## Cottages on a Straight Road

There are four cottages on a straight road. The distance between Ted’s and Alice’s cottages is $3$ km. Both Bob’s and Carol’s cottages are twice as far from Alice’s as they are from Ted’s. Find the distance between Bob’s and Carol’s cottages in kilometers.
Source: NCTM Mathematics Teacher, August 2006

Solution

Suppose the four cottages of Bob, Ted, Carol, and Alice are located on a straight line with respective coordinates  $b,t,c$, and $a$ such that $0.
Recall that $|x-y|=$ distance between real numbers $x$ and $y$ on the number line.

$|t-a|=$ distance between Ted’s and Alice’s cottages
$|b-a|=$ distance between Bob’s and Alice’
s cottages
$|c-a|=$ distance between Carol’s and Alice’s cottages
$|c-t|=$ distance between Carol’s and Ted’s cottages
Given that $|t-a|=3,|b-a|=2|b-t|$, and $|c-a|=2|c-t|$, we want to find $|b-c|$ the distance between Bob’s and Carol’s cottages.
$|t-a|=3$ implies $a-t=3\qquad\qquad\:\: (1)$
Calculate $|b-t|$ the distance between Bob’s and Ted’s cottages
$|b-a|=2|b-t|$ implies $a-b=2(t-b)$
Add $(b-t)$ to both sides of the equation
$a-b+(b-t)=2t-2b+(b-t)$
$a-t=t-b$
Substitute the value of $(a-t)$ from Eq. $(1)$
$3=t-b$ which implies $|b-t|=3\qquad (2)$
Calculate $|t-c|$ the distance between Ted’s and Carol’s cottages
$|t-c|+|c-a|=3$
$|t-c|+2|c-t|=3$
$3|t-c|=3$
$|t-c|=1\qquad\qquad (3)$
Calculate $|b-c|$ the distance between Bob’s and Carol’s cottages. From Eqs. $(2)$ and $(3)$
$|b-c|=|b-t|+|t-c|=3+1=4$ km

Answer: $4$ km