Each element in a data set is multiplied by 2, and each resulting product is then increased by 5. If 'm' is the mean of the final data set, which of the following expressions gives the mean of the original set in terms of 'm'?
~why is the answer (1/2)(m-5)?
To understand why the mean of the original set is given by the expression (1/2)(m-5), let's break down the steps involved in the given operations:
1. Each element in the original data set is multiplied by 2.
This means that if we have a data set [a, b, c], it becomes [2a, 2b, 2c].
2. Each resulting product is increased by 5.
Now, the data set becomes [2a+5, 2b+5, 2c+5].
3. 'm' is the mean of the final data set.
The mean (average) is calculated by summing up all the elements and dividing it by the total number of elements. Let's assume there are 'n' elements in the original data set.
Now, we need to find the mean of the original set in terms of 'm'. Let's calculate it step by step:
Mean of the final data set = m
To find the mean, we add up all the elements and divide by the number of elements:
(2a+5 + 2b+5 + 2c+5) / 3 = m
Simplifying the expression:
(2a + 2b + 2c + 15) / 3 = m
Next, let's substitute 'm' with (1/2)(m-5), as per the given answer choice. We substitute 'm' as (1/2)(m-5) in the expression above:
(2a + 2b + 2c + 15) / 3 = (1/2)(m-5)
Now, we can simplify this equation to solve for the mean of the original data set:
2(a+b+c) + 15 = (1/2)(m-5) * 3
2(a+b+c) + 15 = (m-5) * 3/2
2(a+b+c) + 15 = (3/2)m - 15/2
2(a+b+c) = (3/2)m - 15/2 - 15
2(a+b+c) = (3/2)m - 30/2 - 15/2
2(a+b+c) = (3/2)m - 45/2
(a+b+c) = (3/4)m - 45/4
Therefore, the mean of the original data set, denoted as (a+b+c), is equal to (3/4)m - 45/4.
To find the mean of the original data set, we need to reverse the operations that were done to each element.
Let's go step by step:
1. Each element in the data set is multiplied by 2: This means that each element is doubled. To reverse this operation, we need to divide each element by 2.
2. Each resulting product is then increased by 5: This means that after multiplying each element by 2, 5 is added to each element. To reverse this operation, we need to subtract 5 from each element.
Since we want to find the mean of the original set in terms of 'm', we need to find an expression that represents these operations in reverse order.
Starting with 'm', which is the mean of the final data set, we reverse the two operations:
1. Divide each element by 2.
2. Subtract 5 from each element.
Therefore, the expression that gives the mean of the original set in terms of 'm' is: (1/2)(m-5).
Let's double-check by applying this expression to find the mean of the original data set:
1. Multiply each element by 2: (1/2)(m-5) * 2 = (m-5).
2. Add 5 to each element: (m-5) + 5 = m.
As we can see, the expression (1/2)(m-5) correctly gives us the mean of the original set, which is 'm'.