# GRE Rate and Work Word Problems: Worked Examples | topin

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## Choose the amount, rate and time in the same units

[ETS](https://www.ets.org/gre/test-takers/general-test/prepare/content/quantitative-reasoning.html) includes ratio, rate and algebraic word problems within GRE Quant. The essential relationship is distance = speed × time, or more generally amount completed = rate × time. A rate of 60 kilometres per hour cannot be multiplied by 20 minutes without a conversion. Either convert 20 minutes to one third of an hour or convert the speed to one kilometre per minute.

Our first step is to draw a three-column table for amount, rate and time. Fill only stated values, then express the unknown consistently. This avoids deciding too early that every motion problem needs a memorised relative-speed shortcut. The story determines whether objects move towards each other, in the same direction, at different starting times or over separate portions of a journey.

Rate questions are often [select-one multiple choice](/gre/quant/multiple-choice), and the choices let you check the model before you finish the algebra. Two workers together must beat the faster one alone, and cannot be more than twice as fast as the faster one. If the faster worker takes six hours, the joint time lies between three and six hours, which usually rules out two or three choices at once. Under a constant rate, doubling time doubles output; if the rate changes partway, split the timeline and add the outputs of each interval. The [GRE Quant section guide](/gre/quant) sets out all five question formats.

## Match the rate model to the relationship

Our method table separates adding rates from averaging rates. Two workers completing the same job at constant independent rates can have their job rates added. Two journeys need their distances and times combined to obtain average speed. The arithmetic average of two speeds is valid for equal time at each speed, but generally not for equal distance.

Check the rate units before combining terms. “One job in six hours” is a time, whereas “one sixth of a job per hour” is a rate. Adding six and ten hours does not calculate joint completion time. Convert each worker’s completion time into a job fraction per hour, add those fractions, then take the reciprocal to find the time for one job.

__Choose the model from the stated relationship__
| Situation                         | Useful equation                  | Condition                                 |
| --------------------------------- | -------------------------------- | ----------------------------------------- |
| Constant travel speed             | Distance = speed × time          | Match distance and time units             |
| Combined work                     | Joint rate = sum of job rates    | Rates remain constant and additive        |
| Average speed                     | Total distance / total time      | Include all stated travel or waiting time |
| Moving towards each other         | Closing rate = speed 1 + speed 2 | Same interval and compatible directions   |
| Catching up in the same direction | Closing rate = faster − slower   | Account for any initial head start        |

## Original problem: equal distances do not mean equal times

Original question: a student travels 60 kilometres at 30 kilometres per hour and returns the same distance at 60 kilometres per hour. Find the average speed for the round trip, excluding any stop. The outward time is 60/30 = two hours and the return time is 60/60 = one hour. Total distance is 120 kilometres and total time is three hours.

The average speed is 120/3 = 40 kilometres per hour, not 45\. The slower leg occupies twice as much time, so it has greater influence on the time-weighted average. For equal distances at positive speeds a and b, the expression simplifies to 2ab/(a + b). This shortcut is derived from total distance over total time; retain that reasoning when the distances differ.

If the student instead travels for one hour at each speed, the total distance is 30 + 60 = 90 kilometres in two hours, giving 45 kilometres per hour. The change from equal distance to equal time is the entire reason the answer changes. This weight choice connects with [weighted averages and mixtures](/articles/gre-weighted-averages-mixtures), where the denominator behind the average decides the weight.

## Original problem: add work rates, then take the time

Original question: printer A completes a batch in six hours and printer B completes the same batch in ten hours. They work together at constant additive rates. How long does one batch take? A’s rate is 1/6 of a batch per hour and B’s is 1/10\. The joint rate is 1/6 + 1/10 = 4/15 of a batch per hour.

The time is 1/(4/15) = 15/4 hours, or three hours 45 minutes. It should be shorter than six hours, since both positive rates contribute. An answer of eight hours, obtained by averaging the solo times, fails that check. The familiar product-over-sum expression, ab/(a + b), is valid for these two constant additive job rates, not for every story containing two times.

If A completes only half a batch before B joins, A works alone for three hours. The remaining half takes (1/2)/(4/15) = 15/8 hours together. Total elapsed time is 39/8 hours, or four hours 52 minutes 30 seconds. Split the timeline at the joining point. Joint completion time for a full batch is not the amount of time they work together in this altered problem.

