GMAT Rates and Work

Start with units and the requested quantity

GMAT rate problems describe how quickly something changes or gets completed. The basic relationship is rate = amount / time, so amount = rate x time and time = amount / rate. For travel, distance = speed x time. These equations are simple; the challenge is identifying which quantities belong together and converting units before using them.

Write down what the question asks before calculating. If it asks how long a trip takes, the target is time. If it asks for average speed over the full trip, use total distance divided by total time. If it asks how much work two people complete together, add their rates over the shared interval. A correct formula aimed at the wrong quantity still gives a wrong answer.

Check the units explicitly. A speed of 48 miles per hour multiplied by 2 hours gives 96 miles because hours cancel. A rate of 3 pages per minute over 20 minutes gives 60 pages. If a problem mixes minutes and hours, convert before combining. Three hours and 20 minutes is not 3.20 hours; it is 10/3 hours or 3.333... hours.

Travel: distance, speed and time

For a constant speed, distance equals speed times time. If a car travels 150 miles at 50 miles per hour, travel time is 150 / 50 = 3 hours. If the trip has multiple legs, compute each leg's distance or time, then add totals. Do not average leg speeds unless the time or distance conditions justify that operation.

Suppose a driver covers 60 miles at 30 mph and another 60 miles at 60 mph. The times are 2 hours and 1 hour, so average speed is 120 miles / 3 hours = 40 mph. The arithmetic mean of 30 and 60 is 45 mph, which is incorrect for equal distances because the driver spends more time at the slower speed.

If instead the driver travels one hour at 30 mph and one hour at 60 mph, the distances are 30 and 60 miles. Total distance is 90 miles over 2 hours, so average speed is 45 mph. Equal time makes the average of the speeds appropriate. The wording determines the weights.

Combined work rates

If one worker completes a job in a hours, that worker's rate is 1/a job per hour. If another completes it in b hours, the second worker's rate is 1/b job per hour. Working together, their rate is 1/a + 1/b. Divide one whole job by the combined rate to find the time.

For example, a pump empties a tank in 6 hours and another in 3 hours. Their rates are 1/6 and 1/3 tank per hour, totaling 1/2 tank per hour. Together they empty the tank in 2 hours. Adding 6 and 3 to get 9 hours confuses completion times with work completed per hour.

When rates are given in units rather than jobs, add the rates directly if the units match. A machine making 12 parts per hour and another making 8 parts per hour produce 20 parts per hour together. If each machine works a different number of hours, calculate each contribution separately and add the parts.

A leak or opposing flow reduces net rate. If a pipe fills a tank at 1/4 tank per hour while a leak drains 1/10 tank per hour, net rate is 1/4 - 1/10 = 3/20 tank per hour. The time to fill one tank is 20/3 hours, or 6 hours 40 minutes. Use addition for work in the same direction and subtraction for opposing work.

Original worked problems

Two people working together

Ava paints a room in 5 hours. Ben paints the same room in 7.5 hours. How long do they need working together at constant rates? Ava's rate is 1/5 room per hour. Ben's is 1/7.5 = 2/15. Their combined rate is 3/15 + 2/15 = 1/3 room per hour, so they finish in 3 hours.

A wrong answer of 12.5 hours adds the two individual times. A wrong answer of 6.25 hours averages them, even though the question asks about simultaneous work. A useful check is that working together should take less than either individual time, assuming neither slows the other.

Catch a unit conversion

A cyclist rides 18 kilometers in 45 minutes. What is the average speed in kilometers per hour? Forty-five minutes is 0.75 hour. Speed is 18 / 0.75 = 24 kilometers per hour. Dividing 18 by 45 gives 0.4 kilometers per minute, which is equivalent but not the requested unit.

The incorrect answer 18/45 = 0.4 km/h comes from labeling minutes as hours. Another tempting answer is 40 km/h, from treating 45 minutes as 0.45 hour. Converting the time first prevents both mistakes.

A delayed start

Machine A produces 10 components per hour. Machine B starts two hours later and produces 15 per hour. How many hours after B starts will their combined output reach 80 components? A has already made 20 components during the delay, leaving 60. Together they produce 25 per hour, so they need 60/25 = 2.4 hours after B starts.

The key is to separate the initial period from the period when both machines operate. Multiplying 25 by the full time ignores B's delayed start. Another error subtracts two hours from the final result even though the question measures from B's start.

Relative motion and meeting problems

When two objects move toward each other, their closing rate is the sum of their speeds. If they move in the same direction and one catches the other, the closing rate is the difference. For example, two cyclists 42 miles apart ride toward each other at 12 mph and 9 mph. They meet in 42/(12+9) = 2 hours.

If one cyclist follows another in the same direction at 12 mph while the leader travels at 9 mph, the gap closes at 3 mph. A 42-mile head start takes 14 hours to close. A diagram with arrows can clarify which speeds add and which subtract.

Relative speed works only when the time interval is shared. If one traveler starts later, calculate the head start distance first, then use the closing rate. Keep the reference point clear: a question may ask when they meet, when one passes the other or how far one has traveled by that time.

Set up a table for repeated quantities

A distance-rate-time table can keep multi-leg trips organized. Use rows for each leg and columns for distance, speed and time. Fill in the two known quantities, then use distance = rate x time. For work problems, make columns for worker, individual rate, operating time and work completed.

For a two-stage trip, suppose a train goes 90 miles at 45 mph and then 120 miles at 60 mph. The times are 2 and 2 hours, so the average speed is 210/4 = 52.5 mph. The simple average of 45 and 60 would be 52.5 here only because the times are equal. Equal distances would produce a different average.

Tables also help with partial jobs. If a worker completes 3/8 of a job in 2 hours, the rate is 3/16 job per hour. The remaining work is 5/8. At the same rate, the remaining time is (5/8)/(3/16) = 10/3 hours. Do not assume the remaining task has the same size as the completed part.

Common rate traps

• Adding completion times instead of work rates.
• Using the arithmetic mean for unequal time or distance intervals.
• Combining rates expressed in incompatible units.
• Ignoring a delayed start or a break in the work period.
• Adding an opposing leak instead of subtracting it.
• Answering in the wrong unit or measuring time from the wrong event.
• Assuming a worker's rate stays constant when the prompt changes it.

The GMAT Quant section has no calculator, so simplify before multiplying. For combined work, 1/6 + 1/3 can be recognized as 1/2. For travel, convert 45 minutes to 3/4 hour and divide by a fraction by multiplying by its reciprocal. Estimation can rule out answers greater than an individual completion time when two workers cooperate.

How to practice

Practice one relationship at a time, then mix travel and work questions. Before calculating, name the rate, amount and time, and write units beside each. After solving, check whether the direction makes sense: two workers together should finish sooner, an opposing leak should lengthen filling time, and equal time intervals make speed weights equal.

Review the setup, not just arithmetic. If you used the wrong average, identify whether the problem gave equal times, equal distances or neither. If you missed a delayed start, draw a timeline. If a unit error caused the miss, convert before inserting values. A new problem with different numbers verifies that you learned the relationship.

Practice without a calculator for Quant, then use the Data Insights calculator only when useful. The underlying math is the same, but DI may put rates in a chart or table with units that must be interpreted. In either section, precise setup usually saves more time than pressing keys faster.