Henry's mother is three times as old as he is. His grandmother is 35 years older than his mother. If Henry's grandmother was 7 years younger, she would be 5 times as old as Henry. How old is Henry's grandmother?

m =3h

g = m + 35
g-7 =5h

g =3h +35
g = 5h + 7

5h+7 = 3h +35
2h =28
h =14
m = 42
g =77

To solve this problem, let's assign variables to the age of Henry, his mother, and his grandmother.

Let's say Henry's age is H.
According to the problem, Henry's mother is three times as old as he is, so his mother's age is 3H.
His grandmother is 35 years older than his mother, so his grandmother's age is 3H + 35.

Now, let's use the second part of the problem. If Henry's grandmother was 7 years younger, she would be 5 times as old as Henry. So, her age would be (H - 7) * 5.

Now we can set up an equation using the information we have:
(H - 7) * 5 = 3H + 35

Let's solve for H:

5H - 35 = 3H + 35
5H - 3H = 35 + 35
2H = 70
H = 70 / 2
H = 35

Now that we know the value of H, we can find the age of Henry's grandmother:
Grandmother's age = 3H + 35
Grandmother's age = 3 * 35 + 35
Grandmother's age = 105 + 35
Grandmother's age = 140

So, Henry's grandmother is 140 years old.