Exercise 67

The average of three numbers is \( x \). If one of the three numbers is \( x + 4 \), then the average of the other two is:

  1. \( 2x - 4 \)

  2. \( \frac{x - 4}{2} \)

  3. \( x - 4 \)

  4. \( x - 2 \)

  5. \( \frac{x + 2}{2} \)

Try to solve it before checking the answer.

  1. \( x – 2 \)

Let \(a\), \(b\) and \(c\) be the three numbers.

We are told that   \( \cfrac{a + b + c}{3} = x \),   hence:

\(\boldsymbol{(1)} \hspace{2em} a + b + c = 3x \)

Let’s assume that:

\(\boldsymbol{(2)} \hspace{2em} c = x + 4 \)

We are asked that \( \frac{a + b}{2} \), so, replacing (2) in (1):

\[ \begin{aligned} a + b + x + 4 = 3x &\Rightarrow a + b = 2x – 4 \\[2em] &\Rightarrow \frac{a + b}{2} = \frac{2x – 4}{2} = \frac{2(x – 2)}{2} = \boldsymbol{x – 2} \end{aligned} \]
\[ \begin{aligned} a + b + x &+ 4 = 3x \\[2em] &\Rightarrow a + b = 2x – 4 \\[2em] &\Rightarrow \frac{a + b}{2} = \frac{2x – 4}{2} \\ &\hspace{3em}= \frac{2(x – 2)}{2} \\[2em] &\hspace{3em}= \boldsymbol{x – 2} \end{aligned} \]

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