A train has a capacity of 154 seats. Tickets for seats are sold at $8 and
$12. There are 1/5 more $8-seats than $12-seats on the train. During a trip the amount collected from the sales of $8 tickets was twice the amount collected from the $12 tickets. The total amount collected was $540. How
many $8-seats were not taken during the trip?
Let's call the number of $12 seats on the train x. Then we know that there are 1/5 more $8 seats, which means there are 6/5 times as many $8 seats as $12 seats. So the number of $8 seats on the train is 6/5x.
Next, we're told that during the trip, the amount collected from $8 tickets was twice the amount collected from $12 tickets. Let's call the number of $12 tickets sold y. Then the number of $8 tickets sold must be 2y, since the amount collected from $8 tickets was twice as much. We can write an equation based on the total amount collected:
8(2y) + 12y = 540
16y + 12y = 540
28y = 540
y = 19.29
We can't sell a fraction of a ticket, so let's round y down to the nearest whole number, which is 19. That means the number of $12 seats sold was 19, and the number of $8 seats sold was 2(19) = 38.
Now we can figure out how many $8 seats were not taken. We know that the total number of seats on the train is 154, so the number of $12 seats plus the number of $8 seats must equal 154:
x + 6/5x = 154
11/5x = 154
x = 70.909
Again, we can't have a fraction of a seat, so let's round x up to the nearest whole number, which is 71. That means the number of $8 seats on the train is 6/5(71) = 85.2, which we'll round down to 85. We also know that 38 $8 seats were sold, so the number of $8 seats not taken is:
85 - 38 = 47
Therefore, 47 $8-seats were not taken during the trip.
To solve this problem, let's break it down step by step:
Step 1: Set up variables:
Let's assume the number of $8-seats on the train is x.
Then, the number of $12-seats on the train is 1/5 less than x, which is (x - (1/5)x) or (4/5)x.
Step 2: Calculate the total amount collected from $8 tickets:
The cost of each $8 ticket is $8, and the number of $8 seats is x.
So, the total amount collected from $8 tickets is 8x.
Step 3: Calculate the total amount collected from $12 tickets:
The cost of each $12 ticket is $12, and the number of $12 seats is (4/5)x.
So, the total amount collected from $12 tickets is 12 * (4/5)x.
Step 4: Write and solve the equation for the total amount collected:
According to the problem, the total amount collected from both types of tickets is $540.
So, we can write the equation:
8x + 12 * (4/5)x = 540
Step 5: Simplify and solve the equation:
8x + (48/5)x = 540
(40/5)x + (48/5)x = 540
(88/5)x = 540
To isolate x, we can multiply both sides of the equation by (5/88):
x = (540 * (5/88))
Step 6: Calculate the value of x:
x ≈ 30
So, there are approximately 30 $8-seats on the train.
Step 7: Calculate the number of $12-seats:
The number of $12-seats is (4/5)x, which is (4/5) * 30 = 24.
Step 8: Calculate the number of $8-seats not taken during the trip:
The total number of seats on the train is 154, so the number of $8 seats not taken is:
154 - 30 = 124.
Therefore, there were 124 $8-seats that were not taken during the trip.
To solve this problem, let's break it down step by step:
Step 1: Set up the equations
Let's assume the number of $12-seats on the train is x.
Since there are 1/5 more $8-seats than $12-seats, the number of $8-seats would be x + (1/5)x = (6/5)x.
Step 2: Calculate the total number of seats
The total number of seats on the train is the sum of $8-seats and $12-seats:
Total seats = (6/5)x + x = (11/5)x.
Step 3: Calculate the total amount collected from $8 tickets
Given that the amount collected from the sales of $8 tickets was twice the amount collected from $12 tickets, we can write:
2(8 * (6/5)x) = 16 * (6/5)x.
Step 4: Calculate the total amount collected from $12 tickets
The amount collected from the sales of $12 tickets is 12 * x.
Step 5: Set up the equation
Total amount collected = 16 * (6/5)x + 12 * x = 540.
Step 6: Solve the equation
16 * (6/5)x + 12 * x = 540.
Simplify and solve:
(96/5)x + 12x = 540,
(96/5 + 12/1)x = 540,
(96 + 60)x/5 = 540,
(156/5)x = 540.
Multiply both sides by 5/156:
x = (540 * 5)/156,
x = 17.5.
Since we can't have half a seat, we can assume there are 17 $12-seats on the train.
Step 7: Find the number of $8-seats
To find the number of $8-seats, substitute the value of x into the expression: (6/5)x = (6/5) * 17.5 = 21.
Step 8: Calculate the number of $8-seats that were not taken
To find the number of $8-seats that were not taken, subtract the number of $8-seats from the total capacity:
Number of $8-seats not taken = 154 - 21 = 133.
Therefore, there were 133 $8-seats that were not taken during the trip.