The sum of kate's age and her mother's age is 52. kates mother is 20 years older than kate. how old is each

The sum of Kate age and her mother's age is 52, so 
K + M = 52 
Kate's mother is 20 years older then Kate, so 
M = 20 + K 
How old is each person? 
Since M = 20 + K, substitute it into the first equation, that is, replace M with 20 + K 

So, K + (20 + K) = 52 
K + 20 + K = 52 
2K + 20 = 52 
Subtract 20 from both sides: 
2K = 52 - 20 
2K = 32 
divide both sides by 2 to get 
K = 16 
Now we can find M, since M = 20 + K = 20 + 16 = 36 
Kate is 16 years old and her mother is 36 years old.
Hope this helps!!1