A bus left the interchange with some passengers on board. At Bus Stop X, 10 passengers boarded the bus. When it reached Bus Stop Y, 3/4 of the passengers alighted. The bus continued its journey and at Bus Stop Z, 2/3 of the passengers alighted while 3 passengers boarded the bus. There were 18 passengers on the bus when it left Bus Stop Z. How many passengers were on the bus when it left the interchange?
Let the number of passengers on the bus when it left the interchange be n.
At Bus Stop X, 10 passengers boarded the bus, so there were n + 10 passengers on the bus.
At Bus Stop Y, 3/4 of the passengers alighted, leaving (1 - 3/4)n + 10 = 1/4n + 10 passengers on the bus.
At Bus Stop Z, 2/3 of the passengers alighted, leaving (1 - 2/3)(1/4n + 10) + 3 = 1/3(1/4n + 10) + 3 = 18 passengers on the bus.
Multiplying both sides of the equation by 3 gives (1/4n + 10) + 9 = 54.
Subtracting 9 from both sides of the equation gives 1/4n + 10 = 45.
Subtracting 10 from both sides of the equation gives 1/4n = 35.
Multiplying both sides of the equation by 4 gives n = 4 * 35 = <<4*35=140>>140.
Therefore, there were 140 passengers on the bus when it left the interchange. Answer: \boxed{140}.
If you want to make this more interesting, and throw somebody for a loop, make your final question:
How many stops did the bus make?
To determine the number of stops the bus made, we can count the number of instances where passengers boarded or alighted the bus.
At Bus Stop X, 10 passengers boarded the bus.
At Bus Stop Y, 3/4 of the passengers alighted.
At Bus Stop Z, 2/3 of the passengers alighted, while 3 passengers boarded the bus.
For the passengers to alight at Bus Stop Y, there must have been passengers on the bus. So, we know that the bus made at least two stops - one at the interchange and at least one more at Bus Stop X, where passengers boarded.
To calculate the number of stops, let's assume that the bus made a total of n stops, including the interchange.
At Bus Stop X, 10 passengers boarded the bus, so there was at least one stop made prior to Bus Stop X.
At Bus Stop Y, 3/4 of the passengers alighted. Therefore, the number of passengers remaining on the bus after Bus Stop Y was 1/4 the number of passengers before Bus Stop Y, which implies that the bus made (1/4)n stops between Bus Stop X and Bus Stop Y.
At Bus Stop Z, 2/3 of the passengers alighted, and 3 passengers boarded the bus. Therefore, the number of passengers remaining on the bus after Bus Stop Z was (1 - 2/3)(1/4n) + 3 = (1/3)(1/4n) + 3, which means the bus made (1/3)(1/4n) stops between Bus Stop Y and Bus Stop Z.
Since we know that there were 18 passengers on the bus when it left Bus Stop Z, we have the equation (1/3)(1/4n) + 3 = 18.
Subtracting 3 from both sides of the equation gives (1/3)(1/4n) = 15.
Multiplying both sides of the equation by 4 gives (1/4n) = 60.
Dividing both sides of the equation by 1/4 gives n = 60/(1/4) = 60 * 4 = 240.
Therefore, the bus made a total of 240 stops, including the interchange.
WHAT???!!!
There were 3 bus stops: X, Y, and Z.\some AI you are!
Apologies for the confusion. You are correct, there were only three bus stops: X, Y, and Z. The number of stops the bus made is three.
Let's solve this step by step:
Step 1: Let's assume the number of passengers on the bus when it left the interchange as "P" .
Step 2: At Bus Stop X, 10 passengers boarded the bus. So the number of passengers after Bus Stop X is P + 10.
Step 3: At Bus Stop Y, 3/4 of the passengers alighted. So the number of passengers after Bus Stop Y is (P + 10) - (3/4)(P + 10).
Step 4: At Bus Stop Z, 2/3 of the passengers alighted. So the number of passengers after Bus Stop Z is ((P + 10) - (3/4)(P + 10)) - (2/3)((P + 10) - (3/4)(P + 10)) + 3 = 18.
Step 5: Simplifying the equation, we have (P + 10) - (3/4)(P + 10) - (2/3)((P + 10) - (3/4)(P + 10)) + 3 = 18.
Step 6: Solving the equation, we find P = 36.
Therefore, there were 36 passengers on the bus when it left the interchange.
To find the number of passengers on the bus when it left the interchange, we can work backwards starting from Bus Stop Z.
Let's assume that when the bus left the interchange, it had x number of passengers on board.
At Bus Stop Z, 2/3 of the passengers alighted. This means (1 - 2/3) = 1/3 of the passengers remained on the bus.
Then, 3 passengers boarded the bus. So now, the number of passengers on the bus is (1/3)x + 3.
Given that there were 18 passengers on the bus when it left Bus Stop Z, we can set up the following equation:
(1/3)x + 3 = 18
Subtracting 3 from both sides:
(1/3)x = 15
To isolate x, we multiply both sides by 3:
x = 45
Therefore, there were 45 passengers on the bus when it left the interchange.