How to rearrange: $16 = \frac{1}{n} 25 + \frac{n−1}{n} 218.75$

Question

Can anyone here help me out to rearrange the following formula and solve for $n$?

$$16 = \frac{1}{n} 25 + \frac{n−1}{n} 218.75$$

Answer 1

$$16 = \frac{1}{n} 25 + \frac{n−1}{n} 218.75$$

$$16=\frac{25}{n}+\frac{218.75n-218.75}{n}$$

$$16=\frac{25+218.75n-218.75}{n}$$

$$16n=25+218.75n-218.75$$

$$202.75n=193.75$$

$$\boxed{\color{blue}{ n=\frac { 775 }{ 811 } }}$$

Answer 2

$$\frac { 25 }{ n } +218.75\frac { n-1 }{ n } =16 $$ multiply both side to n then and get: $$ 16n=25+218.75\left( n-1 \right) \\ \Downarrow \\ 16n=25+\quad 218.75n-193.75\\ \Downarrow \\ 202.75n=193.75\\ \Downarrow \\ n=\frac { 775 }{ 811 } $$

Answer 3

take $\frac{1}{n}$ as a common factor and so $$16 = \frac{1}{n} \big(25 + (n−1)218.75\big)$$

now multiply by $n$ each side to get $$16n = 25 + (n-1)218.75$$

now expand $$(n-1)218.75 = n218.75 - 218.75$$

So now you have $$16n = 25 + n218.75 - 218.75$$

and subtract $n218.75$ each side to get $$16n - n218.75 = 25 -218.75$$

Now take $n$ as common factor again to get $$n(16 - 218.75) = 25 - 218.75$$

Finally divide by $(16 - 218.75)$ on each side to get $$\color{red}{n = \frac{25 -218.75}{16 - 218.75} = \frac{775}{811}}$$

and you are done !

Answer 4

multiply by n

$$16{n} = 25 + 218.75({n−1})$$

$$16{n} = 25 + 218.75{n}−218.75$$

move n to the right and the constants to the left hand side of the equation

$$218.75 - 25 = 218.75{n}−16{n}$$

solve for n

$$193.75 = 202.75{n}$$

$$193.75/202.75 = {n}$$

simplify by making the top and bottom integers

$$775/811 = {n}$$

Done as 811 is a prime