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?

Correct result:

n =  75

Solution:

$$\begin{gathered} a:b:c=6+5+4 \\ a+b=45\mathrm{\mid }\mathrm{\mid }b+c=45\mathrm{\mid }\mathrm{\mid }c+a=45 \\ {k}_1={\displaystyle \frac{6}{6+5+4}}={\displaystyle \frac{2}{5}}=0.4 \\ {k}_2={\displaystyle \frac{5}{6+5+4}}={\displaystyle \frac{1}{3}}\doteq 0.3333 \\ {k}_3={\displaystyle \frac{4}{6+5+4}}={\displaystyle \frac{4}{15}}\doteq 0.2667 \\ {q}_1=k_1+k_2=0.4+0.3333={\displaystyle \frac{11}{15}}\doteq 0.7333 \\ {q}_2=k_1+k_3=0.4+0.2667={\displaystyle \frac{2}{3}}\doteq 0.6667 \\ {q}_3=k_3+k_2=0.2667+0.3333={\displaystyle \frac{3}{5}}=0.6 \\ {n}_1=45\mathrm{/}{q}_1=45\mathrm{/}0.7333={\displaystyle \frac{675}{11}}\doteq 61.3636 \\ {n}_2=45\mathrm{/}{q}_2=45\mathrm{/}0.6667={\displaystyle \frac{135}{2}}=67.5 \\ {n}_3=45\mathrm{/}{q}_3=45\mathrm{/}0.6=75 \\ n=n_3=75 \\ a=k_1\cdot n=0.4\cdot 75=30 \\ b=k_2\cdot n=0.3333\cdot 75=25 \\ c=k_3\cdot n=0.2667\cdot 75=20 \\ n=75 \end{gathered}$$

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