Matematică >> rapoarte și proporții >> 3
Dacă numerele \( \color{red} x_1 \), \( \color{red} x_2 \), \( \color{red} x_3 \), ..., \( \color{red} x_n \) sunt
direct proporționale cu numerele \( \color{blue} y_1 \), \( \color{blue} y_2 \), \( \color{blue} y_3 \), ..., \( \color{blue} y_n \),
atunci și sumele lor sunt direct proporționale,
iar astfel se obține șirul de rapoarte egale:
\( \displaystyle \frac{\color{red} x_1}{\color{blue} y_1} = \frac{\color{red} x_2}{\color{blue} y_2} = \frac{\color{red} x_3}{\color{blue} y_3} = \dots \frac{\color{red} x_n}{\color{blue} y_n} = \frac{\color{red} x_1 + x_2 + x_3 + \dots + x_n}{\color{blue} y_1 + y_2 + y_3 + \dots + y_n} \),
din care se poate determina orice termen, de exemplu:
\( \color{black} * \) \( \displaystyle \frac{\color{red} x_1}{\color{blue} y_1} = \frac{\color{red} x_1 + x_2 + x_3 + \dots + x_n}{\color{blue} y_1 + y_2 + y_3 + \dots + y_n} \Rightarrow \)
\( \displaystyle \color{red} x_1 \color{dimgray} = \frac{\color{blue} y_1 \color{dimgray} \cdot ( \color{red} x_1 + x_2 + x_3 + \dots + x_n \color{dimgray} )}{ \color{blue} y_1 + y_2 + y_3 + \dots + y_n} \)
sau
\( \displaystyle \color{blue} y_1 \color{dimgray} = \frac{\color{red} x_1 \color{dimgray} \cdot ( \color{blue} y_1 + y_2 + y_3 + \dots + y_n \color{dimgray} )}{ \color{red} x_1 + x_2 + x_3 + \dots + x_n} \)
\( \color{black} * \) \( \displaystyle \frac{\color{red} x_2}{\color{blue} y_2} = \frac{\color{red} x_1 + x_2 + x_3 + \dots + x_n}{\color{blue} y_1 + y_2 + y_3 + \dots + y_n} \Rightarrow \)
\( \displaystyle \color{red} x_2 \color{dimgray} = \frac{\color{blue} y_2 \color{dimgray} \cdot ( \color{red} x_1 + x_2 + x_3 + \dots + x_n \color{dimgray} )}{ \color{blue} y_1 + y_2 + y_3 + \dots + y_n} \)
sau
\( \displaystyle \color{blue} y_2 \color{dimgray} = \frac{\color{red} x_2 \color{dimgray} \cdot ( \color{blue} y_1 + y_2 + y_3 + \dots + y_n \color{dimgray} )}{ \color{red} x_1 + x_2 + x_3 + \dots + x_n} \)
...
\( \color{black} * \) \( \displaystyle \frac{\color{red} x_n}{\color{blue} y_n} = \frac{\color{red} x_1 + x_2 + x_3 + \dots + x_n}{\color{blue} y_1 + y_2 + y_3 + \dots + y_n} \Rightarrow \)
\( \displaystyle \color{red} x_n \color{dimgray} = \frac{\color{blue} y_n \color{dimgray} \cdot ( \color{red} x_1 + x_2 + x_3 + \dots + x_n \color{dimgray} )}{ \color{blue} y_1 + y_2 + y_3 + \dots + y_n} \)
sau
\( \displaystyle \color{blue} y_n \color{dimgray} = \frac{\color{red} x_n \color{dimgray} \cdot ( \color{blue} y_1 + y_2 + y_3 + \dots + y_n \color{dimgray} )}{ \color{red} x_1 + x_2 + x_3 + \dots + x_n} \)
Să se determine numerele știind că suma lor este \( 21 \).
