[tex]\large\underline{\sf{Solution-}}[/tex]

Total number of mangoes = 230

The share of x, y, z are as

[tex]\sf \: x : y : z \: = \: \dfrac{1}{2} : \dfrac{1}{3} : \dfrac{1}{8} \\ \\ [/tex]

can be rewritten as

[tex]\sf \: x : y : z \: = \: \dfrac{12}{24} : \dfrac{8}{24} : \dfrac{3}{24} \\ \\ [/tex]

[tex]\bf\implies \:\sf \: x : y : z \: = \: 12 : 8 : 3 \\ \\ [/tex]

So, Sum of ratios = 12 + 8 + 3 = 23

Now,

[tex]\sf \: Share \: of \: x \: = \: \dfrac{12}{23} \times 230 = 12\times 10 = 120 \\ \\ [/tex]

[tex]\sf \: Share \: of \: y \: = \: \dfrac{8}{23} \times 230 = 8\times 10 = 80 \\ \\ [/tex]

[tex]\sf \: Share \: of \: z \: = \: \dfrac{3}{23} \times 230 = 3\times 10 = 30 \\ \\ [/tex]

Thus,

[tex]\bf\implies \:x \: got \: 120 \: mangoes \\ \\ [/tex]

[tex]\bf\implies \:y\: got \: 80 \: mangoes \\ \\ [/tex]

[tex]\bf\implies \:z\: got \: 30 \: mangoes \\ \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Additional Information

[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]

Step-by-step explanation:

Given :-

☞ Quantity of mangoes

To Find :-

☞ To divide these mangoes among x, y and z

Solution :-

❃ Total number of mangoes = 230

❖ Ratio at which mangoes are distributed = [tex]\sf \: \frac{1}{2} : \frac{1}{3} : \frac{1}{8} \\ [/tex]

Now,

☯ LCM of 2, 3 and 8 = 24

So,

[tex]\sf \: \frac{1}{2} : \frac{1}{3} : \frac{1}{8} [/tex] is also written as,

[tex] ☞ \tt \: \frac{1}{2} \times 24 : \frac{1}{3} \times 24 : \frac{1}{8} \times 24 \\ \\ \\ \red{☞ \tt \: 12 : 8 : 3}[/tex]

Sum of ratios = 12 + 8 + 3 = 23

Now,

✞ X's share = [tex]\tt \: \frac{12}{23} \times 230 = 12 \times 10 = 120 \: \rm \: mangoes[/tex]

✞ Y's share = [tex] \sf \: \frac{8}{23} \times 230 = 8 \times 10 = 80 \rm \: mangoes [/tex]

✞ Z's share = [tex]\bf \: \frac{3}{23} \times 230 = 3 \times 10 = 30 \: \rm mangoes[/tex]

So,

In this way

❖ X got 120 mangoes.

❖ Y got 80 mangoes.

❖ Z got 30 mangoes