

Ratios in Police Context

Ratio questions test whether you can scale proportions. A common CAT format presents a shift ratio and asks how many officers are assigned to a specific function.

Worked example — shift staffing: A precinct staffs patrol, traffic, and community policing in a 7:3:2 ratio. If 168 officers total are assigned, how many work patrol?

Step 1: Find total parts — 7 + 3 + 2 = 12. Step 2: Find one part — 168 ÷ 12 = 14 officers per part. Step 3: Multiply by patrol parts — 14 × 7 = 98 patrol officers.

Worked example — ratio comparison: At a checkpoint, the ratio of passenger vehicles to commercial vehicles inspected was 5:2. If 35 commercial vehicles were inspected, how many passenger vehicles were inspected?

Step 1: 2 parts = 35, so 1 part = 17.5. Step 2: Passenger parts = 5 → 17.5 × 5 = 87.5. Since you cannot inspect half a vehicle, recheck — but ratios can yield non-integer parts; round only if the question implies whole units. Here 35 ÷ 2 = 17.5 and 17.5 × 5 = 87.5, which signals the original total was 122.5 vehicles — a clue that the question likely expects a total divisible by 7 (5+2). If the question instead gave 70 commercial vehicles, passenger = (70 ÷ 2) × 5 = 175.

Time, Distance, and Speed

The fundamental formula is distance = speed × time (d = r × t). Rearrange it as needed:

• Speed = distance ÷ time
• Time = distance ÷ speed

On the CAT, watch the units. Speed is often in miles per hour while time is given in minutes — convert minutes to hours before computing.

Worked example — patrol speed and distance: An officer drives at 60 mph for 15 minutes. How far does the unit travel?

Step 1: Convert minutes to hours — 15 ÷ 60 = 0.25 hours. Step 2: Apply formula — 60 × 0.25 = 15 miles.

Worked example — response time: A call comes in 18 miles away. The officer drives at 45 mph. How long until arrival?

Step 1: Use time = distance ÷ speed — 18 ÷ 45 = 0.4 hours. Step 2: Convert to minutes — 0.4 × 60 = 24 minutes.

Worked example — two units converging: Two patrol units respond from opposite ends of a 30-mile stretch. Unit A travels at 50 mph, Unit B at 40 mph. How long until they meet?

Step 1: Combined closing speed — 50 + 40 = 90 mph. Step 2: Time = distance ÷ combined speed — 30 ÷ 90 = 1/3 hour. Step 3: Convert — 1/3 × 60 = 20 minutes.

Worked example — average speed: An officer drives 120 miles round trip: 60 mph outbound, 40 mph return. What is the average speed?

A common trap is averaging 60 and 40 to get 50 mph. Average speed = total distance ÷ total time. Step 1: Outbound time — 60 ÷ 60 = 1 hour. Step 2: Return time — 60 ÷ 40 = 1.5 hours. Step 3: Total — 120 miles ÷ 2.5 hours = 48 mph.

Quick Unit Conversions to Memorize