Question 983538
{{{A}}}= age of A,
{{{B}}}= age of B,
{{{C}}}= age of C,
{{{D}}}= age of D,
{{{E}}}= age of E, and
{{{F}}}= age of F.
We can use the ratios to calculate {{{B}}} , {{{C}}} , {{{D}}} , and {{{E}}} in terms of {{{A}}} :
The ratio of the ages of A:B=6:5. ---> {{{A/B=6/5}}}-->{{{5A=6B}}}-->{{{(5/6)A=B}}} .
The age of each C and D is 9/10 times that Of B. ---> {{{C=D=(9/10)B=(9/10)(5/6)A=(3/4)A}}} .
The ratio of age of B&E is 2:3. ---> {{{B/E=2/3}}}-->{{{2E=3B}}}-->{{{E=(3/2)B}}}-->{{{E=(3/2)(5/6)A=(5/4)A}}} .
For all those ages to be integers, {{{A}}} must be a multiple of {{{6}}} and {{{4}}} ,
so it must be a multiple of {{{12}}} .
We can use "guess and check" from here, trying 12, 24, 36, etc as values for {{{A}}} .
We can also say that {{{A=12N}}} for some whole number {{{N}}} ,
and express everyone's age as a function of {{{N}}} :
{{{A=12N}}} ,
{{{B=(5/6)(12N)=10N}}},
{{{C+D+(3/4)(12N)=9N}}}, and
{{{E=(5/4)(12N)=15N}}}
The age of A is 3 years less than E. ---> {{{12N=15N-3}}}-->{{{15N-12N=3}}}-->{{{3N=3}}}-->{{{N=3/3}}}-->{{{N=1}}} .
So,
{{{A=12N=12*1=12}}} and {{{B=10N=10*1=10}}} .
Since the age of F is less than A but greater than B, {{{F=11}}} ,
and the rati we need is
{{{A/F=highlight(12/11)}}} .