The mean of 18 numbers is denoted by x. The mean of the first 10 numbers is 45, while the mean of the last 5 numbers is 60. The 11th, 12th, and 13th values are given as (2x - 30), (x + 40), and 110, respectively. What is the value of the 12th number?

The correct option is 98.

Step-by-Step Explanation:

Step 1: Express the total sum of all 18 numbers in terms of $x$.
Given that the mean of 18 numbers is $x$, the sum of all 18 numbers is:

$\text{Sum of 18 numbers}=18\times x=18x$

Step 2: Calculate the sums of the sub-groups.
- Mean of the first 10 numbers is 45, so their sum is:

$\text{Sum of first 10 numbers}=10\times 45=450$

- Mean of the last 5 numbers is 60, so their sum is:

$\text{Sum of last 5 numbers}=5\times 60=300$

- The 11th, 12th, and 13th values are given as $(2x-30)$, $(x+40)$, and $110$, respectively.

Step 3: Formulate the total sum equation.
The sum of all 18 numbers is equal to the sum of the first 10 numbers, plus the 11th, 12th, and 13th numbers, plus the sum of the last 5 numbers:

$\text{Total Sum}=450+(2x-30)+(x+40)+110+300$

Simplifying the right-hand side:

$\text{Total Sum}=(450-30+40+110+300)+(2x+x)$

$\text{Total Sum}=870+3x$

Step 4: Solve for $x$.
Equating the total sum expressions:

$18x=870+3x$

$18x-3x=870$

$15x=870$

$x=\frac{870}{15}=58$

Step 5: Find the value of the 12th number.
Substitute $x=58$ into the expression for the 12th number $(x+40)$:

$\text{12th number}=58+40=98$

Hence, the value of the 12th number is 98.