The total cost of the meal including tax and the $\displaystyle 15\%$ tip is:

$\displaystyle 280\times(1+0.15)=322$

They split the total amount equally between four of them, so each of those four people then pays:

$\displaystyle \frac{322}{4}=80.5$

Each of the four friends pays $\displaystyle \$80.50$.

To calculate the mean of a set of data, we sum the values and then divide by the number of values present. In this case we have 6 values, so we add them all together and then divide by 6 to find the arithmetic mean for this set of data:

$\displaystyle \mu=\frac{\sum_{i=1}^{6}x_i}{6}=\frac{13+7+5+6+8+9}{6}=\frac{48}{6}=8$

To calculate the mean of a set of a data, we add up all the entries and then divide the result by the number of entries in the data set. There are seven entries in the given set, so we'll add them up and then divide by seven to find the mean:

$\displaystyle \mu=\frac{7+11+8+13+14+9+8}{7}=\frac{70}{7}=10$

If the arithmetic mean of $\displaystyle x$ and $\displaystyle y$ is 35, then 

$\displaystyle \frac{x+ y}{2} = 35$.

This, along with the statement 

$\displaystyle x - y = 30$,

form a system of equations that can be solved as follows:

$\displaystyle x - y + y = 30 + y$

$\displaystyle x = 30 + y$

Substitute:

$\displaystyle \frac{ 30 + y+ y}{2} = 35$

$\displaystyle \frac{ 30 +2y}{2} = 35$

$\displaystyle \frac{ 30 +2y}{2} \cdot 2 = 35\cdot 2$

$\displaystyle 30 +2y = 70$

$\displaystyle 30 +2y - 30 = 70 - 30$

$\displaystyle 2y = 40$

$\displaystyle 2y \div 2 = 40 \div 2$

$\displaystyle y = 20$

The arithmetic mean of $\displaystyle x$ and $\displaystyle y$ is 64, so

$\displaystyle \frac{x+ y}{2}= 66$

Similarly, 

$\displaystyle \frac{x+ z}{2}= 60$

$\displaystyle \frac{y+ z}{2}= 72$

Add the expressions:

$\displaystyle \frac{x+ y}{2} + \frac{x+ z}{2}+ \frac{y+ z}{2}= 66+ 60 +72$

$\displaystyle \frac{2x+2 y+ 2z}{2} = 198$

$\displaystyle x+y+z= 198$

The arithmetic mean of $\displaystyle x$, $\displaystyle y$, and $\displaystyle z$ is 

$\displaystyle \frac{x+y+z}{3}= \frac{198}{3} = 66$

The arithmetic mean of $\displaystyle x$ and $\displaystyle y$ is 76, so

$\displaystyle \frac{x+ y}{2}= 76$

Similarly, 

$\displaystyle \frac{x+ z}{2}= 63$

Subtract:

$\displaystyle \frac{x+ y}{2}- \frac{x+ z}{2}= 76-63$

$\displaystyle \frac{ y-z}{2}}= 13$

$\displaystyle \frac{ y-z}{2}}\cdot 2= 13 \cdot 2$

$\displaystyle y-z = 26$

$\displaystyle y = z + 26$,

so $\displaystyle y > z$

By similar reasoning:

$\displaystyle \frac{x+ y}{2}= 76$

$\displaystyle \frac{y+ z}{2}= 84$

$\displaystyle \frac{y+ z}{2}- \frac{x+ y}{2}= 84 - 76$

$\displaystyle \frac{ z-x}{2}}= 8$

$\displaystyle \frac{ z-x}{2}}\cdot 2 = 8 \cdot 2$

$\displaystyle z-x = 16$

$\displaystyle z= x+16$

so $\displaystyle z > x$

Therefore, $\displaystyle x< z< y$.

The arithmetic mean of $\displaystyle 2x$, $\displaystyle x+ 4$, $\displaystyle 3x+7$, and $\displaystyle 2x-8$ is 40, so their sum divided by 4 is equal to 40.

$\displaystyle \frac{2x + (x+4)+(3x+7)+(2x-8)}{4} = 40$

$\displaystyle \frac{2x + x +3x +2x +4+7-8}{4} = 40$

$\displaystyle \frac{8x +3}{4} = 40$

$\displaystyle \frac{8x +3}{4} \cdot 4 = 40\cdot 4$

$\displaystyle 8x+3 = 160$

$\displaystyle 8x = 157$

$\displaystyle x = 19 \frac{5}{8}$

This rounds to 20.

The arithmetic mean of $\displaystyle 10 + x$, $\displaystyle x - 12$, $\displaystyle 14 - x$, and $\displaystyle -x$ is the sum of the expressions divided by 4, or:

$\displaystyle \frac{(10 + x )+ (x-12)+ (14-x)+ (-x)}{4}$

$\displaystyle = \frac{x + x-x -x -12+ 14+ 10 }{4}$

$\displaystyle = \frac{12 }{4}$

$\displaystyle = 3$

The arithmetic mean of $\displaystyle x$, $\displaystyle y$, and $\displaystyle z$ is 200, so

$\displaystyle \frac{x+y+z}{3} = 200$

$\displaystyle \frac{x+y+z}{3} \cdot 3 = 200 \cdot 3$

$\displaystyle x+y+z = 600$

The arithmetic mean of $\displaystyle x$ and $\displaystyle y$ is 190, so

$\displaystyle \frac{x+y }{2} = 190$

$\displaystyle \frac{x+y }{2}\cdot 2 = 190 \cdot 2$

$\displaystyle x+ y = 380$

$\displaystyle x+y+z - (x+y) = 600 - 380$

$\displaystyle z = 220$

The arithmetic mean of $\displaystyle y$ and $\displaystyle z$ is 210, so

$\displaystyle \frac{y+z }{2} = 210$

$\displaystyle \frac{y+z }{2}\cdot 2 = 210 \cdot 2$

$\displaystyle y+z = 420$

$\displaystyle x+y+z - (y+z) = 600 - 420$

$\displaystyle x = 180$

The arithmetic mean of $\displaystyle x$ and $\displaystyle z$ is

$\displaystyle \frac{x+z }{2} = \frac{180+220 }{2}= \frac{400}{2} = 200$

To find the mean, simply sum up the numbers and divide by the amount of numbers.

$\displaystyle \textup{mean}=\frac{6+2+5+5+8+4}{6}=5$