SOLUTION: Four cities lie along a straight highway in the order of city A B C and D. the distance from city A to city B is five times the distance from city B to city CD. The distance from

Algebra ->  Length-and-distance -> SOLUTION: Four cities lie along a straight highway in the order of city A B C and D. the distance from city A to city B is five times the distance from city B to city CD. The distance from      Log On


   

Question 1046349: Four cities lie along a straight highway in the order of city A B C and D. the distance from city A to city B is five times the distance from city B to city CD. The distance from city A to city D is two times The distance from city A to city B the distance between cities C to D is 10 miles. What is the distance between city AB?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
There are typos in that question.
I am guessing the question meant states:
The distance from city A to city B is five times the distance from city B to city C.
The distance from city A to city D is two times the distance from city A to city B.
The distance between cities C to D is 10 miles.

A SHORT WRITE UP:
AB=5BC
AD=2AB
CD=10
AD=AB%2BBC%2BCD
2AB=AB%2BBC%2B10
2%285BC%29=5BC%2BBC%2B10
10BC=6BC%2B10
10BC-6BC=10BC=10
BC=10%2F4
BC=2.5
5BC=5%282.5%29
5BC=12.5
AB=12.5
The distance between city A and city B is highlight%2812.5miles%29 .

A LONGER EXPLANATION, WITH A VARIABLE SO IT LOOKS MORE LIKE ALGEBRA:
Let us define a variable x .
x= the distance from city B to city C, in miles.
(We could call it BC , but x is shorter).
Since the distance from city A to city B is five times the distance from city B to city C,
the distance from city A to city B is 5x .
The distance from A to C must be the sum of the two distances listed above:
5x%2Bx=6x .
So far we have this picture:

We also know that the distance from city A to city D is two times the distance from city A to city B.
So, th2 distance from city A to city D is
2%2A%285x%29=10x ,
and we know that the distance between cities C to D is 10 miles.
Now, the picture looks like this:
.
Considering the distances shown by the arrows
between A and C %286x%29, and a
between A and D %2810x%29 ,
the distance between C and D must be their difference
10x%2B6x=4x .
Now, we can represent the situation as

Finally since the distance, in miles, between C and D is 10 and 4x ,
4x=10
x=10%2F4
x=5 is the distance, in miles, between B and C,
and the distance, in miles, between A and B is
5x=5%2A2.5
5x=highlight%2812.5%29 .
The distance between city A and city B is highlight%2812.5miles%29 .