Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1981 AHSME Problems/Problem 12"
 
(3 intermediate revisions by the same user not shown)
Line 6: Line 6:
 
==Solution 1==
 
==Solution 1==
  
−
The problem asks when <math>M\left(1+\frac{p}{100}\right)\left(1-\frac{q}{100}\left) > M</math>.  We can simplify this inequality and isolate <math>p</math> as follows:
+
The goal is to find when <math>M\left(1+\frac{p}{100}\right)\left(1-\frac{q}{100}\right) > M</math>.  Isolate <math>p</math> as follows:
  
−
<math>\left(1+\frac{p}{100}\right)\left(1-\frac{q}{100}\left) > 1</math>
+
<cmath>\left(1+\frac{p}{100}\right)\left(1-\frac{q}{100}\right) > 1</cmath>
  
−
<math>\left(\frac{100+p}{100}\right)\left({100-q}{100}\left) > 1</math>
+
<cmath>\left(\frac{100+p}{100}\right)\left(\frac{100-q}{100}\right) > 1</cmath>
  
−
<math>\frac{(100+p)(100-q)}{10000} > 1</math>
+
<cmath>\frac{(100+p)(100-q)}{10000} > 1</cmath>
  
−
<math>100+p > \frac{10000}{100-q}</math>
+
<cmath>100+p > \frac{10000}{100-q}</cmath>
  
−
<math>p > \frac{10000}{100-q} - 100 = \frac{100q}{100-q} \fbox{(E)}</math>
+
<cmath>p > \frac{10000}{100-q} - 100 = \boxed{(\textbf{E})\ \frac{100q}{100-q}}</cmath>
  
 +
-j314andrews
  
 
==Solution 2 (Answer Choices)==
 
==Solution 2 (Answer Choices)==
−
Answer Choice <math>A</math>: It is obviously incorrect because if <math>M</math> is <math>50</math> and we increase by <math>50</math>% and then decrease <math>49</math>%, <math>M</math> will be around <math>37</math>.  
+
Choice A is incorrect. If <math>(M, p, q) = (100, 50, 40)</math>, then <math>p > q</math>, but <math>M</math> will not increase, as <math>100 \cdot 1.5 \cdot 0.6 = 90</math>.
  
−
Answer Choice <math>B</math>: If <math>p</math> is <math>100</math> and <math>q</math> is <math>50</math>, it should be equal but instead we get <math>100</math> is more than <math>1</math>. This is therefore also incorrect.
+
Choice B is incorrect. If <math>(M, p, q) = (100, 50, 50)</math>, then <math>p > \frac{q}{100-q}</math>, but <math>M</math> will not increase, as <math>100 \cdot 1.5 \cdot 0.5 = 75</math>.
  
−
Answer Choice <math>C</math>: Obviously incorrect since if <math>q</math> is larger than <math>1</math>, this is always valid since <math>\frac {1}{1-q}</math> is less than <math>0</math> which is obviously false.
+
Choice C is incorrect. If <math>(M, p, q) = (100, 50, 50)</math>, then <math>p > \frac{q}{1-q}</math>, but <math>M</math> will not increase, as <math>100 \cdot 1.5 \cdot 0.5 = 75</math>.  
  
−
Answer Choice <math>D</math>: If <math>p</math> is <math>100</math> and <math>q</math> is <math>50</math>, it should be equal but instead we get <math>100</math> is less than <math>\frac {5000}{150}</math>. Therefore, <math>D</math> is also incorrect.
+
Choice D is incorrect. If <math>(M, p, q) = (100, 100, 60)</math>, then <math>p > \frac{100q}{100+q}</math>, but <math>M</math> will not increase, as <math>100 \cdot 2 \cdot 0.4 = 80</math>.
  
−
Answer Choice <math>E</math>: If <math>p</math> is <math>100</math> and <math>q</math> is <math>50</math>, it should be equal, and if we check our equation, we get <math>\frac {5000}{50}</math> = <math>100</math>. Therefore, our answer is <math>\boxed {(E)\dfrac{100q}{100-q}}</math>
+
By process of elimination, <math>\boxed{(\textbf{E})\ \dfrac{100q}{100-q}}</math> must be correct.
  
−
~Arcticturn
+
-Arcticturn, edited by j314andrews
  
 
==See also==
 
==See also==

Latest revision as of 16:52, 18 August 2025

Problem 12

If $p$, $q$, and $M$ are positive numbers and $q<100$, then the number obtained by increasing $M$ by $p\%$ and decreasing the result by $q\%$ exceeds $M$ if and only if

$\textbf{(A)}\ p>q \qquad\textbf{(B)}\ p>\dfrac{q}{100-q}\qquad\textbf{(C)}\ p>\dfrac{q}{1-q}\qquad \textbf{(D)}\ p>\dfrac{100q}{100+q}\qquad\textbf{(E)}\ p>\dfrac{100q}{100-q}$

Solution 1

The goal is to find when $M\left(1+\frac{p}{100}\right)\left(1-\frac{q}{100}\right) > M$. Isolate $p$ as follows:

\[\left(1+\frac{p}{100}\right)\left(1-\frac{q}{100}\right) > 1\]

\[\left(\frac{100+p}{100}\right)\left(\frac{100-q}{100}\right) > 1\]

\[\frac{(100+p)(100-q)}{10000} > 1\]

\[100+p > \frac{10000}{100-q}\]

\[p > \frac{10000}{100-q} - 100 = \boxed{(\textbf{E})\ \frac{100q}{100-q}}\]

-j314andrews

Solution 2 (Answer Choices)

Choice A is incorrect. If $(M, p, q) = (100, 50, 40)$, then $p > q$, but $M$ will not increase, as $100 \cdot 1.5 \cdot 0.6 = 90$.

Choice B is incorrect. If $(M, p, q) = (100, 50, 50)$, then $p > \frac{q}{100-q}$, but $M$ will not increase, as $100 \cdot 1.5 \cdot 0.5 = 75$.

Choice C is incorrect. If $(M, p, q) = (100, 50, 50)$, then $p > \frac{q}{1-q}$, but $M$ will not increase, as $100 \cdot 1.5 \cdot 0.5 = 75$.

Choice D is incorrect. If $(M, p, q) = (100, 100, 60)$, then $p > \frac{100q}{100+q}$, but $M$ will not increase, as $100 \cdot 2 \cdot 0.4 = 80$.

By process of elimination, $\boxed{(\textbf{E})\ \dfrac{100q}{100-q}}$ must be correct.

-Arcticturn, edited by j314andrews

See also

1981 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1981_AHSME_Problems/Problem_12&oldid=258204"