at the pittsburch zoo, children ride a train for 25 cents, adults pay $1 and seniors pay .75. on a given day, 1400 passengers paid a total of $740 for the rides. there were 250 more children riders than adults and senior riders combined.find the nu,ber of each type of rider.
set up a system and solve.
c+a+s = 1400
.25c + a + .75s = 740
c = a+s + 250
(c,a,s) = (825,410,165)
Let's set up a system of equations to represent the given information.
Let:
C = number of children riders
A = number of adult riders
S = number of senior riders
According to the given information:
1) The total number of passengers is 1400. So we can write the first equation as:
C + A + S = 1400
2) The total amount of money collected is $740. The cost for children is $0.25, for adults is $1, and for seniors is $0.75. So the second equation is:
0.25C + 1A + 0.75S = 740
3) The number of children riders is 250 more than the number of adult and senior riders combined. This can be written as:
C = A + S + 250
Now we have a system of three equations with three variables. Let's solve it.
From equation 3, we can express C in terms of A and S:
C = A + S + 250
Substitute this value of C into equation 1 and 2:
(A + S + 250) + A + S = 1400 [equation 1]
0.25(A + S + 250) + A + 0.75S = 740 [equation 2]
Simplify the equations:
2A + 2S + 250 = 1400
0.25A + 0.25S + 62.5 + A + 0.75S = 740
Combine like terms:
2A + 2S = 1150 [equation 4]
1.25A + 1S = 677.5 [equation 5]
Now we have equation 4 and equation 5, which is a system of two equations with two variables.
Let's solve for A using the substitution method:
From equation 5, we can express A in terms of S:
A = 677.5 - 1S
Substitute this value of A into equation 4:
2(677.5 - 1S) + 2S = 1150
Simplify the equation:
1355 - 2S + 2S = 1150
1355 = 1150
This equation is not possible. It seems that there is a mistake in the problem formulation or given information. Please double-check the values provided to ensure accuracy.
To solve this problem, let's define some variables to represent the number of each type of rider:
Let's call the number of children riders "C".
The number of adult riders will be "A".
The number of senior riders will be "S".
Now, let's set up the equations based on the given information:
1) The total number of passengers is 1400:
C + A + S = 1400
2) The total amount paid for the rides is $740:
0.25C + 1A + 0.75S = 740
3) There were 250 more children riders than the combined number of adult and senior riders:
C = A + S + 250
Now we have a system of three equations with three variables. We can use this system to find the values of C, A, and S.
One way to solve this is by substitution. We can start by solving equation 3 for C:
C = A + S + 250
Then, substitute this into equation 1:
(A + S + 250) + A + S = 1400
2A + 2S + 250 = 1400
2A + 2S = 1150 (equation 4)
Next, we can multiply equation 4 by 0.75 to eliminate the decimals:
1.5A + 1.5S = 862.50 (equation 5)
Now, we can subtract equation 5 from equation 2:
0.25C + 1A + 0.75S - (1.5A + 1.5S) = 740 - 862.50
0.25C - 0.5A - 0.75S = -122.50 (equation 6)
Now, we have two equations:
2A + 2S = 1150 (equation 4)
0.25C - 0.5A - 0.75S = -122.50 (equation 6)
We can solve this system using various methods, such as substitution or elimination. Let's solve it using elimination:
Multiply equation 6 by 4 to eliminate decimals:
C - 2A - 3S = -490 (equation 7)
Now, subtract equation 7 from equation 4:
2A + 2S - (C - 2A - 3S) = 1150 - (-490)
2A + 2S - C + 2A + 3S = 1150 + 490
4A + 5S - C = 1640 (equation 8)
Now we have two equations:
4A + 5S - C = 1640 (equation 8)
0.25C - 0.5A - 0.75S = -122.50 (equation 6)
By solving this system of equations, we can find the values of C, A, and S, which represent the number of children, adults, and seniors, respectively.