# Find a Father | Solution

The answer: with the mother.

Did that make you do a double-take? If so, then let’s work this out together.

Asking “where’s the father” is the clue that tells us where to look for the twist in this problem. There’s no way we can know where the father is in the information given to us, so that tells us to change how we’re looking at the problem:

• The only way to know where the father is would be to think of how he might relate to the age of his child at different times.

• The only way we’d ever know where the father is would be if somehow all this math leads us back to the date the child was conceived (in which case, he’d be with the mother).

Let’s test this out:

The mother is 21 years older than her child. Let’s assume the child’s age, in months, is A.

Then the mother’s age is always going to be A + (21 x 12) = A + 252.

In exactly 6 years, both will gain 72 months (6 x 12 = 72). So, the child’s age will be A + 72, and the mother’s age will be A + 324 (since 252 + 72 = 324).

Now, we’re told that at this point, the mother is exactly 5 times as old as her child. This means her age, A + 324, is 5 times her child’s age, A + 72, or A + 324 = 5 x (A + 72).

From here, we’ll use the rules of algebra:

**Step 1.** On the right side, use the distributive law that 5 x (A + 72) = 5xA + 5×72. Simplifying that, we get 5xA + 360. Our equation now is: A + 324 = 5xA + 360.

**Step 2.** Using the rule that we can subtract anything we want from one side of an equation, provided we subtract the exact same thing on the other:

1. Subtract 324 from both sides of the equation to eliminate 324 on the left:

A + 324 – 324 = 5xA + 360 – 324

A = 5xA + 36

2. Subtract 5xA from both sides of the equation to eliminate 5xA on the right:

A – 5xA = 5xA + 36 – 5xA

-4xA = 36

3. Divide both sides by -4 to isolate A on the left:

-4xA / (-4) = 36 / (-4)

A = -9

Putting this all together, we can see that A, the child’s age, is -9 months. In other words, our suspicion is correct: The father is with the mother because this is the moment the child was conceived.

And there you have it: two puzzles whose answers might seem obvious, but only after you discover the twist behind them.

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