Question

The characteristics of two types of solid waste are given in the table.

Type of solid waste	Moisture content (%)	Nitrogen content (%)	Carbon to Nitrogen (C/N) ratio
Leaves	50	0.7	50
Sludge	75	5.6	6.3

The quantity of sludge to be added to leaves to achieve a C/N ratio of 25 (on dry basis) for the mixed waste is ______ kg of sludge/kg of leaves (rounded off to three decimal places).

Hint:

Convert both wastes to their dry mass, work out the nitrogen mass and then the carbon mass (using C/N ratio × nitrogen mass) in each, and set (total carbon)/(total nitrogen) = 25 to solve for the unknown quantity of sludge.

Answer 1

Correct Answer: 0.334

Step 1: Set up the basis for the mix.
We are told moisture content, nitrogen content and C/N ratio of leaves and sludge, and asked how much wet sludge must be added per kg of wet leaves so that the mixed waste, on a DRY basis, has a C/N ratio of 25. Take 1 kg of wet leaves as the fixed reference and let x = kg of wet sludge added to it. We need to find x.

Step 2: Convert the wet quantities to dry solids.
Moisture content tells us what fraction of the wet mass is water, so the remaining fraction is dry solids.
Dry mass of leaves = 1 kg × (1 - 0.50) = 0.50 kg
Dry mass of sludge = x kg × (1 - 0.75) = 0.25x kg
This step matters because nitrogen % and C/N ratio in these problems are always defined with respect to the dry solids, not the wet weight, so mixing wet weights directly would give a wrong ratio.

Step 3: Work out the nitrogen mass carried by each dry fraction.
Nitrogen content (%) is nitrogen mass divided by dry mass, so:
N from leaves = 0.007 × 0.50 = 0.0035 kg
N from sludge = 0.056 × 0.25x = 0.014x kg

Step 4: Work out the carbon mass carried by each dry fraction.
By definition, C/N ratio = (carbon mass)/(nitrogen mass), so carbon mass = (C/N ratio) × (nitrogen mass). This lets us back-calculate carbon even though it was not given directly.
C from leaves = 50 × 0.0035 = 0.175 kg
C from sludge = 6.3 × 0.014x = 0.0882x kg

Step 5: Impose the target C/N ratio on the combined mixture.
For the blended waste, total carbon over total nitrogen must equal 25:
(0.175 + 0.0882x) / (0.0035 + 0.014x) = 25
Cross-multiplying: 0.175 + 0.0882x = 25 × 0.0035 + 25 × 0.014x = 0.0875 + 0.35x
Collecting x terms on one side: 0.175 - 0.0875 = 0.35x - 0.0882x
0.0875 = 0.2618x
x = 0.0875 / 0.2618 = 0.334 kg of sludge per kg of leaves

Step 6: Check that the answer makes physical sense.
Leaves alone have a C/N ratio of 50, much higher than the target of 25, while sludge is nitrogen rich with a low C/N ratio of 6.3. Blending in a moderate amount of sludge (about one-third of a kg per kg of leaves) pulls the combined ratio down to the desired value of 25, which is the kind of proportion used in real composting operations. The final answer is x = 0.334 kg sludge/kg leaves.