A runs 5 times faster than B. If A gives B a start of 100m, how far must the goal be so that A and B reach there at the same time.

125 m

$\text{Speed ratio }A:B=5:1.$

$\text{Let total distance}=D.$

$\text{B gets a start of }100\text{ m, so B runs }(D-100).$

$\text{Equal time condition:}$

$\frac{D}{5}=\frac{D-100}{1}.$

$D=5(D-100).$

$D=5D-500.$

$4D=500.$

$D=125.$

$\text{Distance of goal from A}=125\text{ m}.$