Question

[Thục Uyên]

the ratio of three numbers is 2/5:3/4:1/6.Sum of squares of them is 24309.FInd the sum of them

Answer 1

[Nguyễn Huy Tú]

Giải:
Gọi 3 số đó là a, b, c

Ta có: \(\frac{a}{\frac{2}{5}}=\frac{b}{\frac{3}{4}}=\frac{c}{\frac{1}{6}}\) và \(a^2+b^2+c^2=24309\)

Đặt \(\frac{a}{\frac{2}{5}}=\frac{b}{\frac{3}{4}}=\frac{c}{\frac{1}{6}}=k\Rightarrow\left\{\begin{matrix}a=\frac{2}{5}k\\b=\frac{3}{4}k\\c=\frac{1}{6}k\end{matrix}\right.\)

Mà \(a^2+b^2+c^2=24309\)

\(\Rightarrow\frac{4}{25}k^2+\frac{9}{16}k^2+\frac{1}{36}k^2=24309\)

\(\Rightarrow\frac{2701}{3600}k^2=24309\)

\(\Rightarrow k^2=32400\)

\(\Rightarrow\left\{\begin{matrix}k=180\\k=-180\end{matrix}\right.\)

+) \(k=180\Rightarrow a=72;b=135;c=30\)

\(\Rightarrow a+b+c=237\)

+) \(k=-180\Rightarrow a=-72;b=-135;c=-30\)

\(\Rightarrow a+b+c=-237\)

Vậy \(\left[\begin{matrix}a+b+c=237\\a+b+c=-237\end{matrix}\right.\)