A frog was at the bottom of an 80 m deep well. It attempted to come out of it by jumping. In each jump, it covered 1.15 m but slipped down by 0.75 m. The number of jumps after which it would be out of the well is:

Question

Prepp Team · Accepted Answer

Problem Setup: Frog in the Well

We need to find the number of jumps a frog takes to escape an 80 m deep well.

Well Depth: $80 \, \text{m}$
Jump Height: $1.15 \, \text{m}$
Slip Back Height: $0.75 \, \text{m}$

Calculating Net Progress Per Jump

In each jump cycle (jump up and slip down), the frog makes a net upward progress.

Net gain per jump = Jump Height - Slip Back Height
Net gain = $1.15 \, \text{m} - 0.75 \, \text{m} = 0.40 \, \text{m}$

Determining the Final Jump Condition

The frog escapes the well when its jump takes it to or past the 80 m mark. Importantly, if the frog reaches a height from which the next jump *alone* is sufficient to clear the well, it will not slip back from that final position.

Consider the height the frog needs to reach *before* its last jump. This height must be such that adding the jump height ($1.15 \, \text{m}$) reaches or exceeds the total depth ($80 \, \text{m}$).

Height needed before the last jump $\ge$ Well Depth - Jump Height
Height needed $\ge 80 \, \text{m} - 1.15 \, \text{m} = 78.85 \, \text{m}$

Calculating the Number of Jumps

Let $N$ be the total number of jumps required.

The number of jumps needed to reach or exceed the height of $78.85 \, \text{m}$ is calculated using the net gain per jump.

Number of jumps before the last one, let's call it $J$, must satisfy:

$J \times (\text{Net gain per jump}) \ge \text{Height needed before the last jump}$
$J \times 0.40 \, \text{m} \ge 78.85 \, \text{m}$
$J \ge \frac{78.85}{0.40}$
$J \ge 197.125$

Since the number of jumps must be a whole number, the minimum integer value for $J$ (the number of jumps *before* the final one) is 198.

The total number of jumps $N$ is the sum of these preliminary jumps ($J$) plus the final jump.