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How to prove without calculator that

$$ \frac{1}{1001} + \frac{1}{3001} > \frac{1}{1000}$$

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  • $\begingroup$ Think about what the sum of the fractions produces (don't multiply out the denominator!). How might you make a comparison of that sum to 1/1000? $\endgroup$ May 5, 2013 at 7:00
  • $\begingroup$ Multiply everything by $1001 \cdot 3001$ $\endgroup$
    – parsiad
    May 5, 2013 at 7:04

2 Answers 2

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$$\frac{1}{1001}+\frac{1}{3001}=\frac{3}{3003}+\frac{1}{3001}>\frac{3}{3003}+\frac{1}{3003}=\frac{4}{3003}>\frac{4}{4000}=\frac{1}{1000}$$

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HINT:

$$\frac1{1000}-\frac1{1001}=\frac{1001-1000}{1000\cdot1001}=\frac1{1000\cdot1001}<\frac1{3001}$$

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