# The price of a box of 15 cloud markers is $12.70. The price of a box of 42 cloud markers is$31.60. All prices are without tax, and the price of the boxes is the same. How much would 50 cloud markers in a box cost?

Feb 17, 2018

The cost of 1 box of 50 markers is $37.20 #### Explanation: This is a simultaneous equation type problem. Let the cost of 1 marker be ${C}_{m}$Let the cost of 1 box br ${C}_{b}$15 markers + 1 box =$12.70

color(white)("d")15C_mcolor(white)("ddd")+color(white)("d")C_b=$12.70" "......................Equation(1) 42 markers + 1 box =$31.60

color(white)("dd")42C_mcolor(white)(".d")+color(white) ("d")C_bcolor(white)(".")=$31.60" "......................Equation(2) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $\textcolor{b l u e}{\text{Plan}}$Eliminate ${C}_{b}$by subtraction leaving just some ${C}_{m}$Use the count of ${C}_{m}$to find the cost of just one of them. Substitute the cost of 1 ${C}_{m}$in $E q u a t i o n \left(1\right)$to find the cost of ${C}_{b}$Use all this information to determine the cost of a box of 50 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $\textcolor{b l u e}{\text{Determine the value of 1 box and 1 marker}}$$E q u a t i o n \left(2\right) - E q u a t i o n \left(1\right)$42C_m+C_b=$31.60
ul(15C_m+C_b=$12.70 larr" Subtract") 27C_mcolor(white)("dddd")=$18.90

divide both sides by 27

color(blue)(C_m=($18.90)/27=$0.70)" ".....................Equation(3)

Using $E q u a t i o n \left(3\right)$ substitute for ${C}_{m} \text{ in } E q u a t i o n \left(1\right)$

color(green)(15color(red)(C_m)+C_b=$12.70color(white)("ddd") ->color(white)("ddd")15(color(red)($0.70))+C_b=$12.70) color(white)("ddddddddddddddddddd")->color(white)("ddd")C_b=$12.70-15($0.70) C_b=$12.7-$10.50 color(blue)(C_b=$2.20)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the value 50 markers and 1 box}}$

50($0.70)+$2.20 = $37.20 The cost of 1 box of 50 markers is $37.20

Feb 17, 2018

The total cost of $50$ markers in a box would be:
$2.20+$35 = $37.20 #### Explanation: Check whether we are working with a direct proportion, in which case the comparison between the price and the number of markers would be the same. $12.70 div 15 =$0.85" and "$31.60 div 42 = $0.75 The prices differ, so they are not directly proportional. The reason is that both prices include the price of the box, which is the same. This can be thought of as the constant. If you were to draw a graph of price (on the $y$-axis) and number of markers (on the $x$-axis) you would get a straight line where the $y$-intercept would be the price of the box and the slope would be the price of a marker. (rate of change of price) Let's find the slope. $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$m = (31.60-12.70)/(42-15) = ($18.90)/(27 "markers") = $0.70 "/""marker"  Therefore each marker costs $0.70 or 70c

the price of $50$ markers is:

50 xx0.7 = $35.00" "larr this is just the markers, How much does the box cost? Consider $12.70 " for " 15 markers at $0.70 each $12.70 - 15 xx$0.70 =$12.70-$10.50 =$2.20

The box itself costs $2.20$

The total cost of $50$ markers in a box would be:
$2.20+$35 = 37.20#

Another way to find this answer would be to draw the graph accurately and read off the value where $x = 50$ on the line.

graph{y= 0.7x+2.2 [-4.03, 75.97, -1.72, 38.28]}