Super-Overpriced-Candy comes in two flavors, Cherry for $12 per pound and Grape

Question

Super-Overpriced-Candy comes in two flavors, Cherry for $12 per pound and Grape for $16 per pound. Frank buys a mix which has 3 times as much Cherry as Grape (by weight), while Karen buys a mix which is 60% Grape (also by weight).

(a) Find the average cost per pound of Frank’s order and of Karen’s order. Express both as weighted means of 12 and 16.

(b) Suppose that Frank’s and Karen’s orders will be combined at a party. If Frank bought twice as many pounds as Karen, what is the average cost per pound for the whole pile of candy?

(c) Express your answer to (b) as (i) a weighted mean of 12 and 16, and (ii) a weighted mean of your answers to (a). (For the first part, suppose Frank buys x pounds of grape; figure how many pounds of cherry he buys and how many pounds of each Karen buys in terms of x.)

(d) Now suppose Bart buys $20 worth of Cherry and $10 worth of Grape; while Lisa buys $6 worth of Cherry and $4 worth of Grape. Find the average cost per pound of Bart’s order and of Lisa’s order. Express both as weighted means of 12 and 16.

Any help would be much appreciated. Thank you!

Explanation / Answer

Cherry C= $12 per pound

Grape G= $16 per pound

Frank Ratio for cherry and grape is 3:1

Karen Ration for cherry and grape is 2: 3

(a) average cost per pound of Frank’s order F= 3C + 1G/ (3+1 ) = (3C + G)/ 4 = (3 *12 + 1 * 16)/ 4 = 13 per pound

average cost per pound of karen’s order K= 2C + 3G/ (2+3 ) = (2C + 3G)/5 = (2 *12 + 3* 16)/ 5 = 14.4 per pound

(b)  Suppose that Frank’s and Karen’s orders will be combined at a party. If Frank bought twice as many pounds as Karen, what is the average cost per pound for the whole pile of candy?

Answer: Ratio of weights of Frank's and Karen's orders is = 2:1

Let say Frank order weight is F and Karan order weight is K

so average cost per pound for the whole pile of candy = 2F + K / (2+1) = (2F + K) / 3

average cost per pound for the whole pile of candy = (2 * 13 + 14.2)/3 = 13.4 per pound

(c) Express your answer to (b) as (i) a weighted mean of 12 and 16, and (ii) a weighted mean of your answers to (a). (For the first part, suppose Frank buys x pounds of grape; figure how many pounds of cherry he buys and how many pounds of each Karen buys in terms of x.)

(i) a weighted mean of 12 and 16

Combined order price = [2 *(3C + G)/ 4 + 1 * (2C + 3G)/5]/ 3 = (19C + 11G)/30

(ii) a weighted mean of your answers to (a)

Combined order price = [2 *F + K]/ 3 = (2F+K)/3

(d) Now suppose Bart buys $20 worth of Cherry and $10 worth of Grape; while Lisa buys $6 worth of Cherry and $4 worth of Grape. Find the average cost per pound of Bart’s order and of Lisa’s order. Express both as weighted means of 12 and 16.

Answer: For Bart :

total amount of cherry he bough = 20/ C

and total amount of grape he bought = 10/G

[ Where C = $12 per pound and G = $ 16 per pound

average cost per pound of Bart’s order = Total Cost / Total Pound = ($ 20 + $ 10)/ [ 20/C + 10 /G]

average cost per pound of Bart’s order = 3CG/ ( C + 2G)

by calculation = 3 * 12 * 16 / (12 + 2 * 16) = $13.09 per pound

For Lisa :

total amount of cherry he bough = 6/ C

and total amount of grape he bought = 4/G

[ Where C = $12 per pound and G = $ 16 per pound]

average cost per pound of Bart’s order = Total Cost / Total Pound = ($6 + $4)/ [ 6/C + 4 /G]

average cost per pound of Bart’s order = 5 CG/ (2G + 3C)

by calculation = 5 * 12 * 16 / (12 *2 + 3 * 16) = $13.33 per pound

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