# Hank and his old car

I'm sort of struggling with this riddle told to me by a friend:

Hank owns a car. He has been taking good care of his car; In fact, he has been taking such good care of it that the age of Hank, and the age his car combined is 56 years!

Coincidentally his car is twice as old as Hank was when his car was as old as Hank is now.

How old are Hank and his car?

I'm having trouble figuring out what is easiest way of unraveling the 2nd sentence ("Coincidentally his car(...)").

For example, when representing the age of Hank, his car, and their combined age as $x$, $y$ and $c$ respectively, I get $x+y=c=56$ from the 1st sentence.

Then when trying do the same, the 2nd sentence I get $y = 2(?)$

So what I guess it boils down to is; I simply don't know how to express the 2nd sentence to define $y$.

Please excuse bad grammar and syntax, as english is not my primary language.

• If it's a question of definition, maybe someone can translate the second sentence? What is a better language? – hunter Feb 2 '15 at 10:35

Let Hank's age now be $x$. We get the following: $$\begin{array}{c|c|c} &\mbox{Hank}&\mbox{Car} \\\hline\mbox{Now}&x&56-x \\ \mbox{earlier} & 3x-56&x \end{array}$$ The 'Hank-earlier entry' is obtained as follows: The earlier time of interest was when the car was $x$ (Hank's current age). That occurred $(56-x)-x=56-2x$ years ago. That many year's ago, Hank's age would be $x-(56-2x)=3x-56$.

Now we can sort out the second sentence that says "...car is twice as old as Hank was when his car was as old as Hank is now.": $$56-x=2(3x-56)$$

We can translate the statements about Hank and his car thusly:

\begin{align} \text{hank}_{\text{now}} + \text{car}_{\text{now}} &= 56 &(1)\\ \text{hank}_{\text{now}} &= \text{car}_{\text{then}} &(2)\\ \text{car}_{\text{now}} &= 2\cdot \text{hank}_{\text{then}} &(3) \end{align}

And we can use the passage of time to convert from "now" to "then":

\begin{align} \text{hank}_{\text{now}} &=\text{time} + \text{hank}_{\text{then}} \\ \text{car}_{\text{now}} &= \text{time} + \text{car}_{\text{then}} \end{align}

Writing $h := \text{hank}_{\text{then}}$, $c := \text{car}_{\text{then}}$, $t := \text{time}$, we have \begin{align} (t+h) + (t+c) = 2 t + h + c &= 56 &(1^\prime)\\ t+h &= c &(2^\prime)\\ t + c &= 2h &(3^\prime) \end{align} Solving gives $$t = 8 \qquad h = 16 \qquad c = 24$$

So,

$$\text{hank}_{\text{then}} = 16 \qquad \text{hank}_{now} = 16 + 8 = 24$$ $$\text{car}_{\text{then}} = 24 \qquad \text{car}_{now} = 24 + 8 = 32$$

Let's check:

[T]he age of Hank (24), and the age his car (32) combined is 56 years!

[H]is car is (32) twice as old as Hank was (16) when his car was (24) as old as Hank is now (24).

That's it!

Let the present age of hank and his car be x and y respectively

Then

x+y=56

when the age of his car was x,

then let y-x (the difference between the age of the car now and then) be j

then, y-x=j

=>y-j=x

=>y=x+j--(1)

now hank's age at that time would be x-j

then according to the question 2(x-j)=y (i.e. two times the past age of hank is the present age of the car)

Now,

2(x-j)=y

=>2(x-j)=x+j (eqn 1)

solve this and you get

x=3j

using eqn 1

y=x+j

=>y=3j+j=4j

now x+y=56

=>3j+4j=56

=>7j=56

=>j=8

x=24 (present age of hank)

y=32(present age of car)

How old was Hank when his car was as hold as Hank is now? Well, the car was definitely $x$ years old back then, and it's $y$ years old now, so that was $y - x$ years ago. How old was Hank then? He was $x - (y - x)$ years old, which is to say he was $2x - y$ years old. So your $(?)$ should be a $2x-y$.

As in, let's say Hank is $A$ years old now. Well, back when the car was $A$ years old - as old as Hank is now - Hank was a little younger, say $a$ years old - $a$ is the age Hank was when his car was as old as he is now. The car is currently twice $a$ - it's twice as old as Hank was when the car was as old as he is now.