Problem
Catching up: subtract the two speeds
Two runners set off together and run the same way. In (1) one starts behind the other on a straight road. In (2) they run round a circular track. Find how long until one catches or laps the other.
Reasoning · Grade 6-1 Distance and Speed (1)
Your answer
Watch out Adding the speeds is tempting, but running the same way you must subtract.
How to solve
Strategy Draw a Diagram — Draw the road as a line with two dots on it and ask what happens to the gap between the dots after one minute. That one picture turns a chase into a single subtraction: the gap shrinks by (faster speed) minus (slower speed) every minute. Then the units do the rest — yards of gap divided by yards-per-minute of closing gives minutes. The circular track in (2) looks like a different problem, but it is the same easier problem in disguise: bend the road into a loop and 'catching up' means gaining a full 400-yard lap.
1STEP 1
(1) See what one minute does to the gap
In (1) the gap shrinks by 20 a minute.
200 - 180 = 20 yards closed per minute
This is the whole idea of the chapter in one line: when two people run the same way, what matters is not either speed but the difference of the speeds, because that difference is the rate at which the gap disappears.
6.RP.A.2Draw A DiagramIn one minute the gap shrinks by exactly the difference between the two speeds, not by either speed itself.
Why?
Both runners move forward together, so the part of the speed they share carries the pair along without closing anything between them.
🧱A gap closes at the difference of the ratesWhen two things move the same way, neither speed alone matters — the gap between them shrinks at exactly the difference of the two speeds.
Why?
That leftover speed acts steadily, so twice the time closes twice as much of the gap.
🧱Proportional quantities scale togetherWhen two quantities keep a fixed ratio, doubling one doubles the other, so a ratio you know can stand in for one you do not.
2STEP 2
(1) Divide the gap by the closing rate
Dividing the gap gives 3 minutes.
60 ÷ 20 = 3 minutes
Watch the units: yards divided by yards-per-minute leaves minutes, which is exactly the kind of answer the question wants. The question asks for minutes and seconds, and 3 minutes comes out exact, so the answer is 3 minutes 0 seconds.
6.RP.A.3Analyze The Units3STEP 3
(1) Check by tracking both runners separately
Tracking both separately lands them together.
200 × 3 = 600, 60 + 180 × 3 = 600
Computing each runner's position separately and seeing the two numbers agree is a check a fourth grader can do with multiplication and addition alone — no rate reasoning needed to confirm it.
4.OA.A.3Draw A Diagram4STEP 4
(2) On a loop, 'meeting again' means gaining one whole lap
In (2) meeting again needs a full lap gained.
distance Owen must gain = 400 yards (one full lap)
Cut the loop open and lay it flat: this is now part (1) again, with a 400-yard head start instead of 60. Recognizing the easier problem you already solved is what makes the circular version harmless.
6.RP.A.3Solve An Easier Related Problem5STEP 5
(2) Find how much Owen gains each second
He gains 0.5 a second.
3 - 2.5 = 0.5 yards gained per second
Subtracting 2.5 from 3 is just decimal subtraction lined up on the point; the speeds are given per second, so the gain comes out per second and the units stay honest.
6.NS.B.3Analyze The Units6STEP 6
(2) Divide, then convert seconds into minutes
That is 800 seconds, or 13 minutes 20 seconds.
400 ÷ 0.5 = 800 seconds = 13 × 60 + 20 = 13 min 20 s
Dividing by 0.5 is the same as doubling — that is the sanity check on the biggest-looking step here — and converting seconds to minutes is the Grade 4 unit conversion 60 seconds = 1 minute.
4.MD.A.1Analyze The Units7STEP 7
(2) Check with laps
Counting laps shows exactly one lap of difference.
3 × 800 = 2400 = 6 × 400, 2.5 × 800 = 2000 = 5 × 400
Both distances landing on a whole number of laps is strong evidence the answer is right: if the time were off even slightly, one of the two runners would be stranded partway round the loop.
6.RP.A.3Draw A DiagramAnswer
3 minutes, 13 minutes 20 seconds
60 ÷ 20 = 3, 400 ÷ 0.5 = 800
Units check out in both parts: yards divided by yards-per-minute gives minutes, yards divided by yards-per-second gives seconds. The magnitudes make sense too. In (1) the gap is small (60 yards) and Hana is a good deal faster (20 yards per minute better), so a few minutes is right; if the answer had come out in hours something would be wrong. In (2) the gap to close is much bigger (400 yards) and the speed advantage is much smaller (0.5 yards per second, one fortieth of the 20 yards per minute closing in part 1 when both are put in the same units), so a much longer time is expected — and 13 minutes 20 seconds is indeed far more than 3 minutes. The dangerous wrong turns are both visible from here: adding the speeds (200 + 180, or 3 + 2.5) would be the head-on case, not the same-direction case, and using 200 alone would pretend Ryan is standing still.
Takeaway
When two people run the same way, subtract the speeds — that difference is how fast the gap disappears, and on a circular track the gap you have to close is one whole lap.
- (1) See what one minute does to the gap
- (1) Divide the gap by the closing rate
- (1) Check by tracking both runners separately
- (2) On a loop, 'meeting again' means gaining one whole lap
- (2) Find how much Owen gains each second
- (2) Divide, then convert seconds into minutes
- (2) Check with laps