We assign variables to represent the ages of the children:
Younger boy: `b_1`
Older boy: `b_2`
Younger girl: `g_1`
Older girl: `g_2`
Then we write out the relationships we were given
`b_2-b_1=2` and `g_2-g_1=2`
We can deduce from these clues that the youngest child is the younger girl `g_1`
Now we start substituting. From the equations we can find `b_2=2+b_1` We substitute that into the equation for the sum of the ages and it is now `b_1+(2+b_1)+g_1+g_2=42` Then we substitute in that `b_1=2g_2` and we get `2g_2+(2+2g_2)+g_1+g_2=42` We simplify by combining like terms and find `2+g_1+5g_2=42` Then we take the fact that `g_2=2+g_1` And we have `g_2+5g_2=42` or `6g_2=42` we divide 42 by 6 and we find that `g_2=7` i.e. the older girl is 7 years old. We also know that she is 2 years older than her sister, the youngest child.
Therefore the youngest child is 5 years old.