This is how a factory works!

Since twenty workers can pack two consignments in ten hours, then each worker packs 0.1 consignments in ten hours, i.e. 0.1 consignments per shift. $$ Consider how many consignments have been packed by workers who have completed their shifts.

• After $10$ hours, the first twenty workers complete their shifts, having packed $2$ consignments.

• After $11$ hours, another four workers complete their shifts, so the total number of consignments packed is $2 + 0.4$ .

• After $12$ hours, another four workers complete their shifts, so the total number of consignments packed is $2 + 2(0.4)$ .

$\qquad \vdots$

• After $k$ hours, $(k \ge 10)$ , the total number of consignments packed is $2 + (k-10)(0.4)$ .

Thus after 113 hours, the total number of consignments packed by workers who have completed their shifts is $2 + (103)(0.4) = 43.2$ $$ However, there are still a number of workers who have not yet completed their shifts:

• One group of four workers have been working for nine hours, they've packed $(0.9)(0.4)$ consignments.

• Another group of four workers have been working for eight hours, they've packed $(0.8)(0.4)$ consignments.

$\qquad \vdots$

• The last group of four workers to start have been working for just one hour, they've packed (0.1)(0.4) consignments.

Thus the total number of consignments packed by workers who have not yet completed their shifts is $(0.9 + 0.8 + .... + 0.1)(0.4) = (4.5)(0.4) = 1.8$ $$ So the number of consignments that have been packed after 113 hours is $43.2 + 1.8 = \boxed{45}$