Pick the story type, enter the numbers, and watch it become an equation: understand → set up → solve → verify against the story itself. That last stage is the one most solvers skip — and the one that catches wrong setups.
Two vehicles start apart and drive toward each other. When do they meet?
In short
Distance-rate-time problems all run on one formula, d = rt. The skill is translation: deciding whether speeds add (objects moving toward each other), subtract (a chase), or split across two legs (a round trip). VerifySolve sets the equation up in stages, solves it, then checks the answer against the story’s own distances and times.
Everything here is d = rt. What changes between problems is how the distances combine — the chart the solver builds in Stage 2 is exactly these relationships written out.
The master formula
Moving toward each other
Catch-up
Round trip
Each note re-tests the answer in the story’s own terms: distances adding up, positions coinciding, legs summing. Run them live in the solver above.
Two cars 300 miles apart drive toward each other at 40 mph and 60 mph. When do they meet?
They meet after 3 hours
Verified against the story: in 3 hours they cover 120 and 180 miles — and 120 + 180 = 300, the original gap ✓.
A truck has a 30-mile head start at 40 mph. A car follows at 55 mph. When does the car catch up?
The car catches up after 2 hours
Verified: after 2 hours both are at mile 110 (55·2 = 110 and 30 + 40·2 = 110) ✓.
A trip of 120 miles is driven out at 40 mph and back at 60 mph. Total time?
5 hours total
Verified: the legs reproduce the distance — 40·3 = 120 ✓ and 60·2 = 120 ✓. True average speed: 240/5 = 48 mph, not 50.
Cyclists 100 miles apart ride toward each other at 30 mph and 45 mph.
They meet after 4/3 hours — 1 hour 20 minutes
Verified: 30·(4/3) + 45·(4/3) = 40 + 60 = 100 ✓. Fractions are normal in motion problems — exact answers matter.
In toward-each-other problems the speeds add (both eat the gap at once); the times are the same single t. Writing two different times, or adding travel times, builds an equation the story never asked for.
In catch-up problems the 30 miles is a gap, not a journey. The chaser does not stop at mile 30 — it stops where the leader has reached by then (mile 110 in our example). Setting positions equal is the reliable frame.
40 out and 60 back does not average 50. The slow leg takes more time, so it weighs more: total distance ÷ total time. Any answer that skips the two legs separately is guessing.
d = r · t: distance equals rate times time. Every problem on this page is that one formula wearing a story. The solver’s job — and the skill you are learning — is translating the story into the right version of the equation: adding speeds when objects move toward each other, subtracting when one chases another.
Add when the gap is closing from both sides (driving toward each other) — the gap shrinks at the combined speed. Subtract when both move the same direction (catch-up) — the gap shrinks only by the difference in speeds. If the chaser is not faster, the gap never closes, and the solver will tell you the story cannot happen.
Against the story, not just the equation: for a meeting problem, each vehicle’s distance at the answer time is computed and the two distances must add up to the original gap. If they do not, the answer does not ship. It is the check a careful student does — automated.
Yes — real problems rarely divide evenly. Answers are exact fractions (4/3 hours) with a friendly conversion (1 hour 20 minutes) so they read naturally.
The round-trip solver computes each leg separately — total distance over total time. Notice it is never just the average of the two speeds; the slower leg takes longer and pulls the true average down. The staged steps make that visible.
Yes — free, no sign-up, no app, with the full staged solution and story verification included.
The other classic two-equation story: percents and prices, in stages.
The staged method explained, plus every word-problem family we cover.
Already have the equation? Solve it directly, with plug-back proof.
Motion problems with two unknowns become systems — solve them here.
A note on honest use: VerifySolve is built for learning. Use the steps to understand the method, then close the tab and try one on your own — that is where the grade actually comes from. If your teacher or school has a policy on calculators, follow it.