three numbers are such that the second is the difference of three times the first and 6 while the third is the sum of 2 and 2/3 the second. The sum of the three numbers is 172. Find the largest number.

n1= first number

n2 = second number

n3 = third number

n2 = 3 n1 - 6

n3 = 2 + ( 2 / 3 ) n2

n3 = 2 + ( 2 / 3 ) * ( 3 n1 - 6 )

n3 = 2 + ( 2 / 3 ) * 3 n1 - ( 2 / 3 ) * 6

n3 = 2 + 2 n1 - 4

n3 = 2 n1 - 2

n1 + n2 + n3 = 172

n1 + 3 n1 - 6 + 2 n1 - 2 = 172

6 n1 - 8 = 172 add 8 to both sides

6 n1 - 8 + 8 = 172 + 8

6 n1 = 180 Divide both sides by 6

n1 = 180 / 6 = 30

n2 = 3 n1 - 6

n2 = 3 * 30 - 6 = 90 - 6 = 84

n3 = 2 n1 - 2

n3 = 2 * 30 - 2 = 60 - 2 = 58

The largest number = n2 = 84

To find the largest number among the three given numbers, we need to solve the problem step by step.

Let's assume the first number is "x."

According to the given information:
- The second number is the difference of three times the first number and 6, so it can be expressed as 3x - 6.
- The third number is the sum of 2 and two-thirds (2/3) of the second number, which can be written as 2 + (2/3) * (3x - 6).

The sum of the three numbers is 172, so we can set up the equation:
x + (3x - 6) + [2 + (2/3) * (3x - 6)] = 172

Simplifying the equation:
x + 3x - 6 + 2 + (2/3) * (3x - 6) = 172
4x - 4 + (2/3) * (3x - 6) = 172
4x - 4 + 2x - 4 = 172
6x - 8 = 172
6x = 180
x = 30

Now, we have the value of the first number, x, which is 30.

To find the largest number, we substitute the value of x back into the expressions we derived earlier.

The second number = 3x - 6 = 3 * 30 - 6 = 90 - 6 = 84
The third number = 2 + (2/3) * (3x - 6) = 2 + (2/3) * (3 * 30 - 6) = 2 + (2/3) * (90 - 6) = 2 + (2/3) * 84 = 2 + 56 = 58

So, the three numbers are 30, 84, and 58.

The largest number among these three is 84.