5.3 Rates, Joint Work & Speed Problems

Rates, Work, and Speed on the GAT

Rates, joint work, and speed problems are among the most practical and frequent questions in the Quantitative Reasoning section of the Saudi Arabia Qiyas General Aptitude Test (GAT). These problems evaluate your understanding of rates, which express how one quantity changes with respect to another, typically time. Succeeding in these questions requires mastery of speed formulas, average velocity calculations, relative motion, joint work equations, and multi-variable worker productivity relationships.

Speed, Distance, and Time Basics

The fundamental relationship governing speed, distance, and time is: Distance = Speed * Time From this, you can derive Speed = Distance / Time and Time = Distance / Speed. In GAT questions, you must be careful with unit consistency. If speed is given in kilometers per hour (km/h) and time is in minutes, you must convert minutes to hours by dividing by 60. For example, 45 minutes is 45/60 = 0.75 hours.

Average Speed and the Harmonic Mean Rule

One of the most common GAT traps involves calculating average speed for a round trip or a journey with different legs. Students often compute the simple arithmetic mean of the speeds, which is mathematically incorrect because more time is spent traveling at the slower speed. When the distance for each leg of the journey is equal (e.g., a round trip going from A to B and returning), the average speed is the harmonic mean of the speeds: Average Speed = (2 * v1 * v2) / (v1 + v2) Where v1 and v2 are the speeds for the two legs. Example: A car travels from Riyadh to Dammam at 120 km/h and returns along the same route at 80 km/h. What is the average speed for the round trip? Applying the formula: Average Speed = (2 * 120 * 80) / (120 + 80) = 19,200 / 200 = 96 km/h. Note that the simple average is (120 + 80) / 2 = 100 km/h, which is a trap answer choice.

Relative Speed and Chase Scenarios

Relative speed is the speed of one object with respect to another.

1. Objects moving in opposite directions (towards or away from each other): Add the speeds. Relative Speed = v1 + v2 Example: Two cars start from cities 300 km apart and travel towards each other at speeds of 60 km/h and 90 km/h. In how many hours will they meet? Combined speed = 60 + 90 = 150 km/h. Time = Distance / Speed = 300 / 150 = 2 hours.
2. Objects moving in the same direction (chasing): Subtract the speeds. Relative Speed = |v1 - v2| Example: A runner starts at 6 m/s, and a cyclist starts chasing from 120 meters behind at 10 m/s. How long will it take the cyclist to catch the runner? Relative speed = 10 - 6 = 4 m/s. Time = Distance / Speed = 120 / 4 = 30 seconds.

Joint Work Rates and Pipes Problems

Joint work problems calculate the time required for multiple workers or machines to complete a task together. Since speeds of work are additive, the rate of work (work per unit time) is: Total Rate = Rate1 + Rate2 + ... Since Rate = 1/Time to complete 1 task, the formula for two workers is: 1/Total Time = 1/T1 + 1/T2 Which simplifies to: Total Time = (T1 * T2) / (T1 + T2) Example: Ahmed can paint a room in 4 hours, and Khaled can paint the same room in 6 hours. If they work together, how long will it take them? Applying the formula: Total Time = (4 * 6) / (4 + 6) = 24 / 10 = 2.4 hours. To convert 2.4 hours into minutes: 0.4 hours * 60 minutes/hour = 24 minutes. So the total time is 2 hours and 24 minutes.

Pipes and tanks problems are variation of joint work. If an inlet pipe fills a tank in Tf hours and an outlet drain empties it in Td hours, and both are open, the net rate of filling is: Net Rate = 1/Tf - 1/Td Total time to fill the empty tank is the reciprocal of the net rate. Example: An inlet pipe fills a pool in 6 hours, while a drain empties it in 12 hours. If both are open, how long to fill the pool? Net Rate = 1/6 - 1/12 = 2/12 - 1/12 = 1/12. Time = 12 hours.

Multi-Variable Worker-Time-Work Formula

For complex problems where the number of workers, time, and amount of work all vary, use the Worker-Time-Work formula (Double Proportion): (Workers1 * Time1) / Work1 = (Workers2 * Time2) / Work2 This equation maintains that productivity per worker remains constant. Example: If 6 workers can construct 12 wooden tables in 4 days, how many days will it take 8 workers to construct 24 wooden tables? Setting up the formula: (6 * 4) / 12 = (8 * x) / 24 24 / 12 = 8x / 24 2 = 8x / 24 8x = 48 x = 6 days.

Common Qiyas Rates and Work Traps

• The Speed Averaging Trap: Always use the harmonic mean formula for equal distance journeys; never calculate a simple arithmetic average.
• Decimal Time Trap: Converting decimal hours to minutes incorrectly (e.g., interpreting 1.5 hours as 1 hour and 50 minutes instead of 1 hour and 30 minutes). Always multiply the decimal portion by 60.
• Work Rate Additivity: Never add the times of individual workers directly (e.g., Ahmed takes 4 hours, Khaled takes 6 hours, so together they take 10 hours is false; they must take less time than the fastest worker). Always add their rates.