Question

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

(i) He jumps 3 steps down and then jumps back 2 steps up. In how many jumps will he reach the water level?
(ii) 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?
(iii) If the number of steps moved down is represented by negative integers and the number of steps moved up by positive integers, represent his moves in part (i) and (ii) by completing the following:
(a) $ -3+2-\ \ ...\ \ =-8 $
(b) $ 4-2+\ \ ...\ \ =8 $
In (a), the sum (-8) represents going down by 8 steps. So, what will the sum 8 in (b) represent?

Answer 1

Hint: When the monkey jumps some steps forward and some steps backwards successively, then effectively, in 2 jumps, he would cover a number of steps which is equal to the difference of the number of steps that he jumped forward and backward.
If he jumps $ F $ steps forward and $ B $ steps backwards, then in 2 jumps he would have covered $ F-B $ steps.
The monkey would reach the destination in the immediate next forward jump when his F steps behind the destination.

Complete step by step solution:
Since the monkey is already at the 1st step, the distance to be covered by the monkey is $ D=8 $ steps.
(i) In this case, the number of steps moved forward is $ F=3 $ and the number of steps moved backwards is $ B=2 $ . Effectively, the monkey moves forward $ 3-2=1 $ step in every 2 jumps.
Let us calculate the number of jumps required to reach exactly $ F $ steps behind the destination, i.e. to reach $ 8-3=5 $ steps.
At the rate of 1 step in 2 jumps, it will cover $ 1\times 5 $ steps in $ 2\times 5 $ jumps.
⇒ 5 steps in 10 jumps.
The monkey will reach the water in the immediate next forward jump.
Therefore, the number of jumps required are:
  $ 10+1 $
= 11 jumps.
(ii) Using the same argument as above, $ F=4 $ and $ B=2 $ . Effectively, the monkey moves forward $ 4-2=2 $ steps in every 2 jumps.
The number of jumps required to reach exactly $ D-F=8-4=4 $ steps behind the destination, can be calculated as follows:
At the rate of 2 steps in 2 jumps, it will cover $ 2\times 2 $ steps in $ 2\times 2 $ jumps.
⇒ 4 steps in 4 jumps.
The monkey will reach the top in the immediate next forward jump.
Therefore, the number of jumps required are:
  $ 4+1 $
= 5 jumps.
(iii) The sequence of jumps in both the above cases will be:
(a) $ (-3+2)+(-3+2)+(-3+2)+(-3+2)+(-3+2)-3=-5-3=-8 $
(b) $ (4-2)+(4-2)+(4-2)+(4-2)+4=4+4=8 $
The value of +8 in (b) indicates a movement in the opposite direction as compared to in (a).

Note: The number of required steps (n) to reach a destination D steps away, can also be calculated directly as:
  $ n=2\left( \dfrac{D-F}{F-B} \right)+1 $
In the above formula, if the value of the expression $ \left( \dfrac{D-F}{F-B} \right) $ is not an integer, consider the next higher integer as its value and then multiply by 2 etc.
e.g. $ \left( \dfrac{9-4}{4-2} \right)=\left( \dfrac{5}{2} \right)=2.5=3 $ and the value of $ n=2(3)+1=6+1=7 $ .
The same technique is also applied to calculate the amount of time required to complete a work by a number of people working alternately in a fixed order.