Soluție:
Se poate scrie șirul de rapoarte egale:
\( \displaystyle \frac{\color{red} a}{\color{blue} 2} = \frac{\color{red} b}{\color{blue} 5} = \frac{\color{red} a+b }{\color{blue} 2+5} \), dar
\( \color{red} a + b \color{dimgray} = 21\), iar șirul devine:
\( \displaystyle \frac{\color{red} a}{\color{blue} 2} = \frac{\color{red} b}{\color{blue} 5} = \frac{21}{7} \),
de unde:
\( \displaystyle \frac{\color{red} a}{\color{blue} 2} = \frac{21}{7} \Rightarrow \color{red} a \color{dimgray} = \frac{\color{blue} 2 \color{dimgray} \cdot 21}{7} \Rightarrow \color{red} a \color{dimgray} = 6 \)
și
\( \displaystyle \frac{\color{red} b}{\color{blue} 5} = \frac{21}{7} \Rightarrow \color{red} b \color{dimgray} = \frac{\color{blue} 5 \color{dimgray} \cdot 21}{7} \Rightarrow \color{red} b \color{dimgray} = 15 \),
deci
\( \color{red} a \color{dimgray} = 6 \) și \( \color{red} b \color{dimgray} = 15 \).
2. Trei numere \( \color{red} a \), \( \color{red} b \) și \( \color{red} c \) sunt direct proporționale cu \( \color{blue} 2 \), \( \color{blue} 5 \), respectiv \( \color{blue} 4 \).
Să se determine numerele știind că suma lor este \( 77 \).
Soluție:
Se poate scrie șirul de rapoarte egale:
\( \displaystyle \frac{\color{red} a}{\color{blue} 2} = \frac{\color{red} b}{\color{blue} 5} = \frac{\color{red} c}{\color{blue} 4} = \frac{\color{red} a+b+c }{\color{blue} 2+5+4} \), dar
\( \color{red} a+b+c \color{dimgray} = 77 \), iar șirul devine:
\( \displaystyle \frac{\color{red} a}{\color{blue} 2} = \frac{\color{red} b}{\color{blue} 5} = \frac{\color{red} c}{\color{blue} 4} = \frac{77}{11} \),
de unde:
\( \displaystyle \frac{\color{red} a}{\color{blue} 2} = \frac{77}{11} \Rightarrow \color{red} a \color{dimgray} = \frac{\color{blue} 2 \color{dimgray} \cdot 77}{11} \Rightarrow \color{red} a \color{dimgray} = 14 \),
\( \displaystyle \frac{\color{red} b}{\color{blue} 5} = \frac{77}{11} \Rightarrow \color{red} b \color{dimgray} = \frac{\color{blue} 5 \color{dimgray} \cdot 77}{11} \Rightarrow \color{red} b \color{dimgray} = 35 \)
și
\( \displaystyle \frac{\color{red} c}{\color{blue} 4} = \frac{77}{11} \Rightarrow \color{red} c \color{dimgray} = \frac{\color{blue} 4 \color{dimgray} \cdot 77}{11} \Rightarrow \color{red} c \color{dimgray} = 28 \),
deci
\( \color{red} a \color{dimgray} = 14 \), \( \color{red} b \color{dimgray} = 35 \) și \( \color{red} c \color{dimgray} = 28 \).
3. Patru numere \( \color{red} a \), \( \color{red} b \), \( \color{red} c \) și \( \color{red} d \) sunt direct proporționale cu \( \color{blue} 2 \), \( \color{blue} 5 \), \( \color{blue} 4 \), respectiv \( \color{blue} 7 \).
Să se determine numerele știind că suma lor este \( 72 \).
