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
image
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<b<t<c<a.
Recall that |x-y|= distance between real numbers x and y on the number line.
image
|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

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About mvtrinh

Retired high school math teacher.
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