Question

If the decimal $0.d25d25d\mathrm{25....}$ is expressible in the form $\frac{n}{27}$, then find the value of $d+n$

Answer 1

$\frac{1}{27}=\mathrm{0.037037037...}$

So the multiplication of $\frac{1}{27}$(like $\frac{10}{27},\frac{13}{27},\frac{20}{27}$ etc.)

is in the form of $0.abcabcabcabc...$ where $a,b,c$ are single digit numbers.

By comparing $b$ and $c$, we have $b=2,c=5$,

so $37$ times a number $n=$ a hundred digit number with last two digits of $25$

Trial and error shows that $37\times 25=925$

So, $n=25$

$\frac{n}{27}=\frac{25}{27}=\mathrm{0.925925925...}$

So,$d=9$

And thus, $d+n=9+25=34$