1.

divide 150 into 3 parts such that the second number is five-sixths the first and the third number is four-fifths the second.....

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Solution

Let the three part be a, b and c.

First part = 'x'

SECOND part = \dfrac{5}{6} of 'x' is \dfrac{5}{6}x.

Third part = \dfrac{4}{5} of second part

                ⇒  \dfrac{4}{5} ×  \dfrac{5}{6}x = \dfrac{2}{3}x

                                             

Sum of the three PARTS is 150

According to the question

\implies\dfrac{x+5x}{6}+\dfrac{2x}{3}=150\\\\\\\implies6x+5x+4x=6\times150\\\\\\\implies15x=900\\\\\\\implies x=\dfrac{900}{15}\\\\\\\implies x=\boxed{\underline{\boxed{60}}}

                                             

Let us verify our answer

For verification we have to PUT 60 at the place of 'x'

When we put value of 'x' then we obtain,

\implies\dfrac{60+5\times60}{6}+\dfrac{2\times60}{3}=150\\\\\\\implies\dfrac{360}{6}+\dfrac{120}{3}=150\\\\\\\implies6\times60+5\times60+4\times60=6\times150\\\\\\\implies360+300+240=900\\\\\\\implies900=900

Here, LHS=RHS

Hence, it is VERIFIED

                                                 

First number is 60

Second number is 5/6 of 'x'

                           = 5/6 × 60

                           = 50

Third number is 2/3 × 'x'

                        = 2/3 × 60

                        = 40


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