Question

1. to increase the mean of 4 numbers by 3, by how much would the sum of the 4 numbers have to increase?

f. $\frac{3}{4}$
g. 1
h. $\frac{4}{3}$
j. 7
k. 12

Explanation:

Step1: Recall the formula for the mean

The formula for the mean \(\bar{x}\) of \(n\) numbers \(x_1,x_2,\cdots,x_n\) is \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\). Let the original sum of the \(4\) numbers be \(S\), so the original mean is \(\bar{x}_1=\frac{S}{4}\).

Step2: Define the new mean and new sum

The new mean \(\bar{x}_2=\bar{x}_1 + 3\). Let the new sum be \(S'\). Then \(\bar{x}_2=\frac{S'}{4}\). Since \(\bar{x}_2=\bar{x}_1 + 3\), we have \(\frac{S'}{4}=\frac{S}{4}+3\).

Step3: Solve for \(S'\) in terms of \(S\)

Multiply both sides of the equation \(\frac{S'}{4}=\frac{S}{4}+3\) by \(4\). Using the distributive property \(a(b + c)=ab+ac\) (here \(a = 4\), \(b=\frac{S}{4}\), \(c = 3\)), we get \(S'=S + 12\).

Answer:

K. \(12\)