Difference between revisions of "2023 AMC 10A Problems/Problem 8"

Revision as of 09:56, 11 November 2023

Problem

Barb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function of the Fahrenheit scale. Bread rises at $110$ degrees Fahrenheit, which is $0$ degrees on the Breadus scale. Bread is baked at $350$ degrees Fahrenheit, which is $100$ degrees on the Breadus scale. Bread is done when its internal temperature is $200$ degrees Fahrenheit. What is this in degrees on the Breadus scale?

$\textbf{(A) }33\qquad\textbf{(B) }34.5\qquad\textbf{(C) }36\qquad\textbf{(D) }37.5\qquad\textbf{(E) }39$

Solution 1 (Substitution)

To solve this question, you can use $y = mx + b$ where the $x$ is Fahrenheit and the $y$ is Breadus. We have $(110,0)$ and $(350,100)$ . We want to find the value of $y$ in $(200,y)$ that falls on this line. The slope for these two points is $\frac{5}{12}$ ; $y = \frac{5}{12}x + b$ . Solving for $b$ using $(110, 0)$ , $\frac{550}{12} = -b$ . We get $b = \frac{-275}{6}$ . Plugging in $(200, y), \frac{1000}{12}-\frac{550}{12}=y$ . Simplifying, $\frac{450}{12} = \boxed{\textbf{(D) }37.5}$

~walmartbrian

Solution 2 (Faster)

Let $^\circ B$ denote degrees Breadus. We notice that $200^\circ F$ is $90^\circ F$ degrees to $0^\circ B$ , and $150^\circ F$ to $100^\circ B$ . This ratio is $90:150=3:5$ ; therefore, $200^\circ F$ will be $\dfrac3{3+5}=\dfrac38$ of the way from $0$ to $100$ , which is $\boxed{\textbf{(D) }37.5.}$

~Technodoggo

Solution 3 (Intuitive)

From $110$ to $350$ degrees Fahrenheit, the Breadus scale goes from $1$ to $100$ . $110$ to $350$ degrees is a span of $240$ , and we can use this to determine how many Fahrenheit each Breadus unit is worth. $240$ divided by $100$ is $2.4$ , so each Breadus unit is $2.4$ Fahrenheit, starting at $110$ Fahrenheit. For example, $1$ degree on the Breadus scale is $110 + 2.4$ , or $112.4$ Fahrenheit. Using this information, we can figure out how many Breadus degrees $200$ Fahrenheit is. $200-110$ is $90$ , so we divide $90$ by $2.4$ to find the answer, which is $\boxed{\textbf{(D) }37.5}$

~MercilessAnimations

Solution 4

We note that the range of F temperatures that $0-100$ $\text{Br}^\circ$ represents is $350-110 = 240$ $\text{F}^\circ$ . $200$ $\text{F}^\circ$ is $(200-110) = 90$ $\text{F}^\circ$ along the way to getting to $240$ $\text{F}^\circ$ , the end of this range, or $90/240 = 9/24 = 3/8 = 0.375$ of the way Therefore if we switch to the Br scale, we are $0.375$ of the way to $100$ from $0$ , or at $\boxed{\textbf{(D) }37.5}$ $\text{Br}^\circ$ .

~Dilip ~Minor edits by FutureSphinx

Video Solution by Math-X (First understand the problem!!!)

https://youtu.be/cMgngeSmFCY?si=88mPIms6wdZ6-deq&t=1602 ~Math-X

Video Solution

https://youtu.be/bYzV5B425V4

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

@@ Line 25: / Line 25: @@
 ==Solution 4==
-We note that the range of F temperatures that 0-100 Br represents is 350-110 = 240 degF
+We note that the range of F temperatures that <math>0-100</math> <math>\text{Br}^\circ</math> represents is <math>350-110 = 240</math> <math>\text{F}^\circ</math>.
-degF is (200-110) = 90 degF along the way to getting to 240 degF, the end of this range, or 90/240 = 9/24 = 3/8 = .375 of the way
+<math>200</math> <math>\text{F}^\circ</math> is <math>(200-110) = 90</math> <math>\text{F}^\circ</math> along the way to getting to <math>240</math> <math>\text{F}^\circ</math>, the end of this range, or <math>90/240 = 9/24 = 3/8 = 0.375</math> of the way
-Therefore if we switch to the Br scale, we are .375 of the way to 100 from 0, or at <math>\boxed{\textbf{(D) }37.5}</math> degBr.
+Therefore if we switch to the Br scale, we are <math>0.375</math> of the way to <math>100</math> from <math>0</math>, or at <math>\boxed{\textbf{(D) }37.5}</math> <math>\text{Br}^\circ</math>.
 ~Dilip
+~Minor edits by FutureSphinx
 ==Video Solution by Math-X (First understand the problem!!!)==

2023 AMC 10A (Problems • Answer Key • Resources)
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