10000 kg of wheat with a moisture content of 13% (db) is required. The freshly harvested crop is having a moisture content of 20% (wb). How many kg of wheat are to be procured?

Steps to Solve

1. Identify the Initial and Target Moisture Content

The initial moisture content of the wheat is 30% (wb), and the target moisture content is 13% (wb).

2. Calculate the Dry Basis Moisture Content

To convert moisture content from wet basis (wb) to dry basis (db), we can use the formula:

$$ \text{Moisture content (db)} = \frac{\text{Moisture content (wb)}}{1 - \text{Moisture content (wb)}} $$

For the initial moisture content:

$$ \text{Initial db} = \frac{0.30}{1 - 0.30} = \frac{0.30}{0.70} \approx 0.4286 $$

For the target moisture content:

$$ \text{Target db} = \frac{0.13}{1 - 0.13} = \frac{0.13}{0.87} \approx 0.14943 $$

3. Determine the Mass of Dry Wheat

Let ( m ) be the mass of dry wheat in kilograms. The wet mass of wheat (given) is 1000 kg. To find the dry mass:

$$ \text{Dry mass} = \text{Wet mass} \times (1 - \text{Moisture content (wb)}) $$

Using 1000 kg at 30% moisture content:

$$ \text{Dry mass} = 1000 \times (1 - 0.30) = 1000 \times 0.70 = 700 \text{ kg} $$

4. Calculate the Wet Mass Required for Target Moisture Content

To find how much wet wheat is needed to achieve the target moisture content of 13%:

Let ( W ) be the wet mass required. Using the dry mass we calculated:

$$ \text{Dry mass} = W \times (1 - \text{Target moisture content (wb)}) $$

Rearranging gives:

$$ W = \frac{\text{Dry mass}}{(1 - \text{Target moisture content (wb)})} $$

Substituting the values:

$$ W = \frac{700}{1 - 0.13} = \frac{700}{0.87} \approx 804.6 \text{ kg} $$

5. Calculate the Amount of Additional Wheat Required

To find out how much more wheat is needed beyond the initially available 1000 kg to achieve the required moisture content:

$$ \text{Additional wheat needed} = W - 1000 $$

Using the wet mass calculated:

$$ \text{Additional wheat needed} = 804.6 - 1000 \approx -195.4 \text{ kg} $$

Since the solution is negative, we don't need additional wheat; instead, we have plenty available.