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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The mean of a group of 100 observations was found to be 20. Later it was found that four observations were incorrect, which were recorded as 21, 21, 18 and 20. What is the mean if the incorrect observations are omitted?
