A shop wants to sell a certain quantity (in kg) of grains. It sells half the quantity and an additional 3 kg of these grains to the first customer. Then, it sells half of the remaining quantity and an additional 3 kg of these grains to the second customer. Finally, when the shop sells half of the remaining quantity and an additional 3 kg of these grains to the third customer, there are no grains left. The initial quantity, in kg, of grains is

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Accepted Answer

The problem involves calculating the initial quantity of grains based on sequential sales operations. Here's the step-by-step solution:

**Let the Initial Quantity Be $$x$$ kg:**
   - The shop starts with $$x$$ kg of grains.

**First Sale:**

- The shop sells **half the quantity plus 3 kg** to the first customer:
     $$
     	ext{Grains sold} = \frac{x}{2} + 3
     $$
   - Remaining quantity after the first sale:
     $$
     x - \left(\frac{x}{2} + 3ight) = \frac{x}{2} - 3
     $$

**Second Sale:**

- The shop sells **half of the remaining quantity plus 3 kg** to the second customer:
     $$
     	ext{Grains sold} = \frac{\left(\frac{x}{2} - 3ight)}{2} + 3 = \frac{x}{4} - \frac{3}{2} + 3
     $$
   - Remaining quantity after the second sale:
     $$
     \left(\frac{x}{2} - 3ight) - \left(\frac{x}{4} - \frac{3}{2} + 3ight)
     $$
     Simplify:
     $$
     	ext{Remaining quantity} = \frac{x}{4} - \frac{9}{2}
     $$

**Third Sale:**

- The shop sells **half of the remaining quantity plus 3 kg** to the third customer:
     $$
     	ext{Grains sold} = \frac{\left(\frac{x}{4} - \frac{9}{2}ight)}{2} + 3 = \frac{x}{8} - \frac{9}{4} + 3
     $$
   - After this sale, no grains are left:
     $$
     \left(\frac{x}{4} - \frac{9}{2}ight) - \left(\frac{x}{8} - \frac{9}{4} + 3ight) = 0
     $$

**Solve for $$x$$:**

Simplify the equation:
   $$
   \frac{x}{4} - \frac{9}{2} = \frac{x}{8} - \frac{9}{4} + 3
   $$
   Multiply through by 8 to eliminate fractions:
   $$
   2x - 36 = x - 18 + 24
   $$
   Simplify:
   $$
   x = 42
   $$

### Final Answer:

The initial quantity of grains is **42 kg**.  
Correct option: **D**

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