Question

Kyle works at a donut​ factory, where a​ 10-oz cup of coffee costs 95¢​, a​ 14-oz cup costs​ $1.15, and a​ 20-oz cup costs​ $1.50. During one busy​ period, Kyle served 29 cups of​ coffee, using 444 ounces of​ coffee, while collecting a total of ​$35.90. How many cups of each size did Kyle​ fill?
Kyle filled ___ 10-oz cup(s), ___ 14-oz cup(s), and ___ ​20-oz cup(s).

Answer 1

Answer:

Kyle filled 10 10-oz cups, 6 14-oz cups, and 13 20-oz cups

Step-by-step explanation:

In order to solve this question we first have to create a system of equations. If we let "a" represent the number of 10-oz cups, let "b" represent the number of 14-oz cups, and let "c" represent the number of 20-oz cups, then we can make the following 3 equations...

1.   a + b + c = 29                                      (29 is the total number of cups)

2.  10a + 14b + 20c = 444                       (444 is the total volume of the coffee)

3.  0.95a + 1.15b + 1.50c = 35.90           (35.90 is the total price of the coffee)

Now we can solve this system of equation by first representing "a" in terms of "b" and "c". We use the first equation, to do that...

a + b + c = 29

a = 29 - b - c

Now we substitute "a" with an equal value in terms of "b" and "c" which we found above.

10a + 14b + 20c = 444

10 (29 - b - c) + 14b + 20c = 444

290 - 10b - 10c + 14b + 20c = 444

4b + 10c = 154

Now we use the fully simplified equation above to represent "c" in terms of "b", and we get that...

4b + 10c = 154

10c = 154 - 4b

c = 15.4 - 0.4b

Now we take the third equation and substitute "a" and "c" with equivalent values, and we get an equation from which we can find the value of "b"...

0.95a + 1.15b + 1.50c = 35.90

0.95 (29 - b - c) + 1.15b + 1.50(15.4 - 0.4b) = 35.90

27.55 - 0.95b - 0.95c + 1.15b + 23.1 - 0.6b = 35.90

50.65 - 0.4b - 0.95(15.4 - 0.4b) = 35.90

50.65 - 0.4b - 14.63 + 0.38b = 35.90

36.02 - 0.02b = 35.90

0.02b = 36.02 - 35.90

0.02b = 0.12

b = 6

Now that we know what "b" equals to, we can find the value of "c".

c = 15.4 - 0.4b

(we found this formula when we represented c in terms of "b" previously)

c = 15.4 - 0.4(6)

c = 15.4 - 2.4

c = 13

Now we can find the value of "a" buy using the first formula and substituting "b" and "c" with their values...

a + b + c = 29

a + 6 + 13 = 29

a = 29 - 6 - 13

a = 10

Therefore Kyle filled 10 10-oz cups, 6 14-oz cups, and 13 20-oz cups.