At a bakery. Andy bought 12 cupcakes while Ben bought 20 buns. The price of 3 cupcakes is the same as the price of 4 buns. Andy, Ben and Carl agreed to share the cost of the cupcakes and buns equally among themselves. Carl paid Andy and Ben total of $23.40 for his share.
(a) How many cupcakes can Ben buy with the amount of money he spent on the 20 buns?
(b) How much did Carl pay Andy?
Let the price of one cupcake be $c and the price of one bun be $b. We are given that:
3c = 4b …(1)
To find the cost of each cupcake and each bun, we need one more equation. Let the total cost of the cupcakes be Cc and the total cost of the buns be Cb. Then we have:
Cc + Cb = (12)(c) + (20)(b)
Now we need to express Cc and Cb in terms of c and b using equation (1):
Cc = (4/3)Cb
Substituting this in the previous equation, we get:
(4/3)Cb + Cb = (12)(c) + (20)(b)
Simplifying, we get:
Cb = (60/7)b
Substituting this in equation (1), we get:
c = (4/7)b
Now we can answer the questions:
(a) Ben spent the same amount on 20 buns as Andy spent on 12 cupcakes. Let's call this common amount X. Then we have:
20b = 12c
Substituting c = (4/7)b, we get:
20b = 12(4/7)b
Solving for b, we get:
b = $1.05
Substituting this in 20b, we get:
Ben spent 20($1.05) = $21 on the buns.
The cost of one cupcake is c = (4/7)b = $0.60.
Therefore, Ben could buy $21/$0.6 = 35 cupcakes with the same amount of money.
(b) Carl paid $23.40 for his share of the cost, which is (1/3) of the total cost. Therefore, the total cost of the cupcakes and buns was:
3($23.40) = $70.20
Let's call Andy's share of the cost A and Ben's share B. Then we have:
A + B + 23.40 = 70.20
A + B = 46.80
We know that Andy bought 12 cupcakes, so his share of the cost is:
Cc = 12c = 12($0.60) = $7.20
Substituting B = 46.80 - A in the previous equation, we get:
A + (46.80 - A) = 7.20 + B
Simplifying, we get:
B - A = 39.60
We also know that Ben spent $21 on the buns, so his share of the cost is:
Cb = 20($1.05) = $21
Substituting Cb = (60/7)b in the previous equation, we get:
A = (4/3)Cb = (4/3)($21) = $28
Therefore, Carl paid Andy $28 for his share.
Let's solve the problem step by step:
Step 1: Determine the price of one cupcake and one bun.
Let's assume the price of one cupcake is 'c' and the price of one bun is 'b'.
Step 2: Set up equations based on the given information.
From the problem, we know that the price of 3 cupcakes is the same as the price of 4 buns. So we can set up the equation:
3c = 4b
Step 3: Calculate the ratio of cupcakes to buns.
From the equation above, we can rearrange it to find the ratio of cupcakes to buns:
c/b = 4/3
Step 4: Calculate the cost of one cupcake.
Since Andy bought 12 cupcakes, the cost of 12 cupcakes would be:
12c
Step 5: Calculate the cost of one bun.
Since Ben bought 20 buns, the cost of 20 buns would be:
20b
Step 6: Calculate the total cost of cupcakes and buns.
The total cost of cupcakes and buns would be:
12c + 20b
Step 7: Calculate the amount Carl paid.
We know that Carl paid a total of $23.40 for his share, which is equal to 12c + 20b. So we can set up the equation:
12c + 20b = 23.40
Step 8: Solve the equations simultaneously.
Now we have two equations:
3c = 4b
12c + 20b = 23.40
We can rearrange the first equation:
c = (4/3)b
Substitute this into the second equation:
12((4/3)b) + 20b = 23.40
Step 9: Solve for b.
Multiply through by 3 to get rid of the fraction:
16b + 60b = 70.20
76b = 70.20
b = 70.20/76
b ≈ 0.922
Step 10: Calculate the price of a cupcake.
Substitute the value of b into the first equation:
c = (4/3) * 0.922
c ≈ 1.23
(a) How many cupcakes can Ben buy with the amount of money he spent on the 20 buns?
The amount Ben spent on buns is 20b ≈ 20 * 0.922 = $18.44.
So, Ben can buy 18.44/1.23 ≈ 15 cupcakes.
(b) How much did Carl pay Andy?
Carl paid a total of $23.40. Since the cost of each cupcake is 1.23, Carl bought 23.40/1.23 ≈ 19 cupcakes.
If Carl bought 19 cupcakes and Andy bought 12 cupcakes, then Carl paid Andy for 19 - 12 = 7 cupcakes.
The cost of 7 cupcakes is 7 * 1.23 = $8.61.
Therefore, Carl paid Andy $8.61.