Let's calculate the cost price and selling price to determine the least possible value of x for the shopkeeper to make a profit of at least 10%.

Cost Price (CP) of 100 mangoes = Tk. 12 each Total Cost Price of 100 mangoes = 100 * Tk. 12 = Tk. 1200

Selling Price (SP) of 60 mangoes = Tk. 17.40 each Total Selling Price of 60 mangoes = 60 * Tk. 17.40 = Tk. 1044

Now, we need to find the SP of the remaining (100 - 60 = 40) mangoes to cover the cost price and achieve a profit of at least 10%.

Let x be the number of mangoes sold at Tk. 11.31 each.

Total SP of x mangoes = x * Tk. 11.31

To make a profit of at least 10%, the SP must be at least 110% of the CP.

So, 110% of CP = 1.1 * Tk. 1200 = Tk. 1320

Now, we can set up an equation for the remaining mangoes:

Total SP of x mangoes + Tk. 1044 (SP of the first 60 mangoes) = Tk. 1320

x * Tk. 11.31 + Tk. 1044 = Tk. 1320

Now, subtract Tk. 1044 from both sides:

x * Tk. 11.31 = Tk. 1320 - Tk. 1044

x * Tk. 11.31 = Tk. 276

Now, divide both sides by Tk. 11.31 to find the value of x:

x = Tk. 276 / Tk. 11.31

x ≈ 24.37

Since x must be a whole number (representing the number of mangoes), we can round up to the nearest whole number to ensure a profit of at least 10%.

Rounding up, x = 25

So, the least possible value of x is 25 to achieve a profit of at least 10%.