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When you have two positive integers a and b, and the ratio of a to b is 5 to 2, then a must always be a multiple of 5, and b must always be a multiple of 2. This will always be true when you have a ratio of two integers, provided the ratio is completely reduced (so if you knew, say, the ratio of a to b was 10 to 4, you'd need to reduce that ratio to 5 to 2 first before drawing any conclusions about multiples).

So here, we know that the ratio of a to b is 5 to 2, so a is a multiple of 5. We also know the ratio of a to c is 7 to 5, so a is a multiple of 7. Thus a is a multiple of both 5 and 7, and the smallest possible value of a is 35. If a is 35, then since a/b = 5/2, b would be 14, and 2a + b = 84.
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Re: PS : Algebra/Ratio - High Difficulty [#permalink]

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03 Apr 2011, 21:09

a:b = 5:2 a:c = 7:5 So to form a:b:c we a needs to be the LCM of 7 and 5 i.e. 35 So Multiply first ratio by 7 and second by 5 a:b:c = 35:14:25 So 2*35+14 = 84

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