Question for a math genius?

Question

a bus for a school field trip costs 624 pesos. Since the number of persons making the trip is 13 less the the capacity of the bus, it costs 8 pesos more per student than if the bus is filled. How many students are in the trip?

Answer 1

np = 624
(n-13)(p+8) = 624

-13p + 8n - 104 = 0

8n - 104 = 13p ... p = (8n-104)/13

n(8n - 104)/ 13 = 624 {from the first equation}
{divide by 8}

n^2 - 13n = 13*78
n = 39 .. the number of students if the bus is filled...

thus there are actually 26 students that joined...

§

Answer 2

x pesos is the cost of the trip for each student if the bus were at capacity.

since the bus is not at capacity the cost is x+8 pesos.

y is the capacity of the bus.

y-13 is the actual number of students on the bus.

x*y = 624 (i)

(x+8)*(y-13) = 624
x*y+8y-13x-104 = 624 (ii)

we have to set equations (i) and (ii) equal to eachother
x*y = 624 = x*y+8y-13x-104
x*y = x*y+8y-13x-104
13x+104 = 8y
13x/8 + 13 = y

substituting this in equation (i)
we have:
x*y = 624
x*(13x/8+13) = 624
(13x^2)/8 +13x = 624
(13x^2)/8 +13x - 624 = 0

solve this with the quadratic formula
x = 16 or -24
obviously x can't be negative so it is 16

solve equation (i) for y using x = 16
16*y = 624
y = 624/16
y = 39
since there is actually (y - 13) students on the bus we have:

26 students on the bus.

Each student must pay 24 pesos to go on the trip.

Answer 3

Let´s call C the capacity and $ the unit cost
C*$ =624 ans
(C-13)*($+8) = 624 so+C*$ +8C-13 $ -104 =624
and C*$= 624
8C -13*$ -104=0
and $ = 624/C
8C-8112/C-104 =0 8C^2 -104C-8112=0
So C = 39 and the number of student is 26

Answer 4

x = capacity of the bus
y = fee per student
xy = 624 equation (1)

x-13 = students who will join the trip
(y + 8) (x - 13) = 624
xy - 13y + 8x -104 = 624
or
xy - 13y + 8x = 728 equation (2)

Solve for x using equation (1)
x = 624/y equation (3)

Substitute x in equation (2)
(624/y) (y) - 13y + 8(624/y) = 728
624 - 13y + 4992/y = 728
subtract 624 to both sides of the equation:
-13y + 4992/y = 104
LCD = y
-13y^2 + 4992 = 104y
-13y^2 - 104y + 4992 = 0
Factor:
(13y - 208) (-y - 24)

13y - 208 = 0
13y = 208
y = 16 (fee per student if filled)

y + 8 = 24 (fee per student if less than the capacity of the bus)

Substitute the value of y in equation (3)
x = 624/16
x = 39 (capacity of the bus)

x - 13 = 26 (students who will join the trip)

To check:

Bus rental
16 x 39 = 624 (if filled)
24 x 26 = 624 (if less than the capacity)

Answer 5

I get 26 too (there are enough equations already presented so I'll omit that and just show the check)

624/26 = 16
624/39 = 24

Answer 6

Let x be the capacity of the bus

624/(x-13)=(624/x)+8

From this equation you can find x value.

Answer 7

624 -13*8 = 8x

solve for x

624 - 104 = 8x

520 = 8x

x = 65

v_v

Answer 8

that would mean 624/8 and ur answer would be 78 people are on the bus.