A jar was full with honey. A person used to draw out 20% of the honey from the jar and replaced it with sugar solution. He has repeated the same process 4 times and thus there was only 512 gm of honey left in the jar, the rest part of the jar was filled with the sugar solution. The initial amount of honey in the jar was :

Options

A. 1.25 kg
B. 1 kg
C. 1.5 kg
D. none of these

Answer

To solve the problem step by step, we will denote the initial amount of honey in the jar as \( x \) grams. The process of drawing out 20% of the honey and replacing it with sugar solution is repeated 4 times, and we know that after these operations, 512 grams of honey is left in the jar.

### Step 1: Understand the process
Each time 20% of the honey is drawn out, 80% of the honey remains in the jar. Therefore, after each operation, the amount of honey left can be expressed as a fraction of the previous amount.

### Step 2: Set up the equation
After one operation, the amount of honey left is:
\[ \text{Honey after 1st draw} = x \times \left(1 - \frac{20}{100}\right) = x \times \frac{80}{100} = x \times \frac{4}{5} \]

After the second operation, the amount of honey left is:
\[ \text{Honey after 2nd draw} = \left(x \times \frac{4}{5}\right) \times \frac{4}{5} = x \times \left(\frac{4}{5}\right)^2 \]

Continuing this process for 4 operations, we have:
\[ \text{Honey after 4th draw} = x \times \left(\frac{4}{5}\right)^4 \]

### Step 3: Set the equation to the final amount of honey
According to the problem, after 4 operations, there are 512 grams of honey left:
\[ x \times \left(\frac{4}{5}\right)^4 = 512 \]

### Step 4: Calculate \(\left(\frac{4}{5}\right)^4\)
Calculating \(\left(\frac{4}{5}\right)^4\):
\[ \left(\frac{4}{5}\right)^4 = \frac{4^4}{5^4} = \frac{256}{625} \]

### Step 5: Substitute back into the equation
Now substituting this back into our equation:
\[ x \times \frac{256}{625} = 512 \]

### Step 6: Solve for \( x \)
To find \( x \), we rearrange the equation:
\[ x = 512 \times \frac{625}{256} \]

### Step 7: Simplify the multiplication
Calculating \( 512 \div 256 \):
\[ 512 \div 256 = 2 \]
Thus,
\[ x = 2 \times 625 = 1250 \text{ grams} \]

### Step 8: Convert grams to kilograms
To convert grams to kilograms:
\[ 1250 \text{ grams} = \frac{1250}{1000} \text{ kg} = 1.25 \text{ kg} \]

### Final Answer
The initial amount of honey in the jar was **1.25 kg**.

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