Difference between revisions of "2019 AMC 8 Problems/Problem 22"

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==Problem 22==
 
==Problem 22==
A store increased the original price of a shirt by a certain percent and then decreased the new price by the same amount. Given that the resulting price was <math>84\%</math> of the original price, by what percent was the price increased and decreased?
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A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was <math>84\%</math> of the original price, by what percent was the price increased and decreased<math>?</math>
  
 
<math>\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40</math>
 
<math>\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40</math>
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Suppose the fraction of discount is <math>x</math>. That means <math>(1-x)(1+x)=0.84</math>; so <math>1-x^{2}=0.84</math>, and <math>(x^{2})=0.16</math>, obtaining <math>x=0.4</math>. Therefore, the price was increased and decreased by <math>40</math>%, or <math>\boxed{\textbf{(E)}\ 40}</math>.
 
Suppose the fraction of discount is <math>x</math>. That means <math>(1-x)(1+x)=0.84</math>; so <math>1-x^{2}=0.84</math>, and <math>(x^{2})=0.16</math>, obtaining <math>x=0.4</math>. Therefore, the price was increased and decreased by <math>40</math>%, or <math>\boxed{\textbf{(E)}\ 40}</math>.
  
==Solution 2(Answer options)==
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==Solution 2 (Answer options)==
Let the price be <math>100</math> and then trying for each option leads to <math>\boxed{\textbf{(E)}\ 40}</math>.
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We can try out every option and see which one works out. By this method, we get <math>\boxed{\textbf{(E)}\ 40}</math>.
 
 
~phoenixfire
 
  
 
==Solution 3==
 
==Solution 3==
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Thus <math>8400</math> = <math>(100+x)(100-x)</math>. Solving for <math>x</math> gives us <math>x = 40, -40</math>. But <math>x</math> has to be positive. Thus <math>x</math> = <math>40</math>.
 
Thus <math>8400</math> = <math>(100+x)(100-x)</math>. Solving for <math>x</math> gives us <math>x = 40, -40</math>. But <math>x</math> has to be positive. Thus <math>x</math> = <math>40</math>.
  
~phoenixfire
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==Solution 4 ~ using the answer choices==
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Let our original cost be <math>\$ 100.</math> We are looking for a result of <math>\$ 84,</math> then. We try 16% and see it gets us higher than 84. We try 20% and see it gets us lower than 16 but still higher than 84. We know that the higher the percent, the less the value. We try 36, as we are not progressing much, and we are close! We try <math>\boxed{40\%}</math>, and we have the answer; it worked.
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==Video explaining solution==
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https://www.youtube.com/watch?v=_TheVi-6LWE -  Happytwin
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Associated video - https://www.youtube.com/watch?v=aJX27Cxvwlc
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https://youtu.be/gX_l0PGsQao
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https://www.youtube.com/watch?v=RcBDdB35Whk&list=PLLCzevlMcsWNBsdpItBT4r7Pa8cZb6Viu&index=4 ~ MathEx
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https://youtu.be/h2GlK7itc2g
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~savannahsolver
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https://www.youtube.com/watch?v=DaF8uD8V8u0&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=24
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2019|num-b=21|num-a=23}}
 
{{AMC8 box|year=2019|num-b=21|num-a=23}}
  
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[[Category:Introductory Algebra Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 14:53, 28 July 2021

Problem 22

A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was $84\%$ of the original price, by what percent was the price increased and decreased$?$

$\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40$

Solution 1

Suppose the fraction of discount is $x$. That means $(1-x)(1+x)=0.84$; so $1-x^{2}=0.84$, and $(x^{2})=0.16$, obtaining $x=0.4$. Therefore, the price was increased and decreased by $40$%, or $\boxed{\textbf{(E)}\ 40}$.

Solution 2 (Answer options)

We can try out every option and see which one works out. By this method, we get $\boxed{\textbf{(E)}\ 40}$.

Solution 3

Let x be the discount. We can also work in reverse such as ($84$)$(\frac{100}{100-x})$$(\frac{100}{100+x})$ = $100$.

Thus $8400$ = $(100+x)(100-x)$. Solving for $x$ gives us $x = 40, -40$. But $x$ has to be positive. Thus $x$ = $40$.

Solution 4 ~ using the answer choices

Let our original cost be $$ 100.$ We are looking for a result of $$ 84,$ then. We try 16% and see it gets us higher than 84. We try 20% and see it gets us lower than 16 but still higher than 84. We know that the higher the percent, the less the value. We try 36, as we are not progressing much, and we are close! We try $\boxed{40\%}$, and we have the answer; it worked.

Video explaining solution

https://www.youtube.com/watch?v=_TheVi-6LWE - Happytwin

Associated video - https://www.youtube.com/watch?v=aJX27Cxvwlc

https://youtu.be/gX_l0PGsQao

https://www.youtube.com/watch?v=RcBDdB35Whk&list=PLLCzevlMcsWNBsdpItBT4r7Pa8cZb6Viu&index=4 ~ MathEx

https://youtu.be/h2GlK7itc2g

~savannahsolver

https://www.youtube.com/watch?v=DaF8uD8V8u0&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=24

See Also

2019 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 21
Followed by
Problem 23
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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