How to solve this?

Question

Chas and Ben stop to get funnel cakes and their costs follow a linear model. Chas buys a 3 topping funnel cake that costs $10.85 while Ben can't resist a 7 topping funnel cake that cost $13.65. How much does each topping cost?

Answer 1

The heart of this problem lies in a system of equations. The easiest way to begin to take this word problem and get rid of the words, keep the numbers. Firstly, ask yourself "what am I looking for?" We are looking for the each topping cost, so as with any algebra problem, lets give topping costs a name, "t" sounds good to me. Now we know that we are looking for "t" let's pivot to the second question, "What do I know?" in the problem we can see that a 3 topping cake is 10.85 and a 7 topping cake is 13.65. The cost of either cake can be broken down into 2 parts, the cost of the toppings and the cost of the cake. Well if we remember earlier we said that "t" represented the cost of the toppings, so now we need to name the cost of the cake, "c" will do. We can put this together by saying that the total cost of a funnel cake = cost of toppings + cost of cake. now we have a lot of letters, we need some numbers. We know that there are 2 cakes and so there will be two equations, one for Chas's cake and one for Ben's cake. Chas got bought 1, 3 topping cake for 10.85 So that equation looks something like 10.85 = 3t + 1c. 3 t because there are 3 toppings and 1 c because there is only one cake. Using the same logic we can say the equation for Ben's cake looks like 13.65 = 7t + 1c. So now we have our system, lets do some math. Recall that we are looking for the value of t and Notice that both equations have 1c. There are a few ways to solve this problem but the easiest way is to use a strategy called elimination. Just like we can add and subtract numbers, we can add and subtract equations. The goal of elimination is to add or subtract the equations to get rid of one of the variables. We noticed that both equations had 1c, also notice that 1c-1c = 0. it doesn't matter what c costs we know that statement will always be true. To use elimination the easiest way to write the equations on top of each other like you would if you were subtracting big numbers.

13.65 = 7t + c

10.85 = 3t + c

and just like we subtract big numbers, we are going to subtract the terms of the equation. When we are done we are going to have a new MUCH simpler equation.

subtracting the terms one at a time we get,

13.65-10.85 = 2.80

7t-3t = 4t

c-c = 0

We take these numbers and make a new equation putting all the numbers into the new equation back in the same place we took them from

This equation is

2.80 = 4t + 0 or 2.80 = 4t

Now we have a simple equation and divide by 4 to find that

t = .70

In the context of the problem we can say that

the cost of a topping = .70 dollars

Finally we can say that the cost of a topping is 70 cents!

Answer 2

The price of a cake is so much per topping, t, plus a base overhead cost, b.

3t+b = 10.85

7t+b = 18.65

Subtracting the first equation from the second to eliminate the base cost, b:

4t = 2.8

t = 0.70

3(0.70) +b = 10.85.

b = 10.85-2.10

So each topping costs 70 cents and the overhead cost per cake is $8.75.