Oranges

Mother divided her three children's oranges in a ratio of 6:5:4. Two children gave 45 oranges. How many oranges were there?

Result

n =  75

Solution:

$a:b:c = 6+5+4 \ \\ a+b = 45 || b+c = 45 || c+a = 45 \ \\ \ \\ k_{ 1 } = \dfrac{ 6 }{ 6+5+4 } = \dfrac{ 2 }{ 5 } = 0.4 \ \\ k_{ 2 } = \dfrac{ 5 }{ 6+5+4 } = \dfrac{ 1 }{ 3 } \doteq 0.3333 \ \\ k_{ 3 } = \dfrac{ 4 }{ 6+5+4 } = \dfrac{ 4 }{ 15 } \doteq 0.2667 \ \\ \ \\ q_{ 1 } = k_{ 1 }+k_{ 2 } = 0.4+0.3333 = \dfrac{ 11 }{ 15 } \doteq 0.7333 \ \\ q_{ 2 } = k_{ 1 }+k_{ 3 } = 0.4+0.2667 = \dfrac{ 2 }{ 3 } \doteq 0.6667 \ \\ q_{ 3 } = k_{ 3 }+k_{ 2 } = 0.2667+0.3333 = \dfrac{ 3 }{ 5 } = 0.6 \ \\ \ \\ n_{ 1 } = 45/q_{ 1 } = 45/0.7333 = \dfrac{ 675 }{ 11 } \doteq 61.3636 \ \\ n_{ 2 } = 45/q_{ 2 } = 45/0.6667 = \dfrac{ 135 }{ 2 } = 67.5 \ \\ n_{ 3 } = 45/q_{ 3 } = 45/0.6 = 75 \ \\ \ \\ n = n_{ 3 } = 75 \ \\ \ \\ a = k_{ 1 } \cdot \ n = 0.4 \cdot \ 75 = 30 \ \\ b = k_{ 2 } \cdot \ n = 0.3333 \cdot \ 75 = 25 \ \\ c = k_{ 3 } \cdot \ n = 0.2667 \cdot \ 75 = 20 \ \\ \ \\ n = 75$

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