Understanding the Problem: Mean and Combined Sets

The question asks for the sum of a second set of numbers, given information about its size and the mean of a combined set formed by merging it with a first set. We are given the size and mean of the first set, and the mean of the combined set.

The mean of a set of numbers is calculated by dividing the sum of the numbers by the total count of numbers in the set. Mathematically, this is represented as:

Mean = Sum of numbers / Total count of numbers

From this definition, we can also say:

Sum of numbers = Mean × Total count of numbers

Analyzing the Given Information

Let's break down the information provided for each set:

• First Set:
  • Number of elements = 10
  • Mean = M
• Second Set:
  • Number of elements = M
  • Mean = Unknown
  • Sum = What we need to find. Let's call this S₂.
• Combined Set (First Set + Second Set):
  • Number of elements = (Number of elements in First Set) + (Number of elements in Second Set) = $10 + M$
  • Mean = 10
  • Sum = (Sum of numbers in First Set) + (Sum of numbers in Second Set)

Calculating the Sum of the First Set

Using the formula Sum = Mean × Total count of numbers for the first set:

Sum of First Set = (Mean of First Set) × (Number of elements in First Set)

Sum of First Set = $M \times 10 = 10M$

Calculating the Sum of the Combined Set

Using the same formula for the combined set:

Sum of Combined Set = (Mean of Combined Set) × (Number of elements in Combined Set)

Sum of Combined Set = $10 \times (10 + M)$

Sum of Combined Set = $100 + 10M$

Finding the Sum of the Second Set

The sum of the combined set is also equal to the sum of the individual sets:

Sum of Combined Set = Sum of First Set + Sum of Second Set

We know the Sum of Combined Set is $100 + 10M$ and the Sum of First Set is $10M$. We are looking for the Sum of Second Set (S₂).

So, we can write the equation:

$100 + 10M = 10M + S_2$

To find S₂, we can subtract $10M$ from both sides of the equation:

$100 + 10M - 10M = S_2$

$100 = S_2$

Thus, the sum of the second set of numbers is 100.

Summary of Calculations

Using the relationship: Sum of Combined Set = Sum of First Set + Sum of Second Set

$100 + 10M = 10M + S_2$

Subtracting $10M$ from both sides:

$S_2 = 100$

The sum of the second set of numbers is 100.