Question

C, E and D points divide AB segment by ratio 1:2, 1:3, 1:4 (from point A).

What is the ratio of point E to DC?

please answer with step by step explanation​

Answer 1

`3:5`

Explanation:

`We\ are\ given\ that:\\AB\ is\ a\ Line\ Segment\ that\ has\ Points\ A\ and\ B,\ that\ are\ it's\ two\ End-Points.\\Now,\\Point\ C\ is\ marked\ on\ AB\ that\ divides\ AB\ into\ AC\ and\ CB,\ whose\\ lengths\ are\ in\ a\ Ratio\ 1:2.\\\\Point\ E\ is\ marked\ on\ AB\ that\ divides\ AB\ into\ AE\ and\ EB,\ whose\\ lengths\ are\ in\ a\ Ratio\ 1:3.\\\\Point\ D\ is\ marked\ on\ AB\ that\ divides\ AB\ into\ AD\ and\ DB,\ whose\\ lengths\ are\ in\ a\ Ratio\ 1:4.\\\\`

`Now,\\For\ Point\ C,\\Total\ no.\ of\ parts=1+2=3\\Length\ of\ AC=(1)/(3)x\\Length\ of\ CB=(2)/(3)x\\\\For\ Point\ E,\\Total\ no.\ of\ parts=1+3=4\\Length\ of\ AE=(1)/(4)x\\Length\ of\ EB=(3)/(4)x\\\\For\ Point\ D,\\Total\ no.\ of\ parts=1+4=5\\Length\ of\ AD=(1)/(5)x\\Length\ of\ DB=(4)/(5)x\\`

`Now,\\Lets\ move\ onto\ some\ REAL\ calculations!!\\We\ firstly\ observe\ that,\\AD+DC=AC\\Hence,\\(1)/(5)x+DC=(1)/(3)x\\DC=(1)/(3)x-(1)/(5)x\\DC=(5-3)/(15)=(2)/(15)`

`We\ secondly\ observe\ that,\\AD+DE=AE\\Hence,\\(1)/(5)x+DE=(1)/(4)x\\DE=(1)/(4)x-(1)/(5)x\\DE=(5-4)/(20)=(1)/(20)`

`Now,\\We\ know\ that,\\DE+EC=DC\\Hence,\\(1)/(20)x+EC=(2)/(15)x \\EC=(2)/(15)x-(1)/(20)x\\EC=(40-15)/(300)x\\EC=(25)/(300)x=(1)/(12)x\\\\By\ comparing\ DE\ and\ EC,\\Length\ of\ DE=(1)/(20)x\\Length\ of\ EC=(1)/(12)x\\The\ ratio\ of\ Lengths\ of\ DE\ to\ Length\ of\ EC=(1)/(20)x:(1)/(12)x\\We\ know\ that,\\Ratio's\ are\ merely\ fractions\ represented\ by\ ':'.\\Hence,\\(1)/(20)x:(1)/(12)x\\=((1)/(20)x)/((1)/(12)x)\\`

`=(1)/(20)*12\\=(12)/(20)=(3)/(5)=3:5`