For math genuises?

Question

Solve this problem with 3 variables, and 3 equations:
The Charity Ball Committee sold patron, sponsor, and donor tickets for $10,$5,and $2.50. respectively. The number of donor tickets sold was 24 more than the number of sponsor and patron combined. A total of 326 tickets were sold raising $1432.50 for the charity. How many of each type of ticket were sold?

Answer 1

Let x be the number of patron tickets sold;
     y be the number of sponsor tickets sold; and
     z be the number of donor tickets sold.

From the given problem, we obtain the following system of linear equations with three (3) unknowns as follows:
[1]   x + y + z = 326
[2]   10x + 5y + 2.5z = 1432.5
[3]   z = 24 + x + y

First, we plug equation [3] to equation [1] to obtain the following:
      x + y + (24 + x + y) = 326
      2x + 2y + 24 = 326
      2x + 2y = 302
[4]     x + y = 151 by dividing both sides by 2

Next, we plug equation [3] to equation [2] to obtain the following:
      10x + 5y + 2.5(24 + x + y) = 1432.5
      10x + 5y + 60 + 2.5x + 2.5y = 1432.5
      12.5x + 7.5y = 1432.5 – 60
      12.5x + 7.5y = 1372.5
[5]     5x + 3y = 549 by dividing both sides by 2.5

Observe that from [4] and [5], we get the following system of linear equation with two unknowns:
[4] x + y = 151
[5] 5x + 3y = 549

Now from equation [4], we have an information for x, which is expressed as
[6] x = 151 – y.

If we plug this (equation [6]) to equation [5], we can obtain the value of y. That is
           5(151 – y) + 3y = 549
           755 – 5y + 3y = 549
           – 2y = 549 – 755
           – 2y = – 206
[7]      y = 103.

Going back to equation [6], we can solve for x. That is
          x = 151 – 103
[8]      x = 48.

Finally, going back to equation [3], we can find the value of z in terms of x (in equation [8]) and y (in equation [7]) as follows:
          z = 24 + 48 + 103
[9]           z = 175

Indeed, there were 48 patron tickets, 103 sponsor tickets and 175 donor tickets sold by the Charity Ball Committee.

----

To check, we have the following:

x + y + z ?= 326
48 + 103 + 175 ?= 326
326 = 326 (ok!)

20x + 10y + 5z ?= 2865
20(48) + 10(103) + 5(175) ?= 2865
960 + 1030 + 875 ?= 2865
2865 = 2865 (ok!)

z ?= 24 + x + y
175 ?= 24 + 48 + 103
175 = 175 (ok!)

Answer 2

let # of patron sold--> a
# of sponsor sold --> b
# of donor sold--> c

c = a+b+24 ------ equation# 1
a + b + c = 326

replace c from equation # 1

a + b + a + b + 24 = 326
2*a + 2*b = 302
by division per 2

a + b = 151 --> b = 151-a -------equation # 2

10*a + 5*b + 2.5*c = 1432.5
10*a + 5*b + 2.5*(a+b+24) = 1432.5
10*a + 5*b + 2.5*a + 2.5*b + 60 = 1432.5
12.5*a + 7.5*b = 1372.5
times by 2:
25*a + 15*b = 2745
divide per 5
5*a + 3*b = 549

from equation #2:
5*a + 3*(151-a) = 549
5*a + 453 - 3*a = 549
2*a = 96

SO a = 48;
b = 151-96 = 103
c = 48 + 103 + 24 = 175

Answer 3

First define the variables.

x = Patron tickets Sold, y = Sponsor, z = Donor.

then put in the details.

Z + 24 = X +Y

X + Y + Z = 326

10X + 5Y +2.5Z = 1432.5

Thats a system of equations. When you work all that mess out, you get:

X = 36 (Patron Tickets Sold)

Y = 139 (Sponsor Tickets Sold)

Z = 151 (Donor Tickets Sold)

Hope I could help.

Answer 4

p + s + d = 326
10p + 5s + 2.5d = 1432.50
d = p + s + 24

Rewriting the third equation, you get:
-p - s + d = 24
Add the first equation to this equation:
-p - s + d = 24
+p + s + d = 326
-----------------------
2d = 350
d = 175

Plugging 175 in for d, you'll find that equations 1 and three become the same:
p + s = 151

The second equation is now:
10p + 5s = 995, which, if you divide both sides by 5 becomes:
2p + s = 199

Subtract the first equation from the second to get:
2p + s = 199
p + s = 151
------------------
p = 48

Since p = 48, s must = 103.

