Henry Ernest Dudeney/Puzzles and Curious Problems/152 - The Length of a Lease/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $152$

A friend's property had a $99$ years' lease.

When I asked him how much of this had expired, the reply was as follows:

Two-thirds of the time past was equal to four-fifths of the time to come,

so I had to work it out for myself.

$54$ years have elapsed, leaving $45$ years to go.

Let $t$ be the time elapsed on this lease.

We have:

$\ds \dfrac 2 3 t$

$=$

$\ds \dfrac 4 5 \paren {99 - t}$

Two-thirds of the time past was equal to four-fifths of the time to come

$\ds \leadsto \ \ $

$\ds 10 t$

$=$

$\ds 12 \paren {99 - t}$

$\ds \leadsto \ \ $

$\ds 12 t$

$=$

$\ds 1188$

$\ds \leadsto \ \ $

$\ds t$

$=$

$\ds 54$

and we see that:

$\dfrac 2 3 54 = 36 = \dfrac 4 5 45$

$\blacksquare$