The mean of a set of 10 numbers is M. By combining with it a second set of M numbers, the mean of the combined set becomes 10. What is the sum of the second set of numbers?
100
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
| Set | Number of Elements | Mean | Sum of Numbers |
|---|---|---|---|
| First Set | 10 | M | $10 \times M = 10M$ |
| Second Set | M | Unknown | S2 |
| Combined Set | $10 + M$ | 10 | $10 \times (10 + M) = 100 + 10M$ |
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.
Important Questions from Measures of Central Tendency
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