Concise answer
Three families, one discipline: motion problems run on D = rt with constant speeds, work problems run on rates of 1/t job per hour that add when agents work simultaneously, and mixture problems run on number-times-value bookkeeping whose result is a weighted average. Every setup starts the same way — convert times to rates, lay out a table, and let a rational equation cleared by the LCD do the algebra.
Definitions
- Rate of work
- The fraction of a job done per unit time: a job finished in t hours proceeds at 1/t of the job per hour.
- Uniform motion
- Travel at constant speed, modeled by D = rt; multi-vehicle problems compare two such scenarios sharing a distance, a time, or a meeting point.
- Combined rate
- The sum of the rates of agents working simultaneously on the same job, with opposing agents such as drains or leaks contributing their rate with the opposite sign.
- Mixture value equation
- The bookkeeping identity that the total value of each component (number times value) sums to the total value of the mixture, making the mixture's per-unit value a weighted average of the components'.
Intuition
Rates add because contributions add: in one shared hour, a 6-hour worker delivers 1/6 of the job and a 3-hour worker delivers 1/3, so the pot gains 1/2 of the job. Times do not add — averaging 6 and 3 into 4.5 invents a slower crew than the faster worker alone, which is structurally impossible.
A mixture concentration is a tug-of-war between component concentrations, with amounts as the pulling weights: the result must land strictly between the two and closer to the side with more units. Any computed concentration outside that bracket is wrong before you recheck a single algebra step.
Concept walkthrough
GMAT Quantitative Reasoning — 45 minutes and 21 questions on GMAC's exam-structure page — is a problem-solving section, and rate setups are among its recurring templates. Uniform motion is the cleanest family: with constant speed, D = rt governs each traveler, and problems compare two scenarios — two vehicles in opposite directions splitting a gap, a faster vehicle overtaking a slower one, or one trip at two different speeds. A three-column table (rate, time, distance) per traveler, with the shared quantity identified, turns each variant into one linear or rational equation.
Work problems are motion problems with the 'distance' normalized to one whole job. A worker who finishes in t hours works at 1/t job per hour, and when agents work simultaneously their contributions in a common hour add: the part done by one plus the part done by the other is the amount done together, so 1/a + 1/b = 1/t for two cooperating workers. Opposing agents — a drain while pipes fill, a leak while a pump works — enter the same sum with a minus sign. The resulting rational equations clear via the least common denominator, which is why this family lives beside proportions and variation in the algebra curriculum.
Mixture problems trade time for value: each component contributes number times value — ounces times concentration, pounds times price per pound — and the totals must reconcile, so the components' total values sum to the mixture's. Dividing that identity by the total amount exposes the mixture's per-unit value as a weighted average, which supplies the sanity bracket: between the component values, nearer the bigger batch. The exam's favorite variants — how much pure solution to add, what ratio of two coffees hits a target price — are all one table and one equation away once the value column is written.
After this page, you should be able to
- Translate stated times and speeds into rates and organize motion problems in a rate-time-distance table built on D = rt.
- Solve combined-work problems by adding simultaneous rates, including agents that oppose the job.
- Set up mixture problems with number-times-value rows and solve for an unknown amount or concentration.
- Sanity-check answers against structural bounds: a combined time beats every individual time, and a mixture concentration lands between its components'.
Formulas and assumptions
Distance, rate, and time
Variables
- D: distance traveled
- r: constant rate (speed)
- t: time in motion
Assumptions
- Speed is constant over the interval (uniform motion).
- Units must agree: a rate in miles per hour demands time in hours.
Combined work equation
Variables
- a, b: hours each agent needs alone
- c: hours an opposing agent (drain, leak) needs to undo the whole job alone
- t: hours the group needs together
Assumptions
- Agents work simultaneously at constant individual rates, so rates add.
- For cooperating agents, t is smaller than every individual time.
Mixture value equation
Variables
- a1, a2: amounts of each component
- c1, c2: per-unit values (concentration, price per pound)
- c_mix: the mixture's per-unit value
Assumptions
- Total amount and total value are both conserved when mixing.
- c_mix is the weighted average of c1 and c2, so it lies between them, closer to the larger amount's value.
Worked example
Two pipes and an open drain
Pipe A fills an empty tank in 6 hours and pipe B fills it in 3 hours. With both pipes open and the drain accidentally left open, the tank fills in 4 hours. How long would the open drain alone take to empty a full tank?
- 1Convert every time to a rate: A fills at 1/6 tank per hour, B at 1/3 tank per hour, and the drain empties at an unknown 1/d tank per hour, entering the sum with a minus sign.
- 2Write the combined-rate equation from the observed outcome: 1/6 + 1/3 - 1/d = 1/4.
- 3Simplify the filling side: 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2, so 1/2 - 1/d = 1/4.
- 4Solve for the drain's rate: 1/d = 1/2 - 1/4 = 1/4, so d = 4.
- 5Check by recombining with the LCD 12: 2/12 + 4/12 - 3/12 = 3/12 = 1/4 tank per hour — a 4-hour fill, matching the problem. Sanity check: without the drain the pipes alone work at 1/2 tank per hour (a 2-hour fill), so the drain slowing the fill to 4 hours is consistent.
The drain alone empties a full tank in 4 hours.
Common traps
- Averaging times instead of adding rates: workers who take 6 and 3 hours finish together in 2 hours, not 4.5 — a combined time can never exceed the fastest individual time.
- Mixing units, most often minutes in one rate and hours in another; convert everything before the equation is written.
- Dropping the minus sign on opposing agents — drains, leaks, and headwinds subtract from the combined rate.
- Accepting a mixture concentration outside the range of its components; the weighted average must land between them, nearer the larger batch.
- Computing a round trip's average speed by averaging the two speeds instead of dividing total distance by total time.
Related pages and practice
Question depth and domain coverage vary by exam. Practice answers are checked after submission.
Sources
- The GMAT Focus Edition: Exam Structure, Content & Features — GMAC. Accessed 2026-08-03. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
- Elementary Algebra 2e, Section 3.5: Solve Uniform Motion Applications — OpenStax. Accessed 2026-08-03. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
- Elementary Algebra 2e, Section 8.8: Solve Uniform Motion and Work Applications — OpenStax. Accessed 2026-08-03. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
- Elementary Algebra 2e, Section 3.3: Solve Mixture Applications — OpenStax. Accessed 2026-08-03. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
- Intermediate Algebra 2e, Section 7.5: Solve Applications with Rational Equations — OpenStax. Accessed 2026-08-03. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
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Recheck the GMAC exam-structure page and the OpenStax Elementary and Intermediate Algebra application sections before each major GMAT preparation cycle.