Difference between revisions of "2012 AMC 10A Problems/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? | ||
<math> \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 </math> | <math> \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 </math> | ||
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| + | ==Solution== | ||
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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>\frac{1}{x}=4 \left( \frac{1}{y} \right)</math> | ||
| + | |||
| + | Therefore, <math>4x=y</math> | ||
| + | |||
| + | Replacing <math>y</math> with <math>4x</math> in the equation, | ||
| + | |||
| + | <math>4x^2=9</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> | ||
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| + | 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)== | ||
| + | https://youtu.be/jyA5_tjDOjc | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | == See Also == | ||
| + | |||
| + | {{AMC10 box|year=2012|ab=A|num-b=5|num-a=7}} | ||
| + | {{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?
Solution
Let the two numbers equal 
and 
. From the information given in the problem, two equations can be written:
Therefore, 
Replacing with 
in the equation,
So 
and would then be 
The sum would be 
= 
Video Solution (CREATIVE THINKING)
~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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
