8 girls and 12 boys can complete a piece of work in 4 days, while 10 girls and 6 boys can complete it in 6 days. How long will 5 girls and 3 boys take to complete this same piece of work?

8 girls and 12 boys can complete a piece of work in 4 days, while 10 girls and 6 boys can complete it in 6 days.

$$4\times\ \left(8\ girls+12\ boys\right)\ =\ 6\times\ \left(10\ girls+6\ boys\right)$$

$$2\times\ \left(8\ girls+12\ boys\right)\ =\ 3\times\ \left(10\ girls+6\ boys\right)$$

$$16\ girls+24\ boys\ =\ 30\ girls+18\ boys$$

$$24\ boys-18\ boys\ =\ 30\ girls-16\ girls$$
$$6\ boys\ =\ 14\ girls$$
$$\frac{boys}{girls}\ =\ \frac{14}{6\ }$$
$$\frac{boys}{girls}\ =\ \frac{7}{3\ }$$
Let's assume that each boy and girl can do '7y' and '3y' units of work in a day.
Let's assume the time taken by 5 girls and 3 boys to complete this same piece of work is 'z' days.
$$4\times\ \left(8\ girls+12\ boys\right)\ =\ z\times\ \left(5\ girls+3\ boys\right)$$

$$4\times\ \left(8\ \times3y\ +12\times7y\right)\ =\ z\times\ \left(5\ \times3y+3\times7y\right)$$
$$4\times\ \left(24y\ +84y\right)\ =\ z\times\ \left(15y+21y\right)$$
$$4\times\ 108y\ =\ z\times\ 36y$$
$$4\times3\ =\ z$$
z = 12 days
Short cut ::
10 girls and 6 boys can complete the work in 6 days.
When the number of girls and boys will be half, then the time taken will be double.
time taken by 5 girls and 3 boys to complete this same piece of work = $$6\times\ 2$$
= 12 days