Adult ticket = 8.75
child ticket = 3.50
460 tickets we're sold totaling 3143
how many adults tickets were sold
Adult ticket = 8.75
child ticket = 3.50
460 tickets we're sold totaling 3143
how many adults tickets were sold
Now, the main unknown here is "the # of adult tickets." Another thing that's unknown is the "the # of children's tickets." So, we will assign 2 variables as follows:
Let a = # of adult tickets
and c = # of children's tickets.
The next step is to set up the equations out of the unknowns that follow the word problem. Here they are as follows:
a adult tickets + c children's tickets = 460 tickets altogether, so
a + c = 460.
Now, since we know how much all the tickets cost altogether, $3143, here comes the next equation:
(($8.75/adult ticket) * a adult tickets) + (($3.50/children's ticket) * c children's tickets) = $3143, so
8.75a + 3.50c = 3143 since the units, adult tickets and children's tickets, cancel each other out.
Now, since we are looking for the # of adult tickets, we can express the simple equation, a + c = 460, as c in terms of a:
a + c = 460
-a = -a (i.e. -a on each side of the equation to eliminate a on the left side.)
c = 460 - a
Now that we have the equation c in terms of a, we can plug it into the other equation, 8.75a + 3.50c = 3143. Here it is as follows:
8.75a + 3.50 * (460-a) = 3143 substitute method
8.75a + [(3.50 * 460) - (3.50 * a)] = 3143 distributive property
8.75a + (1610 - 3.50a) = 3143 I multiplied the two terms in parentheses inside the brackets.
8.75a + 1610 - 3.50a = 3143 Everything's out of the parentheses.
5.25a + 1610 = 3143 I combined the like terms above.
- 1610 = -1610 I am eliminating the constant, 1610, on the left side of the equation.
5.25a = 1533
5.25a/5.25 = 1533/5.25
a = 292
So, now I have found the number of adult tickets, 292. Therefore, 292 adult tickets were sold.
Now let's check. Now that we know the value of a of 292, we can plug it into our equations, a + c = 460 and 8.75a + 3.50c = 3143:
292 + c = 460
(8.75 * 292) + 3.50c = 3143
Now, we need to find the value of c in the first equation so that we can plug it into the second equation:
292 + c = 460
-292 = -292
c = 168
(8.75 * 292) + (3.50 * 168) = 3143
2555 + 588 = 3143
3143 = 3143 check!
this is an algebra system of two equations problem. you can read about it here.
http://en.wikipedia.org/wiki/System_of_linear_equations
let a = (adult tickets sold) and c = (childrens tickets sold)
so if 460 tickets total were sold, that means that the sum of the adult and childrens tickets were 460. translated into algebra, it is:
460 = a + c
if there was a total of 3143 dollars, and each adult ticket cost 8.75, each childs ticket cost 3.50, that translates into this equation:
8.75 * a + 3.50 * c = 3143
sub out c and solve for a, and remember order of operations (multiply before you add or subtract)
Let x = the number of adult tickets. Since 460 tickets were sold in all, then the number of children's tickets would be 460 - x.
We then take the number of each ticket, multiply by the price of each ticket, and that gives us total revenue of $3143:
8.75x + 3.50 (460 - x) = 3143
8.75x + 1610 - 3.50x = 3143
5.25 x = 1533
x = 292 adult tickets
If x and y are the number of adult and child tickets then 8.75 x + 3.5 y= 3143....... (1)
and, x + y = 460.......(2)
Multiplying both sides of the equation with 3.5 we get, 3.5 x + 3.5 y = 3.5 x 460 = 1610.....(3)
Subtracting (3) from (1) we have,
8.75 x-3.5 x = 3143-1610 = 1533
5.25 x= 1533
x= 1533/5.25 = 292
y= 460-292= 168
So the number of adult and child tickets sold were 292 and 168 respectively.
Assign variables to each parameter:
x = # of adult tickets sold
y = # of children tickets sold
Since we are given that the total # of tickets sold is 460, then we can arrive at the following equation:
x + y = 460
We are given that the adult tickets were sold for $8.75 per ticket and the children tickets were sold for $3.50 per ticket. With this, we can get the total amount made from the adult tickets sold by multiplying the # of adult tickets sold (x) by the cost of the adult tickets ($8.75). Likewise, the total amount made from the children tickets sold is determined by multiplying the # of children tickets sold (y) by the cost of each children ticket ($3.50). Adding these 2 quantities yield the total amount made from the total tickets sold, which were are given to be $3143. That is,
8.75x + 3.50y = 3143
This generates a system of linear equations, which contain the two equations we found above:
8.75x + 3.50y = 3143
x + y = 460
Since the question only asks you to find how many adult tickets were sold, we only need to solve for x. We can do so by either one of two methods, those being the elimination method and the substitution method. I find the elimination method to be simpler, so I will solve for the # of adult tickets sold (x) by eliminating the other variable which is the # of children tickets sold (y). To eliminate y, multiply the second equation above by -3.50 then add this equation to the first equation.
-3.50(x + y = 460) ==> -3.50·x + -3.50·y = -3.50·460
==> -3.50x - 3.50y = -1610
8.75x + 3.50y = 3143
+ -3.50x - 3.50y = -1610
________________________
8.75x - 3.50x + 3.50y - 3.50y = 3143 - 1610
5.25x = 1533
5.25x/5.25 = 1533/5.25
x = 292
Thus, 292 adult tickets were sold.
x = # of adult tickets sold out
460-x = #of children's tickets sold out
8.75x + 3.50(460-x) = 3143
Solve for x,
x = 292 tickets <==answer