Difference between revisions of "2019 AMC 12A Problems/Problem 2"
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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== | + | ==Solution 1== |
| − | Since <math>a=1.5b</math>, that means <math>b=a | + | 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{200\%}</math> of <math>a</math>. | + | ==Solution 2== |
| + | 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>. | ||
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| + | ==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>. | ||
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| + | ==Solution 4 (similar to Solution 2)== | ||
| + | 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>. | ||
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| + | ==Video Solution 1== | ||
| + | https://youtu.be/CM2PqoCDvqo | ||
| + | |||
| + | ~Education, the Study of Everything | ||
==See Also== | ==See Also== | ||
Latest revision as of 17:31, 30 October 2022
Contents
Problem
Suppose 
is 
of 
. What percent of is 
?
Solution 1
Since 
, that means 
. We multiply by 
to get a term, yielding 
, and 
is 
of .
Solution 2
Without loss of generality, let 
. Then, we have 
and 
. Thus, 
, so is 
of . Hence the answer is .
Solution 3 (similar to Solution 1)
As before, 
. Multiply by 2 to obtain 
. Since 
, the answer is .
Solution 4 (similar to Solution 2)
Without loss of generality, let 
. Then, we have 
and 
. This gives 
, so is of , so the answer is .
Video Solution 1
~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. 
