Problem 10 (Orange Juice Problem)

We have four orange juice mixes with concentrations A₁, A₂, A₃, and A₄ gal orange juice/gal mix and know that we get a concentration of

1. 0.15 gal orange juice/gal mix if we mix 3 gallons of the first mix, 2 gal of the second mix, and 1 gal of the third mix

2. 0.10 gal of orange juice/gal mix if we mix one gallon each of the four mixes

3. 0.10 gal orange juice/gal mix if we mix 1 gallon of the first mix and 1 gallon of the third mix

How much orange juice/gal solution would we get if we add 2 gal of the first mix and 1 gal of the fourth mix ?

Solution

Condition 1 says if we add 3, 2, and 1 gal of orange juice mixes of respective concentrations A₁, A₂, and A₃ gal orange juice/gal mix, then the concentration of the resulting 3 + 2 + 1 = 6 gallons of mix will be 0.15 gal orange juice/gal mix. Hence, we have

3A₁ + 2A₂ + A₃ = 0.15 (3 + 2 + 1) = 0.90 gal orange juice

Likewise, by adding a gallon of each of the four concentrations gives 0.10 gal orange juice/gal mix, and hence

A₁ + A₂ + A₃ + A₄ = 0.10 (1 + 1 + 1 + 1) = 0.40 gal orange juice

And finally by adding one gallon of the first concentration and one gallon of the third concentration gives 0.10 gal orange juice/gal mix, and hence

A₁ + A₃ = 0.10 (1 + 1) = 0.20 gal orange juice

The three previous equations and constitute an underdetermined system of three linear equations with four unknowns A₁, A₂, A₃, and A₄. We solve the three variables A₁, A₂, A₃ in terms of the fourth unknown A₄, which for convenience we call A₄ = k. Using the standard gaussian elimination method for solving linear systems, we find the solutions

A1	= 0.15 + k
A2	= 0.20 - k
A3	= 0.05 - k
A4	= k

where k is an arbitrary real number between 0 and 0.05 (We must restrict k to these values in order to keep all the concentrations nonnegative). Hence, there are many concentrations that satisfy the three given conditions.

Hence, the amount of orange juice in the mix when we mix 2 gal of A₂ and one gal of A₄ cannot be determine exactly from the given data, but we know it is one of the values

2A₁ + A₄ = 0.30 + 3k

And since we have a total of 3 gal of the mix, this means the concentration is one of the values

CONCENTRATION = (0.30 + 3k)/3 = 0.10 + k

gal orange juice/gal solution where k is any number between 0 and 0.05. In other words, the information given does not allow us to specify the concentration of 2A₁ + A₄ any more than saying it is between 0.10 and 0.15 gal orange juice/gal mix.

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Last modified on Wednesday, March 17, 1999