Question

At Carl's Diner ,there are 3 sizes of coffee drinks: regular (300 ml), large(500ml) and
extra - large (800ml), and they cost Rs. 5. Rs.8 and Rs.12 respectively. One evening
the diner sold a total of 37 coffees. The total volume of coffee sold was 19,100ml and the
amount of money made in coffee sales was Rs. 300. How many, of each size of drink,
did customers buy that evening? (Using Cramer's rule)​

Answer 1

Given : At Carl's Diner ,there are 3 sizes of coffee drinks: regular (300 ml), large(500ml) and  extra - large (800ml), and they cost Rs. 5. Rs.8 and Rs.12 respectively.

One evening  the diner sold a total of 37 coffees The total volume of coffee sold was 19,100ml and the  amount of money made in coffee sales was Rs. 300.

 

To Find : How many, of each size of drink,  did customers buy that evening

Solution:

x = 300 ml cup

y = 500 ml cup

z = 800 ml cup

  x  + y  + z   =  37

300x + 500y + 800z  = 19100

=> 3x + 5y + 8z  = 191

   5x + 8y + 12z  =  300

3x  + 3y  + 3z   = 111

3x + 5y + 8z  = 191

=> 2y + 5z  = 80

5x + 8y + 12z  =  300

   5x  + 5y  + 5z   =  185

=> 3y + 7z = 115

2y + 5z  = 80   => 6y + 15z = 240

3y + 7z = 115   => 6y  + 14z  = 230

=> z = 10

2y + 5z  = 80

=> 2y + 50 = 80

=> 2y = 30

=> y = 15

x  + y  + z   =  37

=> x + 15 + 10 = 37

=> x = 12

12 cups of 300 ml

15 cups of  500 ml

10 cups of 800  ml

Cramer rule

 x  + y  + z   =  37  

 3x + 5y + 8z  = 191

  5x + 8y + 12z  =  300

 $\left|\begin{array}{ccc}1&1&1\\3&5&8\\5&8&12\end{array}\right|=-1$

$\left|\begin{array}{ccc}37&1&1\\191&5&8\\300&8&12\end{array}\right|=-12$

=> x = -12/-1  = 12

$\left|\begin{array}{ccc}1&37&1\\3&191&8\\5&300&12\end{array}\right|=-15$

=> y = -15/-1 = 15

$\left|\begin{array}{ccc}1&1&37\\3&5&191\\5&8&300\end{array}\right|=-10$

=> z = -10/-1 = 10

12 cups of 300 ml

15 cups of  500 ml

10 cups of 800  ml

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