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June 3rd, 2015, 06:03 PM  #1 
Senior Member Joined: Jan 2015 From: USA Posts: 107 Thanks: 2  Fraction inequality (hard)
Hi, can someone help me with this problem? I have nooo clue 
June 3rd, 2015, 06:40 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra 
Part 1) is a system of two simultaneous equations. For part 2) you should start by expressing the two fractions using