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

Explanation

To find the initial amount of honey in the jar, we can use the concept of successive replacements. Here’s a step-by-step solution:

### Step 1: Understand the problem
The person draws out 20% of the honey and replaces it with sugar solution. This process is repeated four times, and after the fourth replacement, there are 512 grams of honey left in the jar.

### Step 2: Determine the remaining percentage of honey after each replacement
Each time 20% of the honey is removed, 80% remains. Therefore, after one replacement, the amount of honey left can be expressed as:
- Remaining honey after 1st replacement = 80% of initial honey
- Remaining honey after 2nd replacement = 80% of remaining honey after 1st replacement = \( 80\% \times 80\% \) of initial honey
- Remaining honey after 3rd replacement = \( 80\% \times 80\% \times 80\% \) of initial honey
- Remaining honey after 4th replacement = \( 80\% \times 80\% \times 80\% \times 80\% \) of initial honey

In mathematical terms, if \( x \) is the initial amount of honey, then after 4 replacements, the amount of honey left is:
\[ \text{Remaining honey} = x \times (0.8)^4 \]

### Step 3: Set up the equation
We know that after four replacements, there are 512 grams of honey left:
\[ x \times (0.8)^4 = 512 \]

### Step 4: Calculate \( (0.8)^4 \)
Calculate \( (0.8)^4 \):
\[ (0.8)^4 = 0.8 \times 0.8 \times 0.8 \times 0.8 = 0.4096 \]

### Step 5: Substitute and solve for \( x \)
Now substitute \( (0.8)^4 \) back into the equation:
\[ x \times 0.4096 = 512 \]
To find \( x \), divide both sides by 0.4096:
\[ x = \frac{512}{0.4096} \]

### Step 6: Perform the division
Calculating the division:
\[ x = 1250 \text{ grams} \]

### Conclusion
The initial amount of honey in the jar was **1250 grams**.

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