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Theo đề ta có: \(x:y:z=3:4:5\Rightarrow\frac{x}{3}=\frac{y}{4}=\frac{z}{5}\)
Đặt: \(\frac{x}{3}=\frac{y}{4}=\frac{z}{5}=k\left(k\inℕ^∗\right)\)
Suy ra: \(x=3k;y=4k;z=5k\) Thay vào biểu thức P ta có:
\(P=\frac{3k+8k+15k}{6k+12k+20k}+\frac{6k+12k+20k}{9k+16k+25k}+\frac{9k+16k+25k}{12k+20k+30k}\)
\(P=\frac{26k}{38k}+\frac{38k}{50k}+\frac{50k}{62k}=\frac{13}{19}+\frac{19}{25}+\frac{25}{31}=\frac{33141}{14725}\)
Giải:
Ta có: \(\frac{x}{3}=\frac{y}{4}\Rightarrow\frac{x}{15}=\frac{y}{20}\)
\(\frac{y}{5}=\frac{z}{6}\Rightarrow\frac{y}{20}=\frac{z}{24}\)
\(\Rightarrow\frac{x}{15}=\frac{y}{20}=\frac{z}{24}\)
Đặt \(\frac{x}{15}=\frac{y}{20}=\frac{z}{24}=k\)
\(\Rightarrow\hept{\begin{cases}x=15k\\y=20k\\z=24k\end{cases}}\)
\(\Rightarrow M=\frac{2x+3y+4z}{3x+4y+5z}=\frac{30k+60k+96k}{45k+80k+120k}=\frac{\left(30+60+96\right)k}{\left(45+80+120\right)k}\)
bạn tự tính nốt nhé
Mình giải tiếp cho:
\(M=\frac{\left(30+60+96\right)k}{\left(45+80+120\right)k}=\frac{186k}{245k}=\frac{186}{245}\)
Vậy \(M=\frac{186}{245}\)
\(\frac{x}{3}=\frac{y}{4}\Rightarrow\frac{x}{15}=\frac{y}{20}\left(1\right)\)
\(\frac{y}{5}=\frac{z}{6}\Rightarrow\frac{y}{20}=\frac{z}{24}\left(2\right)\)
từ (1) và (2) => \(\frac{x}{15}=\frac{y}{20}=\frac{z}{24}\)
đặt \(\frac{x}{15}=\frac{y}{20}=\frac{z}{24}=k\Rightarrow x=15k,y=20k,z=24k\)
thay x=15k, y=20k, z=24k vào M ta có:
\(M=\frac{2.15k+3.20k+4.24k}{3.15k+4.20k+5.24k}=\frac{30k+60k+96k}{45k+80k+120k}=\frac{186k}{245k}=\frac{186}{245}\)
vậy M=\(\frac{186}{245}\)
Ta có: \(\frac{x}{3}\)=\(\frac{y}{4}\)=> \(\frac{x}{15}\)=\(\frac{y}{20}\)
\(\frac{y}{5}\)=\(\frac{z}{6}\)=> \(\frac{y}{20}\)=\(\frac{z}{24}\) Vậy \(\frac{x}{15}\)=\(\frac{y}{20}\)=\(\frac{z}{24}\)
đặt \(\frac{x}{15}\)=\(\frac{y}{20}\)=\(\frac{z}{24}\)=k => x=15k; y=20k; z=24k
Thay x=15k; y=20k ; z=24k vào Biểu thức M ta có:
M=\(\frac{2x+3y+4z}{3x+4y+5z}\)=\(\frac{2.15k+3.20k+4.24k}{3.15k+4.20k+5.24k}\)=\(\frac{k\left(30+60+96\right)}{k\left(45+80+120\right)}\)=\(\frac{186}{245}\)
Theo bài ra ta có : \(\frac{x}{3}=\frac{y}{4}\Leftrightarrow x=\frac{3y}{4}\) ; \(\frac{y}{5}=\frac{z}{6}\Leftrightarrow z=\frac{6y}{5}\), Vậy ta có : \(M=\frac{2x+3y+z}{3x+4y+5z}=\frac{2.\frac{3y}{4}+3y+4.\frac{6y}{5}}{3.\frac{3y}{4}+4y+5.\frac{6y}{5}}=\frac{\frac{93y}{10}}{\frac{49y}{4}}=\frac{93}{10}.\frac{4}{49}=\frac{186}{245}\)
Ta có \(\frac{3}{x+3}=\frac{4}{y+4}\Rightarrow3\left(y+4\right)=4\left(x+3\right)\Rightarrow3y=4x\Rightarrow\frac{x}{3}=\frac{y}{4}\)
Tương tự \(\frac{y+10}{5}=\frac{z+12}{6}\Rightarrow6\left(y+10\right)=5\left(z+12\right)\Rightarrow6y=5z\Rightarrow\frac{y}{5}=\frac{z}{6}\)
Từ đó ta có \(\frac{x}{15}=\frac{y}{20}=\frac{z}{16}\)
Đặt \(\frac{x}{15}=\frac{y}{20}=\frac{z}{16}=t\Rightarrow\hept{\begin{cases}x=15t\\y=20t\\z=16t\end{cases}}\)
Vậy thì \(M=\frac{2.15t+3.20t+4.16t}{3.15t+4.20t+5.16t}=\frac{154t}{205t}=\frac{154}{205}.\)
Vì \(\frac{x}{3}=\frac{y}{4}=>x=\frac{3}{4}y\)
\(\frac{y}{5}=\frac{z}{6}=>z=\frac{6}{5}y\)
Ta có
\(M=\frac{2x+3y+4z}{3x+4y+5z}=\frac{2.\frac{3}{4}y+3y+4.\frac{6}{5}y}{3.\frac{3}{4}y+4y+5.\frac{6}{5}y}=\frac{\frac{93}{10}y}{\frac{49}{4}y}=\frac{93}{10}:\frac{49}{4}=\frac{186}{245}\)