# A bowling alley offers special weekly bowling rates. The weekly rates are 5 games for $15, 6 games for$17.55, 7 games for $20.10, and 8 games for$22.65. lf this pattern continues, how much will it cost to bowl 10 games in a week?

$27.75 #### Explanation: The thing to do is experiment with ideas that associate the numbers looking for a pattern of behaviour. Eventually you should find something that works. Lets try dividing cost by count $\left(\text{cost")/("game count}\right) \to \frac{15}{5} , \frac{17.55}{6} , \frac{20.1}{7} , \frac{22.65}{8}$$\text{ "darr" "darr" "darr" } \downarrow$$\text{ " 3", "2.19..", "2.87..", } 2.83 . .$NOTHING AMEDIATELY OBVIOUS! Actually there is something hear but it is hidden. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Lets try rate of change. Set $\text{x-axis "->" the set of numbers } \left\{5 , 6 , 7 , 8\right\}$Set "y-axis "->" the set of numbers "{15", "17.55", "20.1", "22.65} 1st pair to 2nd pair: $\text{ change in x is 1}$$\text{ change in y is } 17.55 - 15 = 2.55$2nd pair to 3rd pair: $\text{ change in x is 1}$$\text{ change in y is } 20.1 - 17.55 = 2.55$3rd pair to 4th pair: $\text{ change in x is 1}$$\text{ change in y is } 22.65 - 20.1 = 2.55$$\textcolor{red}{\text{Found a consistent relationship}}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This is the same thing as a straight line graph For each increase in x by 1 the y increases by 2.55 So 2.55 is the value of $m$in the equation: $y = m x + c$giving: $y = 2.55 x + c$Picking on any pair; I select $\left(x , y\right) = \left(5 , 15\right)$Then $y = 2.55 x + c \text{ "->" } 15 = 2.55 \left(5\right) + c$$c = 2.25$giving $y = 2.55 x + 2.25$where $x$is the number of games and $y$is the cost. We need 10 games so " "y=2.55(10)+2.25" " =" "$27.75 