Difference between revisions of "2017 AMC 8 Problems/Problem 14"

Latest revision as of 12:49, 14 January 2024

Problem

Chloe and Zoe are both students in Ms. Demeanor's math class. Last night, they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only $80\%$ of the problems she solved alone, but overall $88\%$ of her answers were correct. Zoe had correct answers to $90\%$ of the problems she solved alone. What was Zoe's overall percentage of correct answers?

$\textbf{(A) }89\qquad\textbf{(B) }92\qquad\textbf{(C) }93\qquad\textbf{(D) }96\qquad\textbf{(E) }98$

Solution 1

Let the number of questions that they solved alone be $x$ . Let the percentage of problems they correctly solve together be $a$ %. As given, $\frac{80x}{100} + \frac{ax}{100} = \frac{2 \cdot 88x}{100}$ .

Hence, $a = 96$ .

Zoe got $\frac{90x}{100} + \frac{ax}{100} = \frac{186x}{100}$ problems right out of $2x$ . Therefore, Zoe got $\frac{\frac{186x}{100}}{2x} = \frac{93}{100} = \boxed{\textbf{(C) } 93}$ percent of the problems correct.

Solution 2

Assume the total amount of problems is $100$ per half homework assignment since we are dealing with percentages, not values. Then, we know that Chloe got $80$ problems correct by herself and got $176$ problems correct overall. We also know that Zoe had $90$ problems she did correctly alone. We can see that the total amount of correct problems Chloe and Zoe did together was $176-80=96$ . Therefore, Zoe did $96+90=186$ problems out of $200$ problems correctly. This is $\boxed{\textbf{(C) } 93}$ percent.

Solution 3

In the problem, we can see that Chloe solved 80% of the problems she solved alone, but 88% of her answers are correct. If 80 and another number's average is 88, the other number must be 96. Then, Zoe solved 90% of the problems she did alone, but 96% of her answers were correct. Then, the average of 90 and 96 is $\boxed{\textbf{(C) } 93}$ .

Solution 4

(Slightly different Solution) Suppose we said that there were $100$ problems in their assignment. Then, Chloe had $40$ correct and $10$ incorrect on her portion, and $48$ correct and $2$ incorrect on the portion she and Zoe solved. Zoe has $45$ correct and $5$ incorrect on her portion, and $48$ correct and $2$ incorrect on the portion that she and Chloe solved. Then, Zoe has $48 + 45 = 93$ correct answers out of $100$ , so the answer is $\boxed{\textbf{(C) } 93}$ .

-HW73

Solution 5

Let the total number of problems be $t$ . Let the percentage of the number of problems that Chloe and Zoe did together and got right be $x$ . As we can see, Chloe got $80$ % of $\frac {1}{2}$ of the total problems right, hence, ${0.80 \cdot \frac{1}{2}t}$ . We also know that Chloe got $88$ % of $t$ problems right altogether, making it ${0.88 \cdot t}$ total problems right. If we add $x$ to the percentage of correct problems that Chloe solved alone, then that should be equal to the total number of problems that Chloe got right, making the equation: ${0.80 \cdot \frac{1}{2}t} + x = {0.88 \cdot t}$ . Solving that, we get $x = 0.48t$ . We also know that Zoe got $90$ % of $\frac {1}{2}$ of the total problems right, making it ${0.90 \cdot \frac{1}{2}t}$ . We now add that amount to the percentage of problems that Chloe and Zoe got right together, making ${0.90 \cdot \frac{1}{2}t}+ 0.48t$ . Solving that, we get $0.93t$ , which is equal to $93$ %; hence, $\boxed{\textbf{(C) } 93}$ .

