There are four cottages on a straight road. The distance between Ted’s and Alice’s cottages is 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 , and such that .

Recall that distance between real numbers and on the number line.

distance between Ted’s and Alice’s cottages

distance between Bob’s and Alice’s cottages

distance between Carol’s and Alice’s cottages

distance between Carol’s and Ted’s cottages

Given that , and , we want to find the distance between Bob’s and Carol’s cottages.

implies

Calculate the distance between Bob’s and Ted’s cottages

implies

Add to both sides of the equation

Substitute the value of from Eq.

which implies

Calculate the distance between Ted’s and Carol’s cottages

Calculate the distance between Bob’s and Carol’s cottages. From Eqs. and

km

**Answer**: km