Question 344976
Divide n into 3 parts such that the first shall be one-rth of the second and one-kth of the third.
<pre>
Let x be the first part of n
Let y be the second part of n
Ley z be the thitrd part of n

Then 

x + y + z = n

>>...the first shall be one-rth of the second...<<

x = {{{1/r}}}*y

>>...the first shall be...one-kth of the third...<<

x = {{{1/k}}}*z

So we have this system of three equations and three unknowns:

{{{system(x+y+z=n,x=expr(1/r)*y, x=expr(1/k)*z)}}}

Clearing the 2nd and 3rd of fractions:

{{{system(x+y+z=n,rx=y, kx=z)}}}

Substituting rx for y and kx for z in the first equation:

{{{x+rx+kx=n}}}

{{{(1+r+k)x=n}}}

{{{x=n/(1+r+k)}}}

Since {{{y=rx}}},

{{{y=r*expr(n/(1+r+k))}}}

{{{y=rn/(1+r+k)}}}

Since {{{z=kx}}},

{{{y=k*expr(n/(1+r+k))}}}

{{{y=kn/(1+r+k)}}}

So the parts are

{{{n/(1+r+k)}}}, {{{rn/(1+r+k)}}}, {{{kn/(1+r+k)}}}

Edwin</pre>