-fn106068

Video Solution (CREATIVE THINKING + ANALYSIS!!!)

https://youtu.be/6Am5gaLa4JQ

~Education, the Study of Everything

Video Solution

https://youtu.be/WgoAEitW5D4

https://youtu.be/1VWcwRNNJoI

~savannahsolver

@@ Line 1: / Line 1: @@
-==Problem 14==
+==Problem==
-Chloe and Zoe are both students in Ms. Demeanor's math class. Last night they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only <math>80\%</math> of the problems she solved alone, but overall <math>88\%</math> of her answers were correct. Zoe had correct answers to <math>90\%</math> of the problems she solved alone. What was Zoe's  overall percentage of correct answers?
+Chloe and Zoe are both students in Ms. Demeanor's math class. Last night, they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only <math>80\%</math> of the problems she solved alone, but overall <math>88\%</math> of her answers were correct. Zoe had correct answers to <math>90\%</math> of the problems she solved alone. What was Zoe's  overall percentage of correct answers?
 <math>\textbf{(A) }89\qquad\textbf{(B) }92\qquad\textbf{(C) }93\qquad\textbf{(D) }96\qquad\textbf{(E) }98</math>
-I promise that you'll never find another like me
-I know that I'm a handful, baby, uh
-I know I never think before I jump
-And you're the kind of guy the ladies want
-(And there's a lot of cool chicks out there)
-I know that I went psycho on the phone
-I never leave well enough alone
-And trouble's gonna follow where I go
-(And there's a lot of cool chicks out there)
-But one of these things is not like the others
-Like a rainbow with all of the colors
-Baby doll, when it comes to a lover
-I promise that you'll never find another like
-Me-e-e, ooh-ooh-ooh-ooh
-I'm the only one of me
-Baby, that's the fun of me
-Eeh-eeh-eeh, ooh-ooh-ooh-ooh
-You're the only one of you
-Baby, that's the fun of you
-And I promise that nobody's gonna love you like me-e-e
-I know I tend to make it about me
-I know you never get just what you see
-But I will never bore you, baby
-(And there's a lot of lame guys out there)
-And when we had that fight out in the rain
-You ran after me and called my name
-I never wanna see you walk away
-(And there's a lot of lame guys out there)
-'Cause one of these things is not like the others
-Livin' in winter, I am your summer
-Baby doll, when it comes to a lover
-I promise that you'll never find another like
-Me-e-e, ooh-ooh-ooh-ooh
-I'm the only one of me
-Let me keep you company
-Eeh-eeh-eeh, ooh-ooh-ooh-ooh
-You're the only one of you
-Baby, that's the fun of you
-And I promise that nobody's gonna love you like me-e-e
-Hey, kids!
-Spelling is fun!
-Girl, there ain't no I in "team"
-But you know there is a "me"
-Strike the band up, one, two, three
-I promise that you'll never find another like me
-Girl, there ain't no I in "team"
-But you know there is a "me"
-And you can't spell "awesome" without "me"
-I promise that you'll never find another like
-Me-e-e (yeah), ooh-ooh-ooh-ooh (and I want ya, baby)
-I'm the only one of me (I'm the only one of me)
-Baby, that's the fun of me (baby, that's the fun of me)
-Eeh-eeh-eeh, ooh-ooh-ooh-ooh (oh)
-You're the only one of you (oh)
-Baby, that's the fun of you
-And I promise that nobody's gonna love you like me-e-e
-Girl, there ain't no I in "team" (Ooh-ooh-ooh-ooh)
-But you know there is a "me"
-I'm the only one of me (Oh-oh)
-Baby, that's the fun of me
-(Eeh-eeh-eeh, ooh-ooh-ooh-ooh)
-Strike the band up, one, two, three
-You can't spell "awesome" without "me"
-You're the only one of you
-Baby, that's the fun of you
-And I promise that nobody's gonna love you like me-e-e
-stop reading this
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-buoi
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-YA YA
 ==Solution 1==
@@ Line 85: / Line 13: @@
 Zoe got <math>\frac{90x}{100} + \frac{ax}{100} = \frac{186x}{100}</math> problems right out of <math>2x</math>. Therefore, Zoe got <math>\frac{\frac{186x}{100}}{2x} = \frac{93}{100} = \boxed{\textbf{(C) } 93}</math> percent of the problems correct.
-zzzzzzzzzzzzzzzzzzzz
 ==Solution 2==
