How many pounds of each coffee should be used in a blend that sells for ​$12.8212.82 per​ lb?

Question

Three varieties of coffee long dash— Coffee A, Coffee B, and Coffee C long dash— are combined and roasted, yielding a 5050-lb batch of coffee beans. Twice as many pounds of Coffee C, which retails for $12.7612.76 per lb, are needed as Coffee A, which sells for $15.9915.99 per lb. Coffee B retails for $12.3112.31 per lb. 

Answer 1

Let A, B and C be the weights of the coffees.

We have three equations.  The first two are straightforward.

(1)  A+B+C = 50

(2)  C = 2A

Substitute 2A for C in (1)

(1)  A+B+2A = 3A+B = 50

(4)  B = 50-3A   (We'll use this for substitution below.)

For the third equation, you can calculate the total retail value of the batch.  Divide that by 50 and you get the retail value per pound.

(3)  (15.99A+12.31B+12.76C) / 50 = 12.82

(3)  0.3198A+0.2462B+0.2552C = 12.82

Use equations (4) and (2) to substitute for B and C.

(3)  0.3198A+0.2462(50-3A)+0.2552(2A) =12.82

Solve for A

0.3198A+12.31-0.7386A+0.5104A = 12.82

0.0916A = 12.82-12.31 = 0.51

A = 5.568

Now use equations (4) and (2) to find B and C

(4)  B = 50-3A = 50-(3)(5.568)

B = 33.297

(2)  C = 2A = (2)(5.568)

C = 11.136

 

You can check the solutions by plugging the values into equations (1) and (3).

I did, and they agree with a small round-off error.