Difference between revisions of "2015 AMC 10A Problems/Problem 6"

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{{duplicate|[[2015 AMC 12A Problems|2015 AMC 12A #4]] and [[2015 AMC 10A Problems|2015 AMC 10A #6]]}}
 
==Problem==
 
==Problem==
  
 
The sum of two positive numbers is <math> 5 </math> times their difference. What is the ratio of the larger number to the smaller number?
 
The sum of two positive numbers is <math> 5 </math> times their difference. What is the ratio of the larger number to the smaller number?
  
<math> \textbf{(A)}\ \frac{5}{4}\qquad\textbf{(B)}\ \frac{3}{2}\qquad\textbf{(C)}\ \frac{9}{5}\qquad\textbf{(D)}}\ 2 \qquad\textbf{(E)}\ \frac{5}{2} </math>
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<math> \textbf{(A)}\ \frac{5}{4}\qquad\textbf{(B)}\ \frac{3}{2}\qquad\textbf{(C)}\ \frac{9}{5}\qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac{5}{2} </math>
  
==Solution==
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==Solution 1==
  
 
Let <math>a</math> be the bigger number and <math>b</math> be the smaller.
 
Let <math>a</math> be the bigger number and <math>b</math> be the smaller.
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<math>a + b = 5(a - b)</math>.
 
<math>a + b = 5(a - b)</math>.
  
Solving gives <math>\frac{a}{b} = \frac32</math>, so the answer is \boxed{\textbf{(B) }\frac32}$.
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Multiplying out gives <math>a + b = 5a - 5b</math> and rearranging gives <math>4a = 6b</math> and factorised into <math>2a = 3b</math> and then solving gives
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<math>\frac{a}{b} = \frac32</math>, so the answer is <math>\boxed{\textbf{(B) }\frac32}</math>.
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==Solution 2==
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Without loss of generality, let the two numbers be <math>3</math> and <math>2</math>, as they clearly satisfy the condition of the problem. The ratio of the larger to the smaller is <math>\boxed{\textbf{(B) }\frac32}</math>.
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==Video Solution==
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https://youtu.be/JLKoh-Nb0Os
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~savannahsolver
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==See Also==
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{{AMC10 box|year=2015|ab=A|num-b=5|num-a=7}}
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{{AMC12 box|year=2015|ab=A|num-b=3|num-a=5}}
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{{MAA Notice}}

Revision as of 00:56, 17 June 2022

The following problem is from both the 2015 AMC 12A #4 and 2015 AMC 10A #6, so both problems redirect to this page.

Problem

The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number?

$\textbf{(A)}\ \frac{5}{4}\qquad\textbf{(B)}\ \frac{3}{2}\qquad\textbf{(C)}\ \frac{9}{5}\qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac{5}{2}$

Solution 1

Let $a$ be the bigger number and $b$ be the smaller.

$a + b = 5(a - b)$.

Multiplying out gives $a + b = 5a - 5b$ and rearranging gives $4a = 6b$ and factorised into $2a = 3b$ and then solving gives

$\frac{a}{b} = \frac32$, so the answer is $\boxed{\textbf{(B) }\frac32}$.

Solution 2

Without loss of generality, let the two numbers be $3$ and $2$, as they clearly satisfy the condition of the problem. The ratio of the larger to the smaller is $\boxed{\textbf{(B) }\frac32}$.

Video Solution

https://youtu.be/JLKoh-Nb0Os

~savannahsolver

See Also

2015 AMC 10A (ProblemsAnswer KeyResources)
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
2015 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
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

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