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

The correct option is 42.

To find the initial quantity of grains, we can work backward step-by-step from the final state where no grains are left after the third customer.

Step 1: Quantity before the third customer
Let the quantity of grains remaining in the shop before the third customer's purchase be

$R_2$

The shop sells half of this quantity and an additional 3 kg to the third customer, leaving 0 kg of grains. We can write this relation as:

$R_2-\left(\frac{1}{2}{R}_2+3\right)=0$

Simplifying the left-hand side:

$\frac{1}{2}{R}_2-3=0$

Adding 3 to both sides:

$\frac{1}{2}{R}_2=3$

Multiplying by 2:

$R_2=6\text{kg}$

So, there were 6 kg of grains remaining before the third customer made their purchase.

Step 2: Quantity before the second customer
Let the quantity of grains remaining in the shop before the second customer's purchase be

$R_1$

The shop sells half of this quantity and an additional 3 kg to the second customer, leaving the 6 kg we calculated in Step 1. We can write this relation as:

$R_1-\left(\frac{1}{2}{R}_1+3\right)=6$

Simplifying the left-hand side:

$\frac{1}{2}{R}_1-3=6$

Adding 3 to both sides:

$\frac{1}{2}{R}_1=9$

Multiplying by 2:

$R_1=18\text{kg}$

So, there were 18 kg of grains remaining before the second customer made their purchase.

Step 3: Initial quantity before the first customer
Let the initial quantity of grains in the shop be

$Q$

The shop sells half of this initial quantity and an additional 3 kg to the first customer, leaving the 18 kg we calculated in Step 2. We can write this relation as:

$Q-\left(\frac{1}{2}Q+3\right)=18$

Simplifying the left-hand side:

$\frac{1}{2}Q-3=18$

Adding 3 to both sides:

$\frac{1}{2}Q=21$

Multiplying by 2:

$Q=42\text{kg}$

Therefore, the initial quantity of grains was 42 kg.