Thread: Number line problem

On a number line, tick marks are spaced so that the distance from a to b is twice the distance from b to c and the distance from b to c is twice the distance from c to d. If a, b, c, and d are positive integers, what is the smallest possible value of d?

How can we show that it is 8?

Originally Posted by mjoshua

On a number line, tick marks are spaced so that the distance from a to b is twice the distance from b to c and the distance from b to c is twice the distance from c to d. If a, b, c, and d are positive integers, what is the smallest possible value of d?

How can we show that it is 8?

1. The smallest possible tick is at 1. Thus a = 1.

2. Let x denote the distance between a and b: b = a + x

3. Then you get:



4. Obviously the quotient is a positive integer if x = 4, therefore d = 8.

Originally Posted by mjoshua

On a number line, tick marks are spaced so that the distance from a to b is twice the distance from b to c and the distance from b to c is twice the distance from c to d. If a, b, c, and d are positive integers, what is the smallest possible value of d?

How can we show that it is 8?

Could it be that you didn't mention that a < b < c < d ?

Since the distance is a scalar (and not a vector) the ticks could be arranged:

Code:

0--1--2--3--4--5\\
---a--d--c-----b

In this case the smallest value of d = 2.

Maybe there other arrangement possible (?)

Hello, mjoshua!

Did you make a sketch?

On a number line, tick marks are spaced so that:
. . the distance from a to b is twice the distance from b to c
. . and the distance from b to c is twice the distance from c to d.
If a, b, c, and d are positive integers, what is the smallest possible value of d?

How can we show that it is 8?

The number line looks like this:

. .

Since we want the smallest value of , let
. . and let , the smallest positive integer.


Then we have:

. .


Therefore: .


Earboth,
I love your "second thought"!
And I'm impressed that you found a solution . . .

Originally Posted by Soroban

[size=3]...

Earboth,
I love your "second thought"!
And I'm impressed that you found a solution . . .

Soroban,

Thank you very much.

Even though it isn't necessary - I believe - I will only state for completeness that d = 4 is of course a valid result too.