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  • A $15$ cm long column of ants starts crawling.
  • A rebel ant at the end of the column steps out and starts marching forward at a higher speed than the column.
  • On reaching the front of the column, it immediately turns around and marches back at the same speed.
  • When he reaches the end of the column, he finds that the column of the remaining ants has moved exactly $15$ cm.
  • What distance did the rebel ant travel ?.

I tried to solve it by simple equations, but I am stuck. I am getting more variables than equations. Please help !.

  • 1
    Show you solution(equations). – 2017-01-13
  • 0
    1/(v2-v1)+1/(v2+v1)=1/v1 – 2017-01-13
  • 0
    v1=vel of column and v2= vel of ant – 2017-01-13

1 Answers 1

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Let $v_1$ be the speed of the column and $v_2$ the speed of the rebel ant, and let $r=\frac{v_2}{v_1}$. Then $v_2=rv_1$, and

$$\frac{15}{v_1}=\frac{15}{v_2-v_1}+\frac{15}{v_2+v_1}=\frac{15}{(r-1)v_1}+\frac{15}{(r+1)v_1}=\frac{30r}{(r^2-1)v_1}\;.$$

Multiply through by $\frac{v_1}{15}$ to get

$$1=\frac{2r}{r^2-1}$$

and hence $r^2-2r-1=0$. The ratio $r$ is not negative, so

$$r=\frac{2+\sqrt{4+4}}2=1+\sqrt2\;.$$

Thus, in the time that it takes the column to travel $15$ cm., the rebel ant travels $15(1+\sqrt2)$ cm., or a little over $36.2$ cm.