Question

question 1 - 7
the weights of the points a and b, shown on the number line, are 67% and 33%, respectively.
which statement is true?
the weighted average of a and b lies to the right of point b.
the weighted average of a and b is closer to point a than to point b.
the weighted average of a and b is closer to point b than to point a.
the weighted average of a and b lies exactly midway between the points a and b.

Explanation:

Step1: Calculate the weighted average

Let \(A = 3\) (weight \(67\%=0.67\)) and \(B = 6\) (weight \(33\% = 0.33\)).
The formula for the weighted average \(W\) is \(W=A\times w_A + B\times w_B\), where \(w_A\) and \(w_B\) are the weights.
So \(W=3\times0.67 + 6\times0.33\).
First, calculate \(3\times0.67 = 2.01\) and \(6\times0.33=1.98\).
Then \(W=2.01 + 1.98=3.99\approx4\).

Step2: Calculate the distances

The distance from the weighted average \(W = 4\) to \(A = 3\) is \(|4 - 3|=1\).
The distance from the weighted average \(W = 4\) to \(B = 6\) is \(|4 - 6| = 2\).

Answer:

The weighted average of \(A\) and \(B\) is closer to point \(A\) than to point \(B\).