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)?
the transformation is :
multiply by 2, then add 5
to get back to where we started from, we have to not only do the inverse of each operation but also reverse the order in which these operations were done.
the last thing we did was add 5, so the first thing we have to do in the inverse is to subtract 5, thus the (m-5)
the first thing we did was multiply by 2, so the last thing we do in the inverse is divide by 2, thus the (m-5)/2
or
(1/2)(m-5)
thank you
To understand why the answer is (1/2)(m-5), let's break down the given steps and their effects on the mean.
1. Each element in the original data set is multiplied by 2.
Let's call this intermediate data set D1.
2. Each element in D1 is then increased by 5.
This gives us the final data set D2.
The mean of a data set is calculated by adding up all the values and dividing it by the total number of values.
Now, let's express the mean of the original set (let's call it "X") in terms of the mean of the final set (m).
Step 1: Multiplication of each element by 2 (D1)
When each element is multiplied by 2, the mean of D1 would be:
Mean of D1 = 2X
Step 2: Addition of 5 to each element (D2)
When 5 is added to each element of D1, the mean of D2 would be:
Mean of D2 = 2X + 5
Since we want to express the mean of the original set (X) in terms of m, we can set up an equation using the known relationship between the means:
Mean D2 = m
Substituting the expressions we derived for the means of D1 and D2 into the equation, we get:
2X + 5 = m
Now, solving this equation for X gives us:
2X = m - 5
X = (m - 5) / 2
Therefore, the mean of the original set (X) in terms of m is (1/2)(m - 5).
To determine the expression that gives the mean of the original data set in terms of 'm', we need to reverse the operations that were performed on each element in the data set.
Let's break down the operations:
1. Each element in the data set is multiplied by 2.
2. Each resulting product is then increased by 5.
So, to reverse the operations, we need to perform the opposite operations in reverse order.
1. Reverse operation: Subtract 5 from each element.
2. Reverse operation: Divide each resulting difference by 2.
Now, let's start with 'm', the mean of the final data set, and reverse these operations step by step.
1. Reverse operation: Subtract 5 from 'm'.
This gives us 'm - 5'.
2. Reverse operation: Divide 'm - 5' by 2.
This gives us '(m - 5) / 2'.
Therefore, the expression that gives the mean of the original data set in terms of 'm' is '(m - 5) / 2'.
The expression (1/2)(m - 5) is equivalent to '(m - 5) / 2' because dividing by 2 is the same as multiplying by 1/2. Thus, both expressions are correct.