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 "2002 AMC 8 Problems/Problem 25"

(Created page with "Loki, Moe, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-t...")
 
 
(23 intermediate revisions by 13 users not shown)
Line 1: Line 1:
 +
==Problem==
 
Loki, Moe, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have?
 
Loki, Moe, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have?
  
 
<math> \text{(A)}\ \frac{1}{10}\qquad\text{(B)}\ \frac{1}{4}\qquad\text{(C)}\ \frac{1}{3}\qquad\text{(D)}\ \frac{2}{5}\qquad\text{(E)}\ \frac{1}{2} </math>
 
<math> \text{(A)}\ \frac{1}{10}\qquad\text{(B)}\ \frac{1}{4}\qquad\text{(C)}\ \frac{1}{3}\qquad\text{(D)}\ \frac{2}{5}\qquad\text{(E)}\ \frac{1}{2} </math>
 +
 +
==Solution 1==
 +
 +
Since Ott gets equal amounts of money from each friend, we can say that he gets <math>x</math> dollars from each friend. This means that Moe has <math>5x</math> dollars, Loki has <math>4x</math> dollars, and Nick has <math>3x</math> dollars. The total amount is <math>12x</math> dollars, and since Ott gets <math>3x</math> dollars total, <math>\frac{3x}{12x}= \frac{3}{12} = \boxed{\text{(B)}\ \frac14}</math>.
 +
 +
==Solution 2==
 +
We can assign any natural number to the price that Ott's friends gave him. For this example, we will use 5.
 +
 +
Moe gave Ott a fifth of his money, and also five dollars. So in total Moe has 25 dollars.
 +
 +
Loki gave Ott a fourth of his money, and also five dollars. So in total Loki has 20 dollars.
 +
 +
Nick gave Ott a third of his money, and also five dollars. So in total Nick has 15 dollars.
 +
 +
<math>5\cdot3=15</math> which is the total money the group gave Ott.
 +
 +
<math>25+20+15=60</math> is the group's balance. Therefore, the fraction of the group's money that Ott now has is <math>\frac{15}{60} = \boxed{\text{(B)}\ \frac14}</math>.
 +
 +
==Video Solution==
 +
 +
 +
 +
https://www.youtube.com/watch?v=F-ZvPoJdnfk  ~David
 +
 +
==See Also==
 +
{{AMC8 box|year=2002|num-b=24|after=Last <br /> Problem}}
 +
{{MAA Notice}}

Latest revision as of 16:14, 30 December 2024

Problem

Loki, Moe, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have?

$\text{(A)}\ \frac{1}{10}\qquad\text{(B)}\ \frac{1}{4}\qquad\text{(C)}\ \frac{1}{3}\qquad\text{(D)}\ \frac{2}{5}\qquad\text{(E)}\ \frac{1}{2}$

Solution 1

Since Ott gets equal amounts of money from each friend, we can say that he gets $x$ dollars from each friend. This means that Moe has $5x$ dollars, Loki has $4x$ dollars, and Nick has $3x$ dollars. The total amount is $12x$ dollars, and since Ott gets $3x$ dollars total, $\frac{3x}{12x}= \frac{3}{12} = \boxed{\text{(B)}\ \frac14}$.

Solution 2

We can assign any natural number to the price that Ott's friends gave him. For this example, we will use 5.

Moe gave Ott a fifth of his money, and also five dollars. So in total Moe has 25 dollars.

Loki gave Ott a fourth of his money, and also five dollars. So in total Loki has 20 dollars.

Nick gave Ott a third of his money, and also five dollars. So in total Nick has 15 dollars.

$5\cdot3=15$ which is the total money the group gave Ott.

$25+20+15=60$ is the group's balance. Therefore, the fraction of the group's money that Ott now has is $\frac{15}{60} = \boxed{\text{(B)}\ \frac14}$.

Video Solution

https://www.youtube.com/watch?v=F-ZvPoJdnfk ~David

See Also

2002 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
Last
Problem
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

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=2002_AMC_8_Problems/Problem_25&oldid=238473"