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.