Hello ipokeyou Originally Posted by

**ipokeyou** Hi.

Can you help me solve this question?

Thanks in advance!

One layer of tinting material on a window cuts out 1/5 of the sun's UV rays.

(a) What fraction would be cut out by using two layers?

(b) How many layers would be required to cut out at least 9/10 of the sun's UV rays?

(a) If one layer cuts out $\displaystyle \frac15$ of the UV, then it must allow $\displaystyle \frac45$ through. Of this, $\displaystyle \frac45$ will be allowed through by a second layer. So, of the original quantity of UV the fraction allowed through by two layers will be: $\displaystyle \frac45\times\frac45 = \frac{16}{25}$

Therefore $\displaystyle 1 - \frac{16}{25}= \frac{9}{25}$ will have been cut out.

(b) Similarly, if there are $\displaystyle n$ layers, the fraction allowed through will be $\displaystyle \left(\frac45\right)^n$. And the fraction cut out will be $\displaystyle 1-\left(\frac45\right)^n$

So we want the smallest integer value of n for which$\displaystyle 1-\left(\frac45\right)^n\ge\frac{9}{10}$

$\displaystyle \Rightarrow \left(\frac45\right)^n\le\frac{1}{10}$

If you understand how to take logs of both sides, you can solve this inequality that way. Or you can simply use your calculator until you get the answer. Either way, I think it's 11 layers.

Grandad