-Assume the total amount of problems is <math>100</math> per half homework assignment, since we are dealing with percentages, and no values. Then, we know that Chloe got <math>80</math> problems correct by herself, and got <math>176</math> problems correct overall. We also know that Zoe had <math>90</math> problems she did alone correct. We can see that the total amount of correct problems Chloe had when Zoe and she did the homework together is <math>176-80=96</math>, which is the total amount of problems she got correct, subtracted by the number of correct problems she did alone. Therefore Zoe has <math>96+90=186</math> problems out of <math>200</math> problems correct. This is <math>\boxed{\textbf{(C) } 93}</math> percent.Boy I'm really boutta get to yo pickle chin ahh boy, egg head like collard greens head ass boy, oh hell nah boy yo dirt ahh boy stank ahh boy afro head ahh, lip gloss chin ahh boy ugly ahh boi *gagging*
+Assume the total amount of problems is <math>100</math> per half homework assignment since we are dealing with percentages, not values. Then, we know that Chloe got <math>80</math> problems correct by herself and got <math>176</math> problems correct overall. We also know that Zoe had <math>90</math> problems she did correctly alone. We can see that the total amount of correct problems Chloe and Zoe did together was <math>176-80=96</math>. Therefore, Zoe did <math>96+90=186</math> problems out of <math>200</math> problems correctly. This is <math>\boxed{\textbf{(C) } 93}</math> percent.
+==Solution 3==
+In the problem, we can see that Chloe solved 80% of the problems she solved alone, but 88% of her answers are correct. If 80 and another number's average is 88, the other number must be 96. Then, Zoe solved 90% of the problems she did alone, but 96% of her answers were correct. Then, the average of 90 and 96 is <math>\boxed{\textbf{(C) } 93}</math>.
+==Solution 4==
+(Slightly different Solution)
+Suppose we said that there were <math>100</math> problems in their assignment. Then, Chloe had <math>40</math> correct and <math>10</math> incorrect on her portion, and
+<math>48</math> correct and <math>2</math> incorrect on the portion she and Zoe solved. Zoe has <math>45</math> correct and <math>5</math> incorrect on her portion, and <math>48</math> correct and <math>2</math> incorrect on the portion that she and Chloe solved. Then, Zoe has <math>48 + 45 = 93</math> correct answers out of <math>100</math>, so the answer is <math>\boxed{\textbf{(C) } 93}</math>.
+-HW73
+==Solution 5==
+Let the total number of problems be <math>t</math>. Let the percentage of the number of problems that Chloe and Zoe did together and got right be <math>x</math>. As we can see, Chloe got <math>80</math>% of <math> \frac {1}{2} </math> of the total problems right, hence,  <math>{0.80 \cdot \frac{1}{2}t}</math> . We also know that Chloe got <math>88</math>% of <math>t</math> problems right altogether, making it <math>{0.88 \cdot t}</math> total problems right. If we add <math>x</math> to the percentage of correct problems that Chloe solved alone, then that should be equal to the total number of problems that Chloe got right, making the equation:  <math>{0.80 \cdot \frac{1}{2}t} + x = {0.88 \cdot t}</math>. Solving that, we get <math>x = 0.48t</math>. We also know that Zoe got <math>90</math>% of <math> \frac {1}{2} </math> of the total problems right, making it <math>{0.90 \cdot \frac{1}{2}t}</math>. We now add that amount to the percentage of problems that Chloe and Zoe got right together, making <math>{0.90 \cdot \frac{1}{2}t}+ 0.48t</math>. Solving that, we get <math>0.93t</math>, which is equal to <math>93</math>%; hence, <math>\boxed{\textbf{(C) } 93}</math>.
+-fn106068
+==Video Solution (CREATIVE THINKING + ANALYSIS!!!)==
+https://youtu.be/6Am5gaLa4JQ
+~Education, the Study of Everything
+==Video Solution==
+https://youtu.be/WgoAEitW5D4
+https://youtu.be/1VWcwRNNJoI
+~savannahsolver
 ==See Also==

2017 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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

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