Kate's father is 32 years older than kate is. in 5 years, he will be five times older. how old is kate?

In 5 years, the ratio of their ages will be 5:1, so the difference in their ages will be 4 ratio units. Those 4 units correspond to 32 years, so each unit corresponds to 32/4 = 8 years.

Thus, in 5 years, Kate will be 8 and her father will be 40. Right now, Kate is ...

... 8 - 5 = 3 years old.

Using equations

Let k represent Kate's current age. Then in 5 years Kate will be k+5, and we will have ...

... 5(k+5) = (k+5)+32 . . the relationship betewen their ages in 5 years

... 5k +25 = k +37 . . . . eliminate parentheses

... 4k = 12 . . . . . . . . . . . add -25-k

... k = 3 . . . . . . . . . . . . . divide by 4

Kate is 3 years old.