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 Mean and Combined Sets
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. The formula for the mean ($\bar{x}$) is:
$$ \bar{x} = \frac{\text{Sum of numbers}}{\text{Number of elements}} $$
Mean of the First Set
We are given that the first set has 10 numbers, and its mean is M.
Let the sum of the first set be S1.
Using the mean formula:
$$ M = \frac{S_1}{10} $$
Multiplying both sides by 10, we find the sum of the first set:
$$ S_1 = 10 \times M = 10M $$
Information about the Second Set
The second set is combined with the first set. The number of elements in the second set is given as M.
Let the sum of the second set be S2. This is what we need to find.
Combined Set Analysis
When the two sets are combined:
- The total number of elements in the combined set is the sum of the number of elements in each set: $10 + M$.
- The total sum of the numbers in the combined set is the sum of the sums of the individual sets: $S_1 + S_2 = 10M + S_2$.
We are also given that the mean of the combined set is 10.
Calculating the Sum of the Second Set
Using the mean formula for the combined set:
$$ \text{Mean of combined set} = \frac{\text{Sum of combined set}}{\text{Total number of elements}} $$
Substitute the known values:
$$ 10 = \frac{10M + S_2}{10 + M} $$
Now, we need to solve this equation for S2.
Multiply both sides by $(10 + M)$:
$$ 10 \times (10 + M) = 10M + S_2 $$
Distribute the 10 on the left side:
$$ (10 \times 10) + (10 \times M) = 10M + S_2 $$
$$ 100 + 10M = 10M + S_2 $$
To isolate S2, subtract 10M from both sides of the equation:
$$ 100 + 10M - 10M = 10M + S_2 - 10M $$
$$ 100 = S_2 $$
So, the sum of the second set of numbers is 100.
Final Answer Check
We found that the sum of the second set is 100. Let's quickly check if this makes sense in the equation:
$$ 10 = \frac{10M + 100}{10 + M} $$
$$ 10(10 + M) = 10M + 100 $$
$$ 100 + 10M = 10M + 100 $$
This equation is true, confirming our calculation for S2 is correct.
The sum of the second set of numbers is 100.
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