Word problem - I think it's to do with lowest common multiple

Question

this is the problem I have:

"If hard drive capacity doubles every five years, and the rate of reading data from a disc doubles every three years, then how long from now will it be 1024 times quicker to read an entire disc?"

.......... After trying to work it out, I basically got this far, though not sure if it's right: the lowest common multiple of 5 and 3 is 15, hence in a 15 year period, size doubles thrice and read rate doubles five times, hence in 15 years it becomes twice as quick to read an entire disc. now since 15 years is twice as quick, then 1024 times as quick is 15 x 512 = 7680 years.

However, the answer I have from someone whose mathematical ability I trust much more than my own is 75 years

Any worked solutions? Thanks

Answer 1

From MJD's remark, here's a worked solution. In 75 years, size has doubled $\frac {75} 5=15$ times so we have $2^{15}$ times more capacity. However, read time has doubled $\frac {75} 3=25$ times so we have $2^{25}$ times faster read times. $\frac {2^{25}} {2^{15}}=2^{10}=1024$ time faster after 75 years.

Answer 2

This has nothing to do with LCM, but with exponential growth. According to the story $$C(t)=C_0\cdot 2^{t/5},\qquad R(t)=R_0\cdot 2^{t/3}\ ,$$ where $t$ is measured in years, and $T_0={C_0\over R_0}$ is the time to read an entire disk as of now. It follows that the time to read an entire disk in $t$ years from now is given by $$T(t)={C(t)\over R(t)}=T_0\cdot {2^{t/5}\over 2^{t/3}}=T_0\cdot 2^{-2t/15}\ .$$ Since we want ${T(t)\over T_0}=2^{-10}$ we have to solve $-{2t\over15}=-10$, which then gives $t=75$ [years].