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 "2019 AMC 12A Problems/Problem 2"

m (→‎Solution 1)
 
(2 intermediate revisions by 2 users not shown)
Line 3: Line 3:
 
Suppose <math>a</math> is <math>150\%</math> of <math>b</math>. What percent of <math>a</math> is <math>3b</math>?
 
Suppose <math>a</math> is <math>150\%</math> of <math>b</math>. What percent of <math>a</math> is <math>3b</math>?
  
−
<math>\textbf{(A) } 50 \qquad \textbf{(B) } 66\frac{2}{3} \qquad \textbf{(C) } 150 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 450</math>
+
<math>\textbf{(A) } 50 \qquad \textbf{(B) } 66+\frac{2}{3} \qquad \textbf{(C) } 150 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 450</math>
  
 
==Solution 1==
 
==Solution 1==
−
Since <math>a=1.5b</math>, that means <math>b=a/1.5</math>. We multiply by 3 to get a <math>3b</math> term, to yield <math>3b=2a</math>.
+
Since <math>a=1.5b</math>, that means <math>b=\frac{a}{1.5}</math>. We multiply by <math>3</math> to get a <math>3b</math> term, yielding <math>3b=2a</math>, and <math>2a</math> is <math>\boxed{\textbf{(D) }200\%}</math> of <math>a</math>.
−
 
 
−
<math>2a</math> is <math>\boxed{\textbf{(D) }200\%}</math> of <math>a</math>.
 
−
 
 
−
-- eric2020
 
  
 
==Solution 2==
 
==Solution 2==
−
WLOG, let <math>b=100</math>. Then, we have <math>a=150</math> and <math>3b=300</math>. Thus, <math>\frac{3b}{a}=\frac{300}{150}=2</math> so <math>3b</math> is <math>200\%</math> or <math>a</math> so the answer is <math>\boxed{\textbf{(D) }200\%}</math>.
+
Without loss of generality, let <math>b=100</math>. Then, we have <math>a=150</math> and <math>3b=300</math>. Thus, <math>\frac{3b}{a}=\frac{300}{150}=2</math>, so <math>3b</math> is <math>200\%</math> of <math>a</math>. Hence the answer is <math>\boxed{\textbf{(D) }200\%}</math>.
  
−
-21jzhang
+
==Solution 3 (similar to Solution 1)==
 +
As before, <math>a = 1.5b</math>. Multiply by 2 to obtain <math>2a = 3b</math>. Since <math>2 = 200\%</math>, the answer is <math>\boxed{\textbf{(D) }200\%}</math>.
  
−
==Solution 3 (Like Solution 1)==
+
==Solution 4 (similar to Solution 2)==
−
<math>a = 1.5b</math>. Multiply by 2 to obtain <math>2a = 3b</math>. Since <math>2 = 200\%</math>, the answer is <math>\boxed{\textbf{(D) }200\%}</math>.
+
Without loss of generality, let <math>b=2</math>. Then, we have <math>a=3</math> and <math>3b=6</math>. This gives <math>\frac{3b}{a}=\frac{6}{3}=2</math>, so <math>3b</math> is <math>200\%</math> of <math>a</math>, so the answer is <math>\boxed{\textbf{(D) }200\%}</math>.
  
−
-DBlack2021
+
==Video Solution 1==
 +
https://youtu.be/CM2PqoCDvqo
  
−
==Solution 4 (Like Solution 2)==
+
~Education, the Study of Everything
−
WLOG, let <math>b=2</math>. Then, we have <math>a=3</math> and <math>3b=6</math>.  Thus, <math>\frac{3b}{a}=\frac{6}{3}=2</math> so <math>3b</math> is <math>200\%</math> of <math>a</math> so the answer is <math>\boxed{\textbf{(D) }200\%}</math>.
 
  
 
==See Also==
 
==See Also==

Latest revision as of 17:31, 30 October 2022

Problem

Suppose $a$ is $150\%$ of $b$. What percent of $a$ is $3b$?

$\textbf{(A) } 50 \qquad \textbf{(B) } 66+\frac{2}{3} \qquad \textbf{(C) } 150 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 450$

Solution 1

Since $a=1.5b$, that means $b=\frac{a}{1.5}$. We multiply by $3$ to get a $3b$ term, yielding $3b=2a$, and $2a$ is $\boxed{\textbf{(D) }200\%}$ of $a$.

Solution 2

Without loss of generality, let $b=100$. Then, we have $a=150$ and $3b=300$. Thus, $\frac{3b}{a}=\frac{300}{150}=2$, so $3b$ is $200\%$ of $a$. Hence the answer is $\boxed{\textbf{(D) }200\%}$.

Solution 3 (similar to Solution 1)

As before, $a = 1.5b$. Multiply by 2 to obtain $2a = 3b$. Since $2 = 200\%$, the answer is $\boxed{\textbf{(D) }200\%}$.

Solution 4 (similar to Solution 2)

Without loss of generality, let $b=2$. Then, we have $a=3$ and $3b=6$. This gives $\frac{3b}{a}=\frac{6}{3}=2$, so $3b$ is $200\%$ of $a$, so the answer is $\boxed{\textbf{(D) }200\%}$.

Video Solution 1

https://youtu.be/CM2PqoCDvqo

~Education, the Study of Everything

See Also

2019 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
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

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=2019_AMC_12A_Problems/Problem_2&oldid=179790"