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Difference between revisions of "2012 AMC 10A Problems/Problem 6"

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== Problem 6 ==
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== Problem ==
  
 
The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers?
 
The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers?
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==Solution==
 
==Solution==
  
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Let the two number equal <math>x</math> and <math>y</math>. From the information given in the problem, two equations can be written:
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Let the two numbers equal <math>x</math> and <math>y</math>. From the information given in the problem, two equations can be written:
  
 
<math>xy=9</math>
 
<math>xy=9</math>
  
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<math>\frac{1}{x}=4(\frac{1}{y})</math>
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<math>\frac{1}{x}=4 \left( \frac{1}{y} \right)</math>
  
 
Therefore, <math>4x=y</math>
 
Therefore, <math>4x=y</math>
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<math>4x^2=9</math>
 
<math>4x^2=9</math>
  
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So <math>x=\frac{3}{2}</math> and <math>y</math> would then be <math>\frac{9}{\frac{3}{2}}=6</math>
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So <math>x=\frac{3}{2}</math> and <math>y</math> would then be <math>4 \times</math> <math>\frac{3}{2}=6</math>
  
 
The sum would be <math>\frac{3}{2}+6</math> = <math>\boxed{\textbf{(D)}\ \frac{15}{2}}</math>
 
The sum would be <math>\frac{3}{2}+6</math> = <math>\boxed{\textbf{(D)}\ \frac{15}{2}}</math>
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==Video Solution (CREATIVE THINKING)==
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https://youtu.be/jyA5_tjDOjc
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~Education, the Study of Everything
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== See Also ==
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{{AMC10 box|year=2012|ab=A|num-b=5|num-a=7}}
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{{MAA Notice}}

Latest revision as of 12:55, 1 July 2023

Problem

The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers?

$\textbf{(A)}\ \frac{10}{3}\qquad\textbf{(B)}\ \frac{20}{3}\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ \frac{15}{2}\qquad\textbf{(E)}\ 8$

Solution

Let the two numbers equal $x$ and $y$. From the information given in the problem, two equations can be written:

$xy=9$

$\frac{1}{x}=4 \left( \frac{1}{y} \right)$

Therefore, $4x=y$

Replacing $y$ with $4x$ in the equation,

$4x^2=9$

So $x=\frac{3}{2}$ and $y$ would then be $4 \times$ $\frac{3}{2}=6$

The sum would be $\frac{3}{2}+6$ = $\boxed{\textbf{(D)}\ \frac{15}{2}}$

Video Solution (CREATIVE THINKING)

https://youtu.be/jyA5_tjDOjc

~Education, the Study of Everything

See Also

2012 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
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

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