Question

Reba works at a starbucks® coffee shop where a 12-oz cup of coffee costs $1.85, a 16-oz cup costs $2.10, and a 20-oz cup costs $2.45. during one busy period, reba served 55 cups of coffee, emptying six 144-oz "brewers" while collecting a total of $115.80. how many cups of each size did reba fill?

Answer 1

Let's start with naming our sizes of coffee first off. Let's call the 12 oz "x", the 16 oz "y", and the 20 oz "z". We are looking for the number of cups of coffee Reba served on this particular day. Since we have 3 unknowns we need a system of 3 equations. The first equation will relate the NUMBER of cups of coffee, the second equation the OUNCES of coffee, and the third equation the COST of each. It says that the number of cups of coffee she served that day was 55, a combination of 12, 16, and 20. That means that our first equation is x + y + z =55. The second equation relates the number of ounces in each. The 12 ounce then would be represented as 12x, the 16 ounce as 16y, and the 20 ounce as 20z. It says that Reba emptied 6 144-ounce coffee makers to make her coffee. 6 * 144 = 864. So the second equation is $12x+16y+20z=864$. The last equation relates the cost of each to the money earned in total. The cost of the 12 oz is 1.85, so we represent it as 1.85x; the cost of the 16 oz is 2.10 so we represent it as 2.1y, and the cost of the 20 ounce is 2.45, represented by 2.45z. The total cost of these sales was 115.80. That means that our third equation is $1.85x+2.1y+2.45z=115.80$. Let's start by solving the first of these 3 equations for x: $x=55-y-z$. We can now sub that value for x into the next 2 equations to get rid of one of the variables. Into the second equation we have $12(55-y-z)+16y+20z=864$. Distribute through the parenthesis to get $660-12y-12z+16y+20z=864$. Combining like terms we have 4y+8z=204. Next let's sub into the last equation to get $1.85(55-y-z)+2.1y+2.45z=115.80$. Distributing through the parenthesis we have $101.75-1.85y-1.85z+2.1y+2.45z=115.80$. Combining like terms we have .25y+.6z=14.05. Those 2 bolded equations are now in terms of y and z only and we can solve them by substitution, just like we did for the x above. Solve the first bolded equation for y: $4y=204-8z$ and $y=51-2z$. Sub that value into the second bolded equation to solve for z: $.25(51-2z)+.6z=14.05$. Distributing through the parenthesis gives us $12.75-.5z+.6z=14.05$. Combing like terms to get $.1z=1.3$ and z=13. That means that Reba sold 13 20 ounce coffees that day. Now let's sub that z value into one of the bolded equations to solve for y. If $y=51-2z$ then $y=51-2(13)$ and $y=51-26$. That means that y=25. Reba sold 25 16 ounce coffees that day. Let's go all the way back to the beginning where we learned that Reba sold a total of 55 cups of coffee. That means that x + 25 + 13 = 55 so x = 17. There you go!