Question

A water tank has steps inside it. A monkey is sitting on the top most step (i.e., the first step). The water level is at the ninth step .

(1) He jumps 3 steps down and then jumps back 2 steps up. In how many jumps will he reach the water level ?

(2) After drinking water , he wants to go back . For this he jumps 4 steps up and then jumps back 2 steps down in every move. In how many jumps will he reach back the top step ?

(3) If the no. Of steps moved down is represented by negative integers and the no. Of steps moved up by positive integers , represent his moves in part 1 and 2 by completing the following; (1) -3+2-..... = -8 (2) 4-2+....... =8. In 1 the sum (-8) represent going down by eight steps. So, what will the sum 8 in 2 represent ?

Answer 1

Let the steps moved down be represented by positive integers and the steps moved up be represented by negative integers.

(i) Initially, the monkey was at step = 1

After 1st jump, the monkey will be at step = 1 + 3 = 4

After 2nd jump, the monkey will be at step = 4 + (−2) = 2

After 3rd jump, the monkey will be at step = 2 + 3 = 5

After 4th jump, the monkey will be at step = 5 + (−2) = 3

After 5th jump, the monkey will be at step = 3 + 3 = 6

After 6th jump, the monkey will be at step = 6 + (−2) = 4

After 7th jump, the monkey will be at step = 4 + 3 = 7

After 8th jump, the monkey will be at step = 7 + (−2) = 5

After 9th jump, the monkey will be at step = 5 + 3 = 8

After 10th jump, the monkey will be at step = 8 + (−2) = 6

After 11th jump, the monkey will be at step = 6 + 3 = 9

Clearly, the monkey will be at water level (i.e., 9th step) after 11 jumps.

(ii) Initially, the monkey was at step = 9

After 1st jump, the monkey will be at step = 9 + (−4) = 5

After 2nd jump, the monkey will be at step = 5 + 2 = 7

After 3rd jump, the monkey will be at step = 7 + (− 4) = 3

After 4th jump, the monkey will be at step = 3 + 2 = 5

After 5th jump, the monkey will be at step = 5 + (− 4) = 1

Clearly, the monkey will reach back at the top step after 5 jumps.

(iii) If steps moved down are represented by negative integers and steps moved up are represented by positive integers, then his moves will be as follows.

Moves in part (i)

− 3 + 2 − 3 + 2 − 3 + 2 − 3 + 2 − 3 + 2 − 3 = −8

Moves in part (ii)

4 − 2 + 4 − 2 + 4 = 8

Moves in part (ii) represent going up 8 steps.