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Difference between revisions of "1983 AHSME Problems/Problem 25"

m (Further cleaned up the solution)
m (Improved clarity)
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We have that <math>12=\frac{60}{5}</math>. We can substitute our value for 5, to get
 
We have that <math>12=\frac{60}{5}</math>. We can substitute our value for 5, to get
 
<cmath>12=\frac{60}{60^b}=60\cdot 60^{-b}=60^{1-b}.</cmath>
 
<cmath>12=\frac{60}{60^b}=60\cdot 60^{-b}=60^{1-b}.</cmath>
−
Therefore we have
+
Hence
 
<cmath>12^{(1-a-b)/\left(2\left(1-b\right)\right)}=60^{(1-b)(1-a-b)/\left(2\left(1-b\right)\right)}=60^{(1-a-b)/2}.</cmath>
 
<cmath>12^{(1-a-b)/\left(2\left(1-b\right)\right)}=60^{(1-b)(1-a-b)/\left(2\left(1-b\right)\right)}=60^{(1-a-b)/2}.</cmath>
 
Since <math>4=\frac{60}{3\cdot 5}</math>, we have
 
Since <math>4=\frac{60}{3\cdot 5}</math>, we have
 
<cmath>4=\frac{60}{60^a 60^b}=60^{1-a-b}.</cmath>
 
<cmath>4=\frac{60}{60^a 60^b}=60^{1-a-b}.</cmath>
 
Therefore, we have
 
Therefore, we have
−
<cmath>60^{(1-a-b)/2}=4^{1/2}=\fbox{\textbf{(B)} 2}</cmath>
+
<cmath>60^{(1-a-b)/2}=4^{1/2}=\boxed{\textbf{(B)} 2}</cmath>
  
 
==See Also==
 
==See Also==

Revision as of 00:03, 20 February 2019

Problem 25

If $60^a=3$ and $60^b=5$, then $12^{(1-a-b)/\left(2\left(1-b\right)\right)}$ is

$\textbf{(A)}\ \sqrt{3}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \sqrt{5}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 2\sqrt{3}\qquad$

Solution

We have that $12=\frac{60}{5}$. We can substitute our value for 5, to get \[12=\frac{60}{60^b}=60\cdot 60^{-b}=60^{1-b}.\] Hence \[12^{(1-a-b)/\left(2\left(1-b\right)\right)}=60^{(1-b)(1-a-b)/\left(2\left(1-b\right)\right)}=60^{(1-a-b)/2}.\] Since $4=\frac{60}{3\cdot 5}$, we have \[4=\frac{60}{60^a 60^b}=60^{1-a-b}.\] Therefore, we have \[60^{(1-a-b)/2}=4^{1/2}=\boxed{\textbf{(B)} 2}\]

See Also

1983 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
Problem 26
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


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