Question

A tea wholesaler blends together three types of tea that normally sell for $10, $11 and
$12 per kilogram so as to obtain 100 kilograms of tea worth $11.20 per kilogram. If the
same amounts of the two higher priced teas are used, calculate how much of each type
must be used in the blend.

Answer 1

The wholesaler must blend a 40 % of a type of tea with a price of $ 11 per kilogram, 40 % of a type of tea with a price of $ 12 per kilogram and 20 % of type of tea with a price of $ 10 per kilogram to obtain a kilogram with a unit price of $ 11.20.

Procedure

Blend model

In this question we must use the concept of weighted averages to determine the proportion of each ingredient in the blend, the cost equation ($C$) is represented below:

$C = x_{1}\cdot C_{1} + x_{2}\cdot C_{2} + x_{3}\cdot C_{3}$ (1)

Where:

• $x_{1}, x_{2}, x_{3}$ - Proportion of each ingredient.
• $C_{1}$, $C_{2}$, $C_{3}$ - Unit cost of each ingredient, in monetary units per kilogram.

Blend conditions

If we know that $x_{1} = x_{2} = x$, $x_{3} = 1-2\cdot x$, $C_{1} = 11$, $C_{2} = 12$, $C_{3} = 10$ and $C = 11.20$, then the proportions for each ingredient are:

$11.20 = 11\cdot x + 12\cdot x + (1-2\cdot x)\cdot 10$

$11.20 = 23\cdot x + 10-20\cdot x$

$1.20 = 3\cdot x$

$x = 0.40$

Hence, we have the following proportions: $x_{1} = 0.40$, $x_{2} = 0.40$ and $x_{3} = 0.20$.

The wholesaler must blend a 40 % of a type of tea with a price of $ 11 per kilogram, 40 % of a type of tea with a price of $ 12 per kilogram and 20 % of type of tea with a price of $ 10 per kilogram to obtain a kilogram with a unit price of $ 11.20.

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