A ship, 40 kilometres from the shore, springs a leak which admits $3 \frac{3}{4}$ tonnes of water in 12 minutes. 60 tonnes would suffice to sink her, but the ship's pumps can throw out 12 tonnes of water in one hour. Find the average rate of sailing, so that she may reach the shore just as she begins to sink. (M.A.T. 2006, 2008)

### Concept & Net Rate of Accumulation This problem requires determining the net rate at which water is accumulating in the ship. We find the difference between the inflow (leak) and outflow (pump) rates, then calculate the time it takes to reach the sinking threshold. $$ \text{Net Inflow Rate} = \text{Leak Rate} - \text{Pump Rate} $$ $$ \text{Required Speed} = \frac{\text{Distance to Shore}}{\text{Time to Sink}} $$ ### Step-by-Step Solution 1. Find the water intake rate per hour: The leak admits $3 \frac{3}{4} = \frac{15}{4}$ tonnes in 12 minutes. Since there are $60 / 12 = 5$ twelve-minute periods in an hour, the hourly leak rate is: $$ \frac{15}{4} \times 5 = \frac{75}{4} = 18.75 \text{ tonnes/hour} $$ 2. Find the net water accumulation rate per hour: The pump throws out 12 tonnes per hour. Net inflow = $18.75 - 12 = 6.75$ tonnes/hour = $\frac{27}{4}$ tonnes/hour. 3. Calculate the total time before the ship sinks: The ship sinks at 60 tonnes. Time to sink = $\frac{60}{27/4} = \frac{60 \times 4}{27} = \frac{240}{27} = \frac{80}{9}$ hours. 4. Calculate the required speed: The ship is 40 km from the shore. It must cover this in $\frac{80}{9}$ hours. Speed = $\frac{\text{Distance}}{\text{Time}} = \frac{40}{80/9} = \frac{40 \times 9}{80} = 4.5$ km/hour. 5. $4.5$ km/hour is equivalent to $4 \frac{1}{2}$ km/hour. ### Exam Strategy & Shortcut Work exclusively with fractions to avoid messy decimal division. Leak = $15/4$ tonnes in $1/5$ hr $\rightarrow 75/4$ tonnes/hr. Net = $75/4 - 48/4 = 27/4$. Time = $60 / (27/4) = 80/9$. Speed = $40 / (80/9) = 9/2 = 4.5$. ### Common Pitfall A common pitfall is forgetting to standardize the time units. Students might compare the 12-minute leak rate directly with the 1-hour pump rate without aligning them first. ### Final Answer Therefore, the correct answer is **$4 \frac{1}{2}$ km per hour**.