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Difference between revisions of "2022 AMC 8 Problems/Problem 15"

(→‎Solution)
(→‎Solution)
Line 126: Line 126:
  
 
~MathFun1000
 
~MathFun1000
 +
 +
<math>\textbf{Remark:}</math>
 +
By answer choice we can cross out the points that haven’t a entire number of weight in the diagram which gives us:
 +
 +
<asy>
 +
//diagram by pog
 +
size(5.5cm);
 +
usepackage("mathptmx");
 +
defaultpen(mediumgray*0.5+gray*0.5+linewidth(0.63));
 +
add(grid(6,6));
 +
label(scale(0.7)*"$1$", (1,-0.3), black);
 +
label(scale(0.7)*"$2$", (2,-0.3), black);
 +
label(scale(0.7)*"$3$", (3,-0.3), black);
 +
label(scale(0.7)*"$4$", (4,-0.3), black);
 +
label(scale(0.7)*"$5$", (5,-0.3), black);
 +
label(scale(0.7)*"$1$", (-0.3,1), black);
 +
label(scale(0.7)*"$2$", (-0.3,2), black);
 +
label(scale(0.7)*"$3$", (-0.3,3), black);
 +
label(scale(0.7)*"$4$", (-0.3,4), black);
 +
label(scale(0.7)*"$5$", (-0.3,5), black);
 +
label(scale(0.75)*rotate(90)*"Price (dollars)", (-1,3.2), black);
 +
label(scale(0.75)*"Weight (ounces)", (3.2,-1), black);
 +
dot((1,1.2),black);
 +
dot((1,1.7),black);
 +
dot((1,2),black);
 +
dot((1,2.8),black);
 +
 +
dot((2,2),black);
 +
dot((2,2.9),black);
 +
dot((2,3),black);
 +
dot((2,4),black);
 +
dot((2,4.35),black);
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dot((2,4.8),black);
 +
 +
dot((3,2.5),blue);
 +
dot((3,3.4),black);
 +
dot((3,4.2),black);
 +
 +
dot((4,3.9),black);
 +
dot((4,5.1),black);
 +
 +
dot((5,4.5),black);
 +
dot((5,5),black);
 +
</asy>
  
 
==See Also==  
 
==See Also==  
 
{{AMC8 box|year=2022|num-b=14|num-a=16}}
 
{{AMC8 box|year=2022|num-b=14|num-a=16}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 17:36, 28 January 2022

Problem

Laszlo went online to shop for black pepper and found thirty different black pepper options varying in weight and price, shown in the scatter plot below. In ounces, what is the weight of the pepper that offers the lowest price per ounce?

[asy] //diagram by pog size(5.5cm); usepackage("mathptmx"); defaultpen(mediumgray*0.5+gray*0.5+linewidth(0.63)); add(grid(6,6)); label(scale(0.7)*"$1$", (1,-0.3), black); label(scale(0.7)*"$2$", (2,-0.3), black); label(scale(0.7)*"$3$", (3,-0.3), black); label(scale(0.7)*"$4$", (4,-0.3), black); label(scale(0.7)*"$5$", (5,-0.3), black); label(scale(0.7)*"$1$", (-0.3,1), black); label(scale(0.7)*"$2$", (-0.3,2), black); label(scale(0.7)*"$3$", (-0.3,3), black); label(scale(0.7)*"$4$", (-0.3,4), black); label(scale(0.7)*"$5$", (-0.3,5), black); label(scale(0.75)*rotate(90)*"Price (dollars)", (-1,3.2), black); label(scale(0.75)*"Weight (ounces)", (3.2,-1), black); dot((1,1.2),black); dot((1,1.7),black); dot((1,2),black); dot((1,2.8),black); dot((1.5,2.1),black); dot((1.5,3),black); dot((1.5,3.3),black); dot((1.5,3.75),black); dot((2,2),black); dot((2,2.9),black); dot((2,3),black); dot((2,4),black); dot((2,4.35),black); dot((2,4.8),black); dot((2.5,2.7),black); dot((2.5,3.7),black); dot((2.5,4.2),black); dot((2.5,4.4),black); dot((3,2.5),black); dot((3,3.4),black); dot((3,4.2),black); dot((3.5,3.8),black); dot((3.5,4.5),black); dot((3.5,4.8),black); dot((4,3.9),black); dot((4,5.1),black); dot((4.5,4.75),black); dot((4.5,5),black); dot((5,4.5),black); dot((5,5),black); [/asy]

