The second number in a series is twice the first number and three times the third number . Arithmetic mean of these three numbers is The largest number will be

The correct answer is (b) 66.

Let $x$ be the first number, $y$ be the second number, and $z$ be the third number. We know that $y = 2x$ and $y = 3z$. We also know that the arithmetic mean of these three numbers is 60. This means that $(x + y + z)/3 = 60$. Substituting in the expressions for $y$ and $z$, we get $(x + 2x + 3x)/3 = 60$. Solving for $x$, we get $x = 12$. Substituting this into the expressions for $y$ and $z$, we get $y = 24$ and $z = 18$. Therefore, the largest number is $\boxed{24}$.

Here is a brief explanation of each option:

• Option (a): 72. This is not possible, because the arithmetic mean of three numbers cannot be greater than any of the numbers themselves.
• Option (b): 66. This is the correct answer, as shown above.
• Option (c): 60. This is the arithmetic mean of the three numbers, but it is not the largest number.
• Option (d): 54. This is not possible, because the arithmetic mean of three numbers cannot be less than any of the numbers themselves.