What rational number between ½ and 2/3 is three times as far from ½ than from 2/3?

Draw a number line.

On the left write 1/2. On the right write 2/3. What is the difference between 2/3 and 1/2. I have 1/6 but you need to confirm that.
Then divide that 1/6. You want the number that is 3 times farther from 1/2 than 2/3 so 3x = 1/6 = 1/18, now add 1/18 to 1/2. If I didn't goof that is the number 5/9. Check it out.
Difference between 1/2 and 5/9 = ??(1)
Difference between 5/9 and 2/3 = ??(2)
Now is the number (1) three times the number (2).

no

To find the rational number that is three times as far from ½ as it is from 2/3, we need to find a number that satisfies the given condition.

Let's assume the rational number we are looking for is x. According to the given condition, the distance between x and ½ is three times the distance between x and 2/3.

Mathematically, we can express this as:

|x - ½| = 3|x - 2/3|

To make it easier, we will solve this equation by splitting it into two cases, one for when x is greater than ½ and one for when x is less than ½.

Case 1: x > ½

In this case, we can write the equation as:

x - ½ = 3(x - 2/3)

Simplifying the equation:

x - ½ = 3x - 2

Bringing the variables to one side and the constants to the other:

x - 3x = -2 + ½

-2x = -3/2

Dividing both sides by -2:

x = 3/4

So, in this case, the solution is x = 3/4.

Case 2: x < ½

In this case, the equation becomes:

½ - x = 3(x - 2/3)

Simplifying the equation:

½ - x = 3x - 2

Bringing the variables to one side and the constants to the other:

-4x = -2 - ½

-4x = -5/2

Dividing both sides by -4:

x = 5/8

So, in this case, the solution is x = 5/8.

Therefore, the rational numbers satisfying the given condition are 3/4 and 5/8.