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Here is the expression that I am trying to solve for n:

$$ \frac{4}{16+n} = \frac{10}{16+n} \frac{10}{16+n}$$

I am doing the following:

\begin{align} \frac{4}{16+n} & = \frac{100}{(16 + n)^2} \\[8pt] \frac{4}{16+n} & = \frac{100}{16^2 + 32n + n^2} \\[8pt] \frac{4}{16+n} & = \frac{100}{256 + 32n + n^2} \\[8pt] \frac{1}{16+n} & = \frac{100}{4(256 + 32n + n^2)} \\[8pt] \frac{1}{16+n} & = \frac{20}{256 + 32n + n^2} \\[8pt] 16+n & = \frac{256 + 32n + n^2}{20} \\[8pt] n & = \frac{256 + 32n + n^2 - 320}{20} \\[8pt] n & = \frac{-64 + 32n + n^2}{20} \\[8pt] 20n & = -64 + 32n + n^2 \\[8pt] 20n -32n - n^2 & = -64 \\[8pt] 12n - n^2 & = -64 \\[8pt] n(12 - n) & = -64 \end{align}

Not sure what to do next now... Book says that n = 9 Cannot get 9... Where am I wrong?

Thank you

Michael Hardy
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UserMoon
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    Arithmetic error going from the 4th to 5th line, you did $100/4$ and simplified to $20$. It should have simplified to $25$. As for how you could complete the steps given your work, you can move everything to one side leaving zero on the other side and use the quadratic formula. – JMoravitz Nov 11 '14 at 03:57
  • What he said, plus one more arithmetic error near the end: $20n-32n=-12n$ – Mike Nov 11 '14 at 04:01
  • I am dumb idiot..jeez :((((( – UserMoon Nov 11 '14 at 04:02

2 Answers2

7

There's simple method

$$\frac{4}{16+n} = \frac{10}{16+n} \times \frac{10}{16+n}$$

$n\neq-16$

$${4} = \frac{10\times10}{16+n} $$

$$64+4n = 100 $$

$$36=4n$$

$$n=9$$

Well let's figure out your mstake, $$\frac{4}{16+n} = \frac{100}{n^2+32n+256} $$ $n\neq-16$

$$n^2+32n+256=25(16+n)$$ $$n^2+7n-144=0$$ $$(n-9)(n+16)$$ since $n\neq-16$ solution is $$n=9$$

Aditya Hase
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5

You can cancel one of the $16 + n$ terms on each side, greatly simplifying the problem. Multiply both sides by $16 + n$. This gives

$$4 = \frac{10 \times 10}{16 + n}$$

Multiply both sides by $16 + n$ again. This gives

$$4(16 + n) = 100$$

Divide both sides by $4$:

$$16 + n = 25$$

Subtract $16$ from both sides:

$$n = 9$$

Null
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