Question

A solution needs to contain between 46% glucose and 50% glucose. Find the least and greatest amount of a 60% glucose solution that should be mixed with a 40% glucose solution in order to

end up with 30 kilograms of a solution containing the allowable percentage of glucose.

Answer 1

• least: 9 kg
• greatest: 15 kg

Explanation:

You want the range of amounts of 60% solution that can be added to a 40% solution to achieve 30 kg of a solution that lies in the range of 46% to 50%.

Setup

Let x represent the mass in kg of 60% solution added. The fraction of the total solution that is glucose will be ...

0.46 ≤ (0.60x +0.40(30 -x))/30 ≤ 0.50

Solution

Multiplying by 30 and simplifying the inequality, we have ...

13.8 ≤ 0.20x +12 ≤ 15

1.8 ≤ 0.2x ≤ 3 . . . . . . . . subtract 12

9 ≤ x ≤ 15 . . . . . . . . . . divide by 0.2

The least amount of 60% solution that should be added is 9 kg. The greatest amount is 15 kg.

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Check

9 kg of 60% +21 kg of 40% has 5.4 +8.4 = 13.8 kg of glucose, the minimum.

15 kg of 60% +15 kg of 40% has 9 +6 = 15 kg of glucose, the maximum.

(The minimum and maximum values are seen in the solutions steps after the initial multiplication by 30.)