Large, medium and small ships are used to bring water. 4 large ships carry as much water as 7 small ships, 3 medium ships carry the same amount of water as 2 large ships and 1 small ship. 15 large, 7 medium and 14 small ships, each made 36 journeys and brought a certain quantity of water. In how many journeys would 12 large, 14 medium and 21 small ships bring the same quantity? (M.A.T., 2007)

### Concept & Ratio Equivalence To compare different capacities, establish a common ratio for the capacities of Large ($L$), Medium ($M$), and Small ($S$) ships. Then calculate the total "capacity units" moved per journey. $$ \text{Total Water} = (\text{Fleet Capacity per Journey}) \times (\text{Number of Journeys}) $$ ### Step-by-Step Solution * **Establish Capacities:** * Given: $4L = 7S \implies L = \frac{7}{4}S$ * Given: $3M = 2L + 1S$. Substitute $L$: $3M = 2(\frac{7}{4}S) + S = \frac{7}{2}S + S = \frac{9}{2}S \implies M = \frac{3}{2}S = \frac{6}{4}S$ * **Create Integer Ratios:** Let $S = 4$ units. * $L = 7$ units. * $M = 6$ units. * $S = 4$ units. * **Calculate First Fleet Capacity:** $15L + 7M + 14S$ * Capacity $= 15(7) + 7(6) + 14(4) = 105 + 42 + 56 = 203$ units per journey. * Total Water $= 203 \times 36$. * **Calculate Second Fleet Capacity:** $12L + 14M + 21S$ * Capacity $= 12(7) + 14(6) + 21(4) = 84 + 84 + 84 = 252$ units per journey. * **Find Journeys ($N$):** Equate the total water. * $252 \times N = 203 \times 36$ * $N = \frac{203 \times 36}{252}$ * Recognize that $252 = 36 \times 7$. Thus, $N = \frac{203 \times 36}{36 \times 7} = \frac{203}{7} = 29$. ### Exam Strategy & Shortcut By assigning an arbitrary integer value to the smallest unit (setting $S=4$ to clear the denominators), you avoid complex fractional arithmetic. Treat the entire fleet as a single "machine" that processes a certain number of units per trip. ### Common Pitfall A common mistake is trying to solve for exact numerical capacities without realizing that only the relative ratios matter to find the number of journeys. ### Final Answer Therefore, the correct answer is **29**.