7. When I am as old as my father is now, I will be five times as old as my son is now. By then, my son will be eight years older than I am now. The sum of my father’s age and my age is 100 years. How much older am I than my son?

This is an "age" type word problem. (

__here__).

Let's start by picking variables for the author's, the father's, and the son's ages:

. . .ages now:

. . . . .author: a

. . . . .father: f

. . . . .son: s

In however many years ("By then") y, all the ages will be greater by the same amount:

. . .ages then:

. . . . .author: a + y

. . . . .father: f + y

. . . . .son: s + y

Let's try restating the original exercise, to perhaps make things a bit more clear:

I'm thinking about my age, my son's age, and my father's age, all on the same date "y" years from now.

(i) On that date, my age then (a + y) will be equal to my father's age today (f): a + y = f.

(ii) On that date, my age then (a + y) will be equal to five times my son's age today (5s): a + y = 5s.

(iii) On that date, my son's age then (s + y) will be my current age, plus another eight years (a + 8): s + y = a + 8.

(iv) Today, while I'm thinking about this, my current age (a) and my father's current age (f) sum to one hundred: a + f = 100.

The equations in (i) and (ii) have the same left-hand side, so we can set the right-hand sides equal, to get:

(v) On that date, my father's age today (f) will be equal to five times my son's age today (5s): f = 5s.

(The more natural way of stating this is that, today, the father's age is fives times the son's age. They made things complicated by putting these current ages in terms of future ages.)

So we have three equations:

. . . . .(iii) s + y = a + 8

. . . . .(iv) a + f = 100

. . . . .(v) f = 5s

Unfortunately, this is three equations in FOUR unknowns, which we can't solve uniquely. So let's see if we can restate anything to get rid of a variable.

The expression author's future age is the father's current age. This means that the author's current age is the father's current age, less those "y" years. That is, a = f - y. Let's try plugging this into Equation (iii), in place of "a", and see if that leads anywhere nice:

. . . . .s + y = (f - y) + 8

Hmm... That doesn't seem much better. What if we plugged in from Equation (v) for the "f"?

. . . . .s + y = 5s - y + 8

. . . . .2y = 4s + 8

. . . . .y = 2s + 4

. . . . .Eqn. (vi)

Okay... Using Equation (iii), we can plug in for "y" and get:

. . . . .s + (2s + 4) = a + 8

. . . . .3s + 4 = a + 8

. . . . .3s - 4 = a

. . . . .Eqn. (vii)

We can plug the right-hand side of Equation (v) in for "f" in Equation (iv), giving us:

. . . . .a + (5s) = 100

. . . . .a = 100 - 5s

. . . . .Eqn. (viii)

Now let's put the left-hand side of Equation (vii) equal to the right-hand side of Equation (viii):

. . . . .3s - 4 = 100 - 5s

. . . . .8s = 104

. . . . .s = 13

Yay!

Now back-solve for the other ages.

Note that the exercise does not ask for anybody's ages. It asks only for the difference between the author's and son's ages. So plug the values into "a - s", and simplify to get the answer they're wanting. Remember to put the appropriate units on your answer.