2.6 Quantitative Word Problems & Speed-Distance-Time Calculations

Quantitative word problems test your ability to apply mathematical concepts to real-world scenarios. They require not just calculation skills, but reading comprehension, logical translation, and structured problem-solving methodologies. A significant portion of these questions in the Nigeria Police Exam involves Speed, Distance, and Time calculations. This section will outline a universal framework for attacking word problems and delve specifically into kinematics equations, equipping you with the strategies needed to interpret and solve complex narrative questions accurately.

Speed, Distance, and Time Formula Reference

• Distance Formula: $\text{Distance} = \text{Speed} \times \text{Time}$ ($D = S \times T$)
• Speed Formula: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ ($S = \frac{D}{T}$)
• Time Formula: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$ ($T = \frac{D}{S}$)
• Average Speed: $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$

A Universal Framework for Word Problems

Many candidates struggle with word problems because they jump straight to calculating without fully understanding the scenario. To maximize success, adopt a structured four-step methodology.

Step 1: Read and Comprehend. Read the entire problem carefully without writing anything down. What is the overall situation? What specific value is the question asking you to find? Identifying the final goal prevents you from calculating irrelevant information.

Step 2: Translate and Organize. Go through the problem sentence by sentence and translate English phrases into mathematical expressions. Define variables clearly (e.g., "Let $x$ be the number of cars"). If the problem involves categories or changes over time, organize the data into a table. If it involves geometry or physical movement, draw a quick diagram. Visualizing the problem often reveals the correct equation.

Step 3: Formulate and Solve. Use the translated expressions to set up an equation or a system of equations. Apply the algebraic rules discussed in previous sections to isolate the variable and solve the equation meticulously.

Step 4: Verify and Contextualize. Do not simply select the answer that matches your calculation. Does your answer make logical sense in the context of the real-world scenario? If a problem asks for the number of people, an answer of 4.5 is logically impossible, indicating an error. Finally, re-read the specific question to ensure you are answering exactly what was asked, not an intermediate step.

Speed, Distance, and Time Calculations

Problems involving motion are incredibly common. They all revolve around a single foundational relationship: Distance equals Speed multiplied by Time. The core formula is $D = S \times T$. From this, using simple algebra, we can derive the other two formulas: Speed = Distance / Time ($S = D / T$) and Time = Distance / Speed ($T = D / S$).

The most critical aspect of these problems is unit consistency. If speed is given in kilometers per hour (km/h) and time is given in minutes, you must convert the time into hours before calculating. To convert minutes to hours, divide by 60. For example, 45 minutes is $45 / 60$ hours, which simplifies to $3/4$ or 0.75 hours.

Let us apply the framework to a standard problem: "A police patrol car travels from Station A to Station B, a distance of 120 kilometers, at an average speed of 80 km/h. How long does the journey take in minutes?"

• Step 1: The goal is to find time, specifically in minutes.
• Step 2: We know Distance ($D$) = 120 km. We know Speed ($S$) = 80 km/h. Units are consistent so far (km and km/h).
• Step 3: The formula for Time is $T = D / S$. Substitute the values: $T = 120 / 80$. This simplifies to $12 / 8$, and further to $3 / 2$ or 1.5 hours.
• Step 4: The question specifically asks for the time in minutes. We must convert our answer. Since 1 hour is 60 minutes, multiply by 60: $1.5 \times 60 = 90$ minutes. The answer makes logical sense for a 120km journey at that speed.

Complex Motion Scenarios

Exam questions often feature more complex scenarios, such as two objects moving towards each other, moving away from each other, or calculating average speed for a multi-part journey.

Relative Speed: When two objects are moving towards each other, their relative speed (the speed at which the distance between them is closing) is the sum of their individual speeds. If Car A travels at 60 km/h and Car B travels towards it at 40 km/h, they are closing the distance at $60 + 40 = 100$ km/h. Conversely, if they are moving in the same direction, the relative speed is the difference between their speeds.

Average Speed: A common trap is assuming average speed is simply the arithmetic mean of different speeds. This is incorrect. The true formula for average speed over an entire journey is: Average Speed = Total Distance / Total Time. For example, if a car travels 100 km at 50 km/h, and then another 100 km at 100 km/h, the average speed is not 75 km/h. First, find total distance: $100 + 100 = 200$ km. Next, find the time for each leg. Leg 1 time = $100 / 50 = 2$ hours. Leg 2 time = $100 / 100 = 1$ hour. Total time = $2 + 1 = 3$ hours. Finally, calculate Average Speed = Total Distance / Total Time = $200 / 3$, which is approximately 66.67 km/h.

Mastering word problems requires patience and strict adherence to structured translation. By rigorously applying the $D=S\times T$ formula, ensuring unit consistency, and correctly handling complex motion scenarios like relative and average speed, you will be capable of dissecting and solving the most challenging narrative questions on the exam.