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 "1991 AHSME Problems/Problem 9"

(Added a solution with explanation)
(→‎Solution)
 
(3 intermediate revisions by 2 users not shown)
Line 10: Line 10:
  
 
== Solution ==
 
== Solution ==
−
<math>\fbox{D}</math> The scale factors for the increases are <math>1+\frac{i}{100}</math> and <math>1+\frac{j}{100}</math>, so the overall scale factor is <math>(1+\frac{i}{100})(1+\frac{j}{100}) = 1 + \frac{i}{100} + \frac{j}{100} + \frac{ij}{100^2}</math>. To convert this to a percentage, we subtract 1 and then multiply by 100, giving <math>i + j + \frac{ij}{100}</math>.
+
The scale factors for the increases are <math>1+\frac{i}{100}</math> and <math>1+\frac{j}{100}</math>, so the overall scale factor is <math>(1+\frac{i}{100})(1+\frac{j}{100}) = 1 + \frac{i}{100} + \frac{j}{100} + \frac{ij}{100^2}</math>. To convert this to a percentage, we subtract 1 and then multiply by 100, giving <math>i + j + \frac{ij}{100}</math>, or <math>\boxed{\text{D}}</math>.
  
 
== See also ==
 
== See also ==

Latest revision as of 12:29, 14 March 2023

Problem

From time $t=0$ to time $t=1$ a population increased by $i\%$, and from time $t=1$ to time $t=2$ the population increased by $j\%$. Therefore, from time $t=0$ to time $t=2$ the population increased by

$\text{(A) (i+j)\%} \quad \text{(B) } ij\%\quad \text{(C) } (i+ij)\%\quad \text{(D) } \left(i+j+\frac{ij}{100}\right)\%\quad \text{(E) } \left(i+j+\frac{i+j}{100}\right)\%$

Solution

The scale factors for the increases are $1+\frac{i}{100}$ and $1+\frac{j}{100}$, so the overall scale factor is $(1+\frac{i}{100})(1+\frac{j}{100}) = 1 + \frac{i}{100} + \frac{j}{100} + \frac{ij}{100^2}$. To convert this to a percentage, we subtract 1 and then multiply by 100, giving $i + j + \frac{ij}{100}$, or $\boxed{\text{D}}$.

See also

1991 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1991_AHSME_Problems/Problem_9&oldid=190937"