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 12A Problems/Problem 10"

(New page: == Problem == Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to t...)
 
 
(8 intermediate revisions by 5 users not shown)
Line 1: Line 1:
 +
{{duplicate|[[2002 AMC 12A Problems|2002 AMC 12A #10]] and [[2002 AMC 10A Problems|2002 AMC 10A #17]]}}
 +
 +
 
== Problem ==
 
== Problem ==
 
Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?
 
Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?
  
 
<math> \mathrm{(A) \ } \frac{1}{4}\qquad \mathrm{(B) \ } \frac13\qquad \mathrm{(C) \ } \frac38\qquad \mathrm{(D) \ } \frac25\qquad \mathrm{(E) \ } \frac12 </math>
 
<math> \mathrm{(A) \ } \frac{1}{4}\qquad \mathrm{(B) \ } \frac13\qquad \mathrm{(C) \ } \frac38\qquad \mathrm{(D) \ } \frac25\qquad \mathrm{(E) \ } \frac12 </math>
 +
 +
== Solution ==
 +
===Solution 1===
 +
We will simulate the process in steps.
 +
 +
In the beginning, we have:
 +
* <math>4</math> ounces of coffee in cup <math>1</math>
 +
* <math>4</math> ounces of cream in cup <math>2</math>
 +
 +
In the first step we pour <math>4/2=2</math> ounces of coffee from cup <math>1</math> to cup <math>2</math>, getting:
 +
* <math>2</math> ounces of coffee in cup <math>1</math>
 +
* <math>2</math> ounces of coffee and <math>4</math> ounces of cream in cup <math>2</math>
 +
 +
In the second step we pour <math>2/2=1</math> ounce of coffee and <math>4/2=2</math> ounces of cream from cup <math>2</math> to cup <math>1</math>, getting:
 +
* <math>2+1=3</math> ounces of coffee and <math>0+2=2</math> ounces of cream in cup <math>1</math>
 +
* the rest in cup <math>2</math>
 +
 +
Hence at the end we have <math>3+2=5</math> ounces of liquid in cup <math>1</math>, and out of these <math>2</math> ounces is cream. Thus the answer is <math>\boxed{\text{(D) } \frac 25}</math>.
 +
 +
===Solution 2===
 +
Let's consider this in steps.
 +
We have 4 ounces of coffee in the first cup.
 +
We hace 4 ounces of cream in the second cup.
 +
We take half of the coffee in the first cup(2 ounces), and add it to the second cup, yielding 6 ounces in total in the second cup(in a 1:2 ratio between coffee and cream, respecitvely).
 +
We then take half of the second cup and pour it into the first cup. <math>6/2=3</math>, so there is now 5 ounces in the first cup, 2 coffee and 3 the mixture.
 +
Remember that the mixture is in a 1:2 ratio between coffee and cream. So, coffee has one ounce and cream has 2 ounces.
 +
In total, there is 2 ounces in the 5 ounce first cup. Putting 2 over 5, we get the answer. Therefore, the answer is <math>\boxed{\text{(D) } \frac 25}</math>.
 +
 +
~MathKatana
 +
 +
==Video Solution==
 +
 +
https://www.youtube.com/watch?v=3Zri7R40wFM    ~David
 +
 +
==Video Solution by Daily Dose of Math==
 +
 +
https://youtu.be/H8A6o_O6SRY
 +
 +
~Thesmartgreekmathdude
 +
 +
== See Also ==
 +
 +
{{AMC12 box|year=2002|ab=A|num-b=9|num-a=11}}
 +
{{AMC10 box|year=2002|ab=A|num-b=16|num-a=18}}
 +
 +
[[Category:Introductory Algebra Problems]]
 +
{{MAA Notice}}

Latest revision as of 17:40, 15 October 2024

The following problem is from both the 2002 AMC 12A #10 and 2002 AMC 10A #17, so both problems redirect to this page.


Problem

Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?

$\mathrm{(A) \ } \frac{1}{4}\qquad \mathrm{(B) \ } \frac13\qquad \mathrm{(C) \ } \frac38\qquad \mathrm{(D) \ } \frac25\qquad \mathrm{(E) \ } \frac12$

Solution

Solution 1

We will simulate the process in steps.

In the beginning, we have:

  • $4$ ounces of coffee in cup $1$
  • $4$ ounces of cream in cup $2$

In the first step we pour $4/2=2$ ounces of coffee from cup $1$ to cup $2$, getting:

  • $2$ ounces of coffee in cup $1$
  • $2$ ounces of coffee and $4$ ounces of cream in cup $2$

In the second step we pour $2/2=1$ ounce of coffee and $4/2=2$ ounces of cream from cup $2$ to cup $1$, getting:

  • $2+1=3$ ounces of coffee and $0+2=2$ ounces of cream in cup $1$
  • the rest in cup $2$

Hence at the end we have $3+2=5$ ounces of liquid in cup $1$, and out of these $2$ ounces is cream. Thus the answer is $\boxed{\text{(D) } \frac 25}$.

Solution 2

Let's consider this in steps. We have 4 ounces of coffee in the first cup. We hace 4 ounces of cream in the second cup. We take half of the coffee in the first cup(2 ounces), and add it to the second cup, yielding 6 ounces in total in the second cup(in a 1:2 ratio between coffee and cream, respecitvely). We then take half of the second cup and pour it into the first cup. $6/2=3$, so there is now 5 ounces in the first cup, 2 coffee and 3 the mixture. Remember that the mixture is in a 1:2 ratio between coffee and cream. So, coffee has one ounce and cream has 2 ounces. In total, there is 2 ounces in the 5 ounce first cup. Putting 2 over 5, we get the answer. Therefore, the answer is $\boxed{\text{(D) } \frac 25}$.

~MathKatana

Video Solution

https://www.youtube.com/watch?v=3Zri7R40wFM ~David

Video Solution by Daily Dose of Math

https://youtu.be/H8A6o_O6SRY

~Thesmartgreekmathdude

See Also

2002 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
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 AMC 12 Problems and Solutions
2002 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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 AMC 10 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=2002_AMC_12A_Problems/Problem_10&oldid=229952"