Soluție:
Se poate scrie șirul de rapoarte egale:
\( \displaystyle \frac{\color{red} a}{\color{blue} 2} = \frac{\color{red} b}{\color{blue} 5} = \frac{\color{red} c}{\color{blue} 4} = \frac{\color{red} d}{\color{blue} 7} = \frac{\color{red} a+b+c+d }{\color{blue} 2+5+4+7} \), dar
\( \color{red} a+b+c+d \color{dimgray} = 72 \), iar șirul devine:
\( \displaystyle \frac{\color{red} a}{\color{blue} 2} = \frac{\color{red} b}{\color{blue} 5} = \frac{\color{red} c}{\color{blue} 4} = \frac{\color{red} d}{\color{blue} 7} = \frac{72}{18} \),
de unde:
\( \displaystyle \frac{\color{red} a}{\color{blue} 2} = \frac{72}{18} \Rightarrow \color{red} a \color{dimgray} = \frac{\color{blue} 2 \color{dimgray} \cdot 72}{18} \Rightarrow \color{red} a \color{dimgray} = 8 \),
\( \displaystyle \frac{\color{red} b}{\color{blue} 5} = \frac{72}{18} \Rightarrow \color{red} b \color{dimgray} = \frac{\color{blue} 5 \color{dimgray} \cdot 72}{18} \Rightarrow \color{red} b \color{dimgray} = 20 \),
\( \displaystyle \frac{\color{red} c}{\color{blue} 4} = \frac{72}{18} \Rightarrow \color{red} c \color{dimgray} = \frac{\color{blue} 4 \color{dimgray} \cdot 72}{18} \Rightarrow \color{red} c \color{dimgray} = 16 \)
și
\( \displaystyle \frac{\color{red} d}{\color{blue} 7} = \frac{72}{18} \Rightarrow \color{red} d \color{dimgray} = \frac{\color{blue} 7 \color{dimgray} \cdot 72}{18} \Rightarrow \color{red} d \color{dimgray} = 28 \),
deci
\( \color{red} a \color{dimgray} = 8 \), \( \color{red} b \color{dimgray} = 20 \), \( \color{red} c \color{dimgray} = 16 \) și \( \color{red} d \color{dimgray} = 28 \).
Trei numere \( \color{red} a \), \( \color{red} b \) și \( \color{red} c \) sunt direct proporționale cu \( \color{blue} 7 \), \( \color{blue} 3 \), respectiv \( \color{blue} 2 \).
Știind că suma numerelor este \( 72 \), numerele sunt:
exercițiu nou
Trei numere \( \color{red} a \), \( \color{red} b \) și \( \color{red} c \) sunt direct proporționale cu \( \color{blue} 7 \), \( \color{blue} 3 \), respectiv \( \color{blue} 2 \).
Știind că suma numerelor este \( 72 \), numerele sunt:
\( \color{red} a \color{dimgray} = 42 \), \( \color{red} b \color{dimgray} = 18 \) și \( \color{red} c \color{dimgray} = 12 \).
Se poate scrie șirul de rapoarte egale:
\( \displaystyle \frac{\color{red} a}{\color{blue} 7} =
\frac{\color{red} b}{\color{blue} 3} =
\frac{\color{red} c}{\color{blue} 2} =
\frac{\color{red} a+b+c }{\color{blue} 7+3+2} \), dar
\( \color{red} a + b + c\color{dimgray} = 72 \), iar șirul devine:
\( \displaystyle \frac{\color{red} a}{\color{blue} 7} =
\frac{\color{red} b}{\color{blue} 3} =
\frac{\color{red} c}{\color{blue} 2} =
\frac{72}{12} \),
de unde:
\( \displaystyle \frac{\color{red} a}{\color{blue} 7} = \frac{72}{12} \Rightarrow
\color{red} a \color{dimgray} = \frac{\color{blue} 7 \color{dimgray} \cdot 72}{12} \Rightarrow
\color{red} a \color{dimgray} = 42 \),
\( \displaystyle \frac{\color{red} b}{\color{blue} 3} = \frac{72}{12} \Rightarrow
\color{red} b \color{dimgray} = \frac{\color{blue} 3 \color{dimgray} \cdot 72}{12} \Rightarrow
\color{red} b \color{dimgray} = 18 \)
și
\( \displaystyle \frac{\color{red} c}{\color{blue} 2} = \frac{72}{12} \Rightarrow
\color{red} c \color{dimgray} = \frac{\color{blue} 2 \color{dimgray} \cdot 72}{12} \Rightarrow
\color{red} c \color{dimgray} = 12 \),
deci
\( \color{red} a \color{dimgray} = 42 \), \( \color{red} b \color{dimgray} = 18 \) și \( \color{red} c \color{dimgray} = 12 \).