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# help

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The arithmetic mean of three numbers x, y and z is 24. The arithmetic mean of x, 2y and z – 7 is 34. What is the arithmetic mean of x and z ? Express your answer as a decimal to the nearest tenth.

May 21, 2020

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The arithmetic mean of three numbers x, y and z is 24.
The arithmetic mean of x, 2y and z – 7 is 34.
What is the arithmetic mean of x and z ?

$$\begin{array}{|rcll|} \hline \mathbf{ \dfrac{x+y+z}{3} } &=& \mathbf{24} \\\\ x+y+z &=& 3*24 \quad | \quad : 2 \\\\ \dfrac{x+y+z}{2} &=& 3*12 \\\\ \dfrac{x+z}{2} + \dfrac{y}{2} &=& 3*12 \\\\ \dfrac{x+z}{2} + \dfrac{y}{2} &=& 36 \\\\ \mathbf{ \dfrac{x+z}{2} } &=& \mathbf{ 36 - \dfrac{y}{2} } \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \mathbf{ \dfrac{x+2y+z-7}{3} } &=& \mathbf{34} \\\\ x+2y+z-7 &=& 3*24 \quad | \quad : 2 \\\\ \dfrac{x+z}{2}+ \dfrac{2y-7}{2} &=& 3*17 \\\\ \dfrac{x+z}{2}+ \dfrac{2y-7}{2} &=& 51 \quad | \quad \mathbf{ \dfrac{x+z}{2} =36 - \dfrac{y}{2} } \\\\ 36 - \dfrac{y}{2}+ \dfrac{2y-7}{2} &=& 51 \\\\ 36 + \dfrac{2y-7-y}{2} &=& 51 \\\\ 36 + \dfrac{y-7}{2} &=& 51 \\\\ \dfrac{y-7}{2} &=& 51-36 \\\\ \dfrac{y-7}{2} &=& 15 \\\\ y-7 &=& 2*15 \\\\ y-7 &=& 30 \\ \mathbf{y} &=& \mathbf{37} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \mathbf{ \dfrac{x+z}{2} } &=& \mathbf{ 36 - \dfrac{y}{2} } \quad | \quad \mathbf{ y=37} \\\\ \dfrac{x+z}{2} &=& 36 - \dfrac{37}{2} \\\\ \dfrac{x+z}{2} &=& \dfrac{35}{2} \\\\ \mathbf{ \dfrac{x+z}{2} } &=& \mathbf{17.5} \\ \hline \end{array}$$

The arithmetic mean of x and z is $$\mathbf{17.5}$$

May 22, 2020