A primary school teacher wants to arrange a visit to the museum which is 600 miles away...
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A primary school teacher wants to arrange a visit to the museum which is 600 miles away from the school. She arranges a school bus for the trip. However, the bus driver does not remember the capacity of the gas tank nor does he remember the efficiency of the engine.
Which of these statements would be sufficient to determine if the bus has enough gas to reach the museum?
Statement 1: The driver is driving at a constant speed.
Statement 2: The driver realizes that he has three-quarters of the full tank of gas left after he drives 150 miles.
Statement 3: The driver realizes that he has three-quarters of the full tank of gas left after he drives 450 miles.
Show answer & explanation
To determine if the bus can travel 600 miles, we need to know the fuel consumption rate relative to the tank capacity. Statement 1 ensures a constant consumption rate, and either Statement 2 or 3 provides a data point to calculate the total range of a full tank.
Step-by-step Derivation:
Step 1: Define variables. Let C be the total capacity of the gas tank and E be the efficiency (miles per unit of fuel). Total range of a full tank = C * E.
Step 2: Analyze Statement 1. Constant speed implies a constant fuel consumption rate (E is constant). This is necessary for the calculations in Statements 2 and 3 to be valid for the remainder of the trip.
Step 3: Analyze Statement 2. After 150 miles, 3/4 of the tank remains. This means 1/4 of the tank was used to cover 150 miles. Therefore, a full tank (4/4) can cover 150 * 4 = 600 miles. Since the museum is 600 miles away and the bus started with a full tank (implied by the context of checking if it has 'enough' gas), it can exactly reach the museum.
Step 4: Analyze Statement 3. After 450 miles, 3/4 of the tank remains. This means 1/4 of the tank was used to cover 450 miles. Therefore, a full tank can cover 450 * 4 = 1800 miles. This is more than enough to cover 600 miles.
Step 5: Conclusion. Statement 1 combined with Statement 2 allows us to determine the bus can reach the museum. Statement 1 combined with Statement 3 also allows us to determine the bus can reach the museum. Thus, (1 and 2) OR (1 and 3) is sufficient.