Depth · one concept from first principles · step 2 of 4

Same way: lap gap; opposite way: lap sum

A and B decide to walk around a lake. While A walks $200$ m, B walks $300$ m. If they start from the same point at the same time and walk around in the same direction at constant speeds, they meet for the first time after $1$ hour $20$ minutes. After that, B starts walking in the opposite direction at a constant speed $1.2$ times B's original speed, and $10$ minutes later there are still $3.6$ km left until the two meet again. How many kilometers is the distance around the lake?

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Your answer

How to solve

Strategy Convert to Algebra — Draw the lake as a circle. Walking the same way, the faster walker must gain one full lap to meet the slower one; walking toward each other from the meeting point, the two distances must add up to one full lap. Each picture gives one equation.

1STEP 1

Name the speeds and the lap

Speeds are in the ratio 200:300 = 2:3, so A walks 2x m and B 3x m per minute. The lap is y m; 1 hour 20 minutes is 80 minutes.

A: 2x, B: 3x, 1.2 × 3x = 3.6x

2STEP 2

Same direction: B gains one lap

Starting together and walking the same way, they next meet when B is exactly one lap ahead.

80 × 3x - 80 × 2x = y, 80x = y

3STEP 3

Opposite directions: together they cover the lap

Walking toward each other, they meet when their distances add to one lap. After 10 minutes they covered 20x + 36x with 3600 m left.

10 × 2x + 10 × 3.6x = y - 3600, 56x = y - 3600

4STEP 4

Solve

Substitute y = 80x into the second equation.

56x = 80x - 3600, 24x = 3600, x = 150, y = 80 × 150 = 12000

5STEP 5

Convert to kilometers

12000 m is 12 km.

12000 m = 12 km

Answer

12 km

A walks 300 m per minute and B 450 m per minute. In 80 minutes B gains 150 × 80 = 12000 m, one lap. Afterward B walks 540 m per minute; in 10 minutes they cover 3000 + 5400 = 8400 m, leaving 12000 - 8400 = 3600 m, as given.

Takeaway

On a loop, chasing means the gap closes by one lap; walking toward each other means the two distances add to one lap.

Name the speeds and the lap
Same direction: B gains one lap
Opposite directions: together they cover the lap
Solve
Convert to kilometers

Where next?

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▶ Practice — 12 problems