Augustus Acevedo

2022-07-15

Can you express this in form $\frac{1}{a}+\frac{1}{b}$
Can you express the fraction $\frac{1949}{1999}$ in the form $\frac{1}{a}+\frac{1}{b}$? Give reasons supporting your answer.
I think the only way to do this is keep trying numbers but then I will never get the answer. I cry every time.

Karla Hull

Expert

$\frac{1}{a}+\frac{1}{b}=\frac{a+b}{ab},$
so you would need $ab$ to divide $1999$. But…

Logan Wyatt

Expert

Given
$\frac{1}{a}+\frac{1}{b}=\frac{1949}{1999}$
Combining the fraction gives
$\frac{a+b}{ab}=\frac{1949}{1999}$
Setting terms gives the system
$\left\{\begin{array}{l}a+b=1949\\ ab=1999\end{array}$
With $a,b$ can be solved by a quadratic. Namely
${b}^{2}-1949b+1999$
${b}^{2}-1949b+1999$
Where $a=1949-b$
Since $1999$ cannot be factored, the roots are really ugly looking numbers. Namely, the two possible values of $b$ are
${b}_{1}=\frac{1949+\sqrt{3790605}}{2}\phantom{\rule{0ex}{0ex}}{b}_{2}=\frac{1949-\sqrt{3790605}}{2}$
and with the $a$ values as
${a}_{1}=\frac{1949-\sqrt{3790605}}{2}\phantom{\rule{0ex}{0ex}}{a}_{2}=\frac{1949+\sqrt{3790605}}{2}$

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