## Original problem: find a missing worker’s rate

Original question: A and B together complete a task in four hours, and A alone completes it in 12 hours. How long would B take alone, assuming constant additive rates? Joint rate is 1/4 job per hour; A’s rate is 1/12\. Subtracting gives B’s rate = 1/4 − 1/12 = 1/6, so B alone takes six hours.

This is subtraction of rates, not subtraction of completion times. Twelve minus four gives eight hours, which is unrelated to B’s required solo time. A forward check adds 1/12 and 1/6 to obtain 1/4\. In the original model, each worker’s positive rate must be smaller than the joint rate. A result implying negative output should trigger a check of the conditions and algebra.

Another original example has a pipe filling a tank in eight hours and a drain emptying a full tank in 24 hours. When both operate at constant rates, the net filling rate is 1/8 − 1/24 = 1/12 tank per hour, so an empty tank fills in 12 hours. The drain’s sign comes from its effect on stored water, not from a special formula.

## Original problem: delayed starts and meeting points

Original question: cyclist A starts along a straight route at 12 kilometres per hour. Cyclist B starts from the same point 30 minutes later at 18 kilometres per hour and follows the same direction. How long after B starts does B catch A? A’s head start is 12 × 0.5 = six kilometres. Their closing speed is 18 − 12 = six kilometres per hour.

B therefore needs one hour after starting. A has then travelled for 1.5 hours and B for one hour: both distances are 18 kilometres. This pair of distances is the clearest check. Answering 1.5 hours would be correct only if the question asked for elapsed time since A started. Keep the requested time origin beside the final number.

For two walkers starting simultaneously 27 kilometres apart and walking towards each other at four and five kilometres per hour, closing speed is nine and meeting time is three hours. The walkers cover 12 and 15 kilometres, whose sum is 27\. Opposite-direction and same-direction shortcuts are consequences of the distance equation; drawing the initial positions keeps their signs clear.

## Check the timeline, the output and the final unit

Our recommended checks are physical and algebraic. A joint positive work rate should beat either worker alone. A catch-up time should close the stated gap. Average speed should agree with total distance divided by total elapsed time. If waiting time is included in the request, include it in the denominator even though the distance accumulated during the wait is zero.

Practise equal-distance travel, equal-time travel, combined work, a missing worker and a delayed start as separate models, then mix them. Use [topin’s free GRE mock](/gre/practice-test), marked on the official scale, for a later transfer check. Record whether a miss came from units, the time origin, averaging the wrong quantities or failing to split the timeline. These are our practice recommendations.

For [Numeric Entry](/gre/quant/numeric-entry), convert the final unit only after identifying what the question requests. Three quarters of an hour is 45 minutes, not 75 minutes. Keep fractions during setup and follow ETS’s rounding instructions at the end. The calculator helps with tedious division, but cannot decide whether you need a time since departure, a working interval or a total duration.

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## FAQs

What is the basic equation for GRE rate problems?

Use amount = rate × time with matching units. For travel, the amount is distance. For work, it is the fraction or number of jobs completed. Changing rates require separate time intervals.

Why can I not average two completion times?

Completion times are reciprocals of job rates. Under a constant additive model, convert each time to a fraction of a job per hour, add rates and take the reciprocal for the combined completion time.

When is average speed the arithmetic average of two speeds?

When the time spent at each speed is equal. Otherwise use total distance divided by total time. Equal distances generally imply unequal times and therefore different weights.

Should relative speeds be added or subtracted?

Add for objects moving towards each other along the route; subtract for a faster object catching a slower one moving in the same direction. Account for head starts and delayed departures before dividing by the closing rate.

What is the best check for a delayed-start answer?

Calculate each object’s distance using its own elapsed travel time. At a catch-up point the distances from the common origin agree. Also confirm whether the answer is measured from the earlier or later departure.

## Sources (checked 5 October 2026)

* [ETS: Quantitative Reasoning overview and calculator guidance (checked 5 October 2026)](https://www.ets.org/gre/test-takers/general-test/prepare/content/quantitative-reasoning.html)
* [ETS: GRE Math Review (checked 5 October 2026)](https://www.ets.org/content/dam/ets-org/pdfs/gre/gre-math-review.pdf)
* [ETS: GRE Mathematical Conventions (checked 5 October 2026)](https://www.ets.org/content/dam/ets-org/pdfs/gre/gre-math-conventions.pdf)
* [ETS: Guidelines specific to the on-screen calculator (checked 5 October 2026)](https://www.ets.org/content/dam/ets-org/pdfs/gre/on-screen-calculator-guidelines.pdf)

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