And you can check your (er, my) work by plugging the numbers back into to the dollar figure equation:
48 ($10) + 103 ($5) + 175 ($2.50) = $1432.50
$480 + $515 + $437.50 = $1432.50

Answer 5

gaussian elimination

we have the followng variables:
p=number of patrons tickets
s=number of sponsor tickets
d=number of donor tickets

from the information given,we have
the following equations:
10p+5s+2.5d=1432.5..divide by 10
p+1/2s+1/4d=143 1/4.......(1)
p+s+d =326.............(2)
p+s-d = -24.............(3)

the above set of equations give the
following augmented matrix:

{1 1/2 1/4 * 143 1/4} r1
{1 1 1 * 326 } r2
{1 1 -1 * -24 } r3

next we operate on the matrix

on r2- (r2-r1)
on r3- (r3-r2),then divide by -2

we now have the upper right
triangular matrix:-

{ 1 1/2 1/4 * 143 1/4 } r1
{ 0 1/2 3/4 * 182 3/4 } r2
{ 0 0 1 * 175 } r3

from r3, read off d=175
from r2, s=103
from r1, p=48

hence 48 patrons tks were sold,
103 sponsor tks were sold
and 175 donor tks were sold

i hope that this helps

Answer 6

Patron = p Sponsor = S24x2.50=60
Tickets = T
10P+5S+2.5T=1432.50
P=10
S=5
T=2.50

T>24 = s+p

T=s+p+24

Total Tickets = 326 = $1432.50

How much each for T, S and P

Answer 7

set the variables:

p = number of patron tickets
s = numer of sponsor tickets
d = number of donor tickets

create equations from the info given:

24 + s + p = d

326 = s + p + d

1432.5 = 10p + 5s + 2.5d

combine the equations to solve individually for each variable:

24+s+p=d
326=s+p+d
so...
326=s+p+24+s+p
326=2s+2p+24
302=2s+2p
151=s+p


24+s+p=d
1432.5=10p+5s+2.5d
so...
1432.5=10p+5s+2.5(24+s+p)
1432.5=10p+5s+60+2.5s+2.5p
1372.5=12.5p+7.5s

151=s+p
or
p=151-s
1372.5=12.5p+7.5s
so...
1372.5=12.5(151-s)+7.5s
1372.5=1887.5-12.5s+7.5s
5s=515
s=103***

plug s into
151=s+p
so...
151=103+p
p=48***

plug s and p into
1432.5=10p+5s+2.5d
1432.5=10(48)+5(103)+2.5d
1432.5=480+515+2.5d
2.5d=437.5
d=175***

number of tickets sold:

175 donor
103 sponsor
48 patron

Answer 8

A = Patron tickets @10
B = Sponsor tickets @ 5
C= Donor tickets @ 2.5
Ticket Number 326= a +b+c and
c=(a+b+24)

Monies Recieved
1432.5=10a+5b+2.5c

solve for A in Ticket numbers
326=2a+2b+24
326-24=2a+2b+24-24
302=2a+2b ..or..2a+2b=302
2a+2b-2b=302-2b
2a=302-2b....divide by two
a=151-b

now substitute into monies equation
1432.5=10a+5b+2.5(a+b+24)
1432.5=10(151-b)+5b+2.5(151-b+b+24)
simplify
1432.5=10(151-b)+5b+2.5(176)
1432.5=1510-10b+5b+437.5
1432.5=1947.5-5b
1432.5-1432+5b=1947.5-1432.5-5b+5b
5b=515
b=103

a=151-b=151-103
a=48

c=a+b+24
c=48+103+24=175

viola !

48 @ $10......480
103@ $5 ......515
175@$2.5.....437.5
total tickets $1,432.5

so where do we go for pizza ?????

Answer 9

d=s+p+24
d+s+p=326
2d=350
d=175
s+p=151
175times 2.50 = 437.50
1432.50-437.50= 995
5s+10p=995
s+p=151
s+2p=199
p=48
s=103
(d=175)

check
48*10 = 480
103*5= 515
175+2.50=437.50
total 1432.50

then place the details