1. The greater of two integers is 4 times the lesser. Of the lesser integer is increased by 12, the result is 3 less than the greater.Find the integers.

2. Tom is three times as old as Leigh Ann. In 7 years, the sum of half of Tom's present age and a third of Leigh Ann's age will be the same as Tom's age now. How old will Tom be in 7 years?

Let the integers be x and x/4.Then

x/4+12=x-3 or x+48=4x-12 by multiplying both sides by 4.
Or -3x=-20-48 or x=20 which is the larger integer. Smaller one is then 20/4=5.
Check:5+12=17=20-3 OK.

To solve these types of problems, we can set up a system of equations based on the given information and then solve them simultaneously. Let's solve each problem step by step:

1. Let's assume the lesser integer is "L" and the greater integer is "G."
- Given: The greater integer is 4 times the lesser integer, so we can write G = 4L.
- Given: If the lesser integer is increased by 12, the result is 3 less than the greater integer. This can be written as L + 12 = G - 3.

Now we have two equations:
1) G = 4L
2) L + 12 = G - 3

To find the integers, we need to find the values of L and G that satisfy both equations. We can substitute the value of G from the first equation into the second equation to eliminate G:

L + 12 = 4L - 3
12 + 3 = 4L - L
15 = 3L
L = 5

Substitute this value back into the first equation to find G:
G = 4L = 4(5) = 20

Therefore, the lesser integer is 5 and the greater integer is 20.

2. Let's assume Leigh Ann's age is "L" and Tom's age is "T."
- Given: Tom is three times as old as Leigh Ann, so we can write T = 3L.
- Given: In 7 years, the sum of half of Tom's present age and a third of Leigh Ann's age will be the same as Tom's age now. This can be written as (T + 7) = (T/2) + (L/3).

Now we have two equations:
1) T = 3L
2) T + 7 = (T/2) + (L/3)

To find Tom's age in 7 years, we need to find the value of T that satisfies both equations. We can substitute the value of T from the first equation into the second equation to eliminate T:

(3L) + 7 = ((3L)/2) + (L/3) (multiply everything by 6 to remove the denominators)
18L + 42 = 9L + 2L
16L = -42
L = -2.625

Substitute this value back into the first equation to find T:
T = 3L = 3(-2.625) = -7.875

Therefore, Leigh Ann's age is -2.625 and Tom's age is -7.875 in the present. But since age cannot be negative, these results don't make sense. It's possible that there is an error in the problem statement or the question needs clarification.