Math Is Fun Forum

iamfriendly

Member

Registered: 2009-07-03

Posts: 72

Plz Help, please !

Hi guys, here is the question:

      The integer numbers x and y satisfy the equation 2x=5y

      Only 1 of the following can be the sum of x and y
 
      2012,         2011,      2010,        2009,           2008

The standard anwser is 2009. The question is why??

Any thought might be a helpful hint to others.

Please help

Last edited by iamfriendly (2010-04-05 13:44:11)

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EYE AM FRIENDLY, THAT'S O U NEED 2 NO,
psst,
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phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,877

Re: Plz Help, please !

2x = 5y
From that we can see that the minimum integer values of x and y are 5 and 2 respectively (2 * 5 = 5 * 2).
The minimum sum of x and y therefore is 5 + 2 = 7.

x and y maintain their integer status if their sum is any multiple of 7...and from the 5 choices only 2009 fits.

Another approach:

2x = 5y
∴ x = 5y/2

Let x + y = z

Plug in x:
5y/2 + y = z
5y + 2y = 2z
7y = 2z

So twice z must be a multiple of 7...only satisfied by 2009.

Same applies if plugging y into the x + y = z equation --> 5 times z must be a multiple of 7 (again only satisfied by 2009).

Last edited by phrontister (2010-04-06 02:14:10)

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iamfriendly

Member

Registered: 2009-07-03

Posts: 72

Re: Plz Help, please !

Hi;

  Thank you for your assistance as well as your patience

  Just last night (in bed, before reading your explanation) I was thinking and I found :

       for 2x = 5y ==> 1x=2.5y==> 1y= 0.4x ===> x+y=1.4x

       Lowest multiple of 1.4 is 7 then yeah.....
 
    a slight variation of your second method,  basically

  Just, like, sharing ideas if you don't mind.
  The 1st is really helpful, not that the 2nd one isn't, Thank you
  'Night, See you around

Last edited by iamfriendly (2010-04-06 13:31:59)

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EYE AM FRIENDLY, THAT'S O U NEED 2 NO,
psst,
Don't trust strangers, EYE AM FRIENDLY, THAT'S O U NEED 2 NO ..........psst, Don't trust strangers......

Monox D. I-Fly

Member

From: Indonesia

Registered: 2015-12-02

Posts: 2,000

Re: Plz Help, please !

2x = 5y
From that we can see that the minimum integer values of x and y are 5 and 2 respectively (2 * 5 = 5 * 2).
The minimum sum of x and y therefore is 5 + 2 = 7.

Ugh... Why have I never thought about this?

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