$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

Solution

[asy] //diagram by pog size(5.5cm); usepackage("mathptmx"); defaultpen(mediumgray*0.5+gray*0.5+linewidth(0.63)); add(grid(6,6)); label(scale(0.7)*"$1$", (1,-0.3), black); label(scale(0.7)*"$2$", (2,-0.3), black); label(scale(0.7)*"$3$", (3,-0.3), black); label(scale(0.7)*"$4$", (4,-0.3), black); label(scale(0.7)*"$5$", (5,-0.3), black); label(scale(0.7)*"$1$", (-0.3,1), black); label(scale(0.7)*"$2$", (-0.3,2), black); label(scale(0.7)*"$3$", (-0.3,3), black); label(scale(0.7)*"$4$", (-0.3,4), black); label(scale(0.7)*"$5$", (-0.3,5), black); label(scale(0.75)*rotate(90)*"Price (dollars)", (-1,3.2), black); label(scale(0.75)*"Weight (ounces)", (3.2,-1), black); dot((1,1.2),black); dot((1,1.7),black); dot((1,2),black); dot((1,2.8),black); dot((1.5,2.1),black); dot((1.5,3),black); dot((1.5,3.3),black); dot((1.5,3.75),black); dot((2,2),black); dot((2,2.9),black); dot((2,3),black); dot((2,4),black); dot((2,4.35),black); dot((2,4.8),black); dot((2.5,2.7),black); dot((2.5,3.7),black); dot((2.5,4.2),black); dot((2.5,4.4),black); dot((3,2.5),blue); dot((3,3.4),black); dot((3,4.2),black); dot((3.5,3.8),black); dot((3.5,4.5),black); dot((3.5,4.8),black); dot((4,3.9),black); dot((4,5.1),black); dot((4.5,4.75),black); dot((4.5,5),black); dot((5,4.5),black); dot((5,5),black); [/asy]

We are looking for a black point, that when connected to the origin, yields the lowest slope. The slope represents the price per ounce. It is clearly the blue point, which can be found visually. Also, it is the only one with a price per once significantly less than $1$. Finally, we see that the blue point is over the category with a weight of $\boxed{\textbf{(C) } 3}$ ounces.

~MathFun1000

$\textbf{Remark:}$ By answer choice we can cross out the points that haven’t a entire number of weight in the diagram which gives us:

[asy] //diagram by pog size(5.5cm); usepackage("mathptmx"); defaultpen(mediumgray*0.5+gray*0.5+linewidth(0.63)); add(grid(6,6)); label(scale(0.7)*"$1$", (1,-0.3), black); label(scale(0.7)*"$2$", (2,-0.3), black); label(scale(0.7)*"$3$", (3,-0.3), black); label(scale(0.7)*"$4$", (4,-0.3), black); label(scale(0.7)*"$5$", (5,-0.3), black); label(scale(0.7)*"$1$", (-0.3,1), black); label(scale(0.7)*"$2$", (-0.3,2), black); label(scale(0.7)*"$3$", (-0.3,3), black); label(scale(0.7)*"$4$", (-0.3,4), black); label(scale(0.7)*"$5$", (-0.3,5), black); label(scale(0.75)*rotate(90)*"Price (dollars)", (-1,3.2), black); label(scale(0.75)*"Weight (ounces)", (3.2,-1), black); dot((1,1.2),black); dot((1,1.7),black); dot((1,2),black); dot((1,2.8),black); dot((2,2),black); dot((2,2.9),black); dot((2,3),black); dot((2,4),black); dot((2,4.35),black); dot((2,4.8),black); dot((3,2.5),blue); dot((3,3.4),black); dot((3,4.2),black); dot((4,3.9),black); dot((4,5.1),black); dot((5,4.5),black); dot((5,5),black); [/asy]

See Also

2022 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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