10 kg of rice costs as much as 20 kg of wheat, 25 kg of wheat costs as much as 2 kg of tea, 5 kg of tea costs as much as 25 kg of sugar. Find the cost of 6 kg of sugar if 4 kg of rice cost Rs 32.

Answer

To solve the problem step by step, we will analyze the relationships between the costs of rice, wheat, tea, and sugar, and then find the cost of 6 kg of sugar.

### Step 1: Establish the cost relationships
1. Let the cost of 1 kg of rice be \( R \).
2. Therefore, the cost of 10 kg of rice is \( 10R \).
3. According to the problem, 10 kg of rice costs as much as 20 kg of wheat. Let the cost of 1 kg of wheat be \( W \).
4. Thus, the cost of 20 kg of wheat is \( 20W \).
5. We can set up the equation:
\[ 10R = 20W \]
Simplifying this gives:
\[ \frac{R}{W} = \frac{20}{10} = 2 \quad \Rightarrow \quad R = 2W \]

### Step 2: Establish the relationship between wheat and tea
1. According to the problem, 25 kg of wheat costs as much as 2 kg of tea. Let the cost of 1 kg of tea be \( T \).
2. Therefore, the cost of 25 kg of wheat is \( 25W \) and the cost of 2 kg of tea is \( 2T \).
3. We can set up the equation:
\[ 25W = 2T \]
Simplifying this gives:
\[ \frac{W}{T} = \frac{2}{25} \quad \Rightarrow \quad W = \frac{2}{25}T \]

### Step 3: Establish the relationship between tea and sugar
1. According to the problem, 5 kg of tea costs as much as 25 kg of sugar. Let the cost of 1 kg of sugar be \( S \).
2. Therefore, the cost of 5 kg of tea is \( 5T \) and the cost of 25 kg of sugar is \( 25S \).
3. We can set up the equation:
\[ 5T = 25S \]
Simplifying this gives:
\[ \frac{T}{S} = \frac{25}{5} = 5 \quad \Rightarrow \quad T = 5S \]

### Step 4: Combine the relationships
Now we have the following relationships:
- \( R = 2W \)
- \( W = \frac{2}{25}T \)
- \( T = 5S \)

### Step 5: Substitute to find the relationship between rice and sugar
1. Substitute \( W \) into the equation for \( R \):
\[ R = 2 \left(\frac{2}{25}T\right) = \frac{4}{25}T \]
2. Substitute \( T \) into the equation:
\[ R = \frac{4}{25}(5S) = \frac{4}{5}S \]
3. Rearranging gives:
\[ S = \frac{5}{4}R \]

### Step 6: Find the cost of sugar using the given information
1. We know that 4 kg of rice costs Rs 32.
\[ \text{Cost of 1 kg of rice} = \frac{32}{4} = 8 \text{ Rs} \]
Therefore, \( R = 8 \).
2. Substitute \( R \) into the equation for \( S \):
\[ S = \frac{5}{4} \times 8 = 10 \text{ Rs} \]

### Step 7: Calculate the cost of 6 kg of sugar
1. The cost of 1 kg of sugar is Rs 10.
2. Therefore, the cost of 6 kg of sugar is:
\[ \text{Cost of 6 kg of sugar} = 6 \times 10 = 60 \text{ Rs} \]

### Final Answer
The cost of 6 kg of sugar is Rs 60.

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