# Solving equations involving powers

I have been solving an OS problem in which I came across this equation where I have to find the value of H as in % but I'm not able to solve it fully. This is not a homework, I'm stuck in between and been left maths from quite some time.

3*10^-8/0.8 = H(2*10^-8 + 10^-8) + (1-H) (10^-8 + 10^-3)

My Take so far -

3*10^-8/0.8 = H(3*10^-8) + (1-H) (10^-11)

3*10^-8/0.8 = H(3*10^-8) + 10^-11 - 10^-11*H

Now I can't able to solve any further.

What I tried more is I cancelled (3*10^-8) from both LHS and RHS and I got

1/0.8 = H + (1-H)*10-11 /// ( I took 10^-11 common from 10^-11 - 10^-11*H)

So what to do after 1/0.8 = H + (1-H)*10-11

• OS = Operating System ?
– user65203
Sep 28, 2018 at 9:18
• The coefficients expressed "in scientific notation" are not really considered as powers. What you have is a linear equation.
– user65203
Sep 28, 2018 at 9:38

Dropping the annoying $$10^{-8}$$,

$$\frac3{0.8} = H(2 + 1) + (1-H) (1 + 100000).$$

Without caring about lower terms, we have

$$1-H\approx\frac3{80000}$$ which gives you an order of magnitude, i.e. $$H$$ very close to $$1$$.

Exact computation is (using $$H=1-(1-H)$$)

$$\frac{15}4=3+(1-H)\,99998,$$

$$1-H=\frac3{4\cdot99998}.$$

(I prefer to evaluate $$1-H$$ as this is more accurate and gives you better understanding of the value.)

So you have

$$\frac{3\cdot10^{-8}}{0.8} = 3\cdot10^{-8}H + (1-H)(10^{-8}+10^{-3})$$

which is

$$\frac{3}{0.8} = 3H + (1-H)(1+10^{5}) = 1+10^{5}-10^{5}H+2H$$

if you divide both sides by $$10^{-8}$$. So

$$H = \frac{\frac{3}{0.8} - 1 - 10^{5}}{2-10^{5}} = 0.9999924998499969999... \approx\mathbf{100\%}$$

• This is wrong, I've made a mistake above 10^-8 + 10^-3 is not 10^-11 Sep 28, 2018 at 3:30
• I edited my answer. Sep 28, 2018 at 8:36
• Now it's perfectly fine. Sep 28, 2018 at 13:06