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 "1989 AJHSME Problems/Problem 18"

(→‎Solution)
(→‎Solution 2)
 
Line 11: Line 11:
 
Thus, we need to iterate the key pressing twice to get the display back to the original <math>\rightarrow \boxed{\text{B}}</math>.
 
Thus, we need to iterate the key pressing twice to get the display back to the original <math>\rightarrow \boxed{\text{B}}</math>.
  
−
==Solution 2==
+
==Solution 2 (simpler, no use of functions)==
 
In terms of mathematics, a number will always be changed back to its original number if you flip, or "reciprocal" it, twice. Therefore, no matter what number it will display, the answer will always be <math>2 = \boxed{\text{B}}.</math>
 
In terms of mathematics, a number will always be changed back to its original number if you flip, or "reciprocal" it, twice. Therefore, no matter what number it will display, the answer will always be <math>2 = \boxed{\text{B}}.</math>
 +
 +
~DuoDuoling0
  
 
==See Also==
 
==See Also==

Latest revision as of 12:25, 28 December 2021

Problem

Many calculators have a reciprocal key $\boxed{\frac{1}{x}}$ that replaces the current number displayed with its reciprocal. For example, if the display is $\boxed{00004}$ and the $\boxed{\frac{1}{x}}$ key is depressed, then the display becomes $\boxed{000.25}$. If $\boxed{00032}$ is currently displayed, what is the fewest number of times you must depress the $\boxed{\frac{1}{x}}$ key so the display again reads $\boxed{00032}$?

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

Solution

Let $f(x)=\frac{1}{x}$. We have \[f(f(x))=\frac{1}{\frac{1}{x}}=x\] Thus, we need to iterate the key pressing twice to get the display back to the original $\rightarrow \boxed{\text{B}}$.

Solution 2 (simpler, no use of functions)

In terms of mathematics, a number will always be changed back to its original number if you flip, or "reciprocal" it, twice. Therefore, no matter what number it will display, the answer will always be $2 = \boxed{\text{B}}.$

~DuoDuoling0

See Also

1989 AJHSME (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1989_AJHSME_Problems/Problem_18&oldid=168746"