Grade 8 - Algebra #5

Algebra Level 3

There is a pond with a perimeter of 3 km.

Tom and Jerry stood at one point on the pond's perimeter and ran towards different directions along the perimeter. It took them 15 minutes to meet each other for the first time.

After meeting each other, Tom turned back and went to the opposite direction of where he had been going, and Jerry continued going to the direction of where he had been going.

Tom bumped into Jerry's back after 40 minutes from when they had started running in the same direction. Then, how long does it take for Jerry to go around the pond's perimeter once completely? Answer in minutes.

Detail: Ignore the width of the path, and the body size of Tom and Jerry. Tom and Jerry run at a constant speed throughout the whole process, where Tom runs faster than Jerry.

Image source: Kasco Maritiem - Pond Health And Beauty

This problem is a part of <Grade 8 - Algebra> series .


The answer is 48.

Boi (보이) by Boi (보이) , 19, South Korea

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1 solution

Chew-Seong Cheong
Jul 10, 2017

Let the speeds of Tom and Jerry be v T v_T km/min and v J v_J km/min respectively.

Starting from the same point running in opposite directions and met 15 minutes later means that the distance covered by Tom and the distance covered by Jerry add up together equals to the perimeter of the pond or 15 v T + 15 v J = 3 15v_T + 15v_J = 3 .

Starting from the same point running in same direction and met 40 minutes later with v T > v J v_T > v_J means that the distance covered by Tom is one pond perimeter or 3 km more than the distance covered by Jerry after 40 minutes that is 40 v T = 40 v J + 3 40v_T = 40v_J+3 .

Therefore, we have:

{ 15 v T + 15 v J = 3 ⟹ v T + v J = 1 5 . . . ( 1 ) 40 v T = 40 v J + 3 ⟹ v T − v J = 3 40 . . . ( 2 ) \begin{cases} 15v_T + 15v_J = 3 & \implies v_T + v_J = \dfrac 15 & ... (1) \\ 40v_T = 40v_J + 3 & \implies v_T - v_J = \dfrac 3{40} & ... (2) \end{cases}

( 1 ) − ( 2 ) : ⟹ 2 v J = 1 5 − 3 40 = 1 8 ⟹ v J = 1 16 \begin{aligned} (1)-(2): \implies 2v_J & = \frac 15 - \frac 3{40} = \frac 18 \\ \implies v_J & = \frac 1{16} \end{aligned}

Therefore, the time for Jerry to go around the pond completely t = 3 v J = 3 1 16 = 48 t = \dfrac 3{v_J} = \dfrac 3{\frac 1{16}} = \boxed{48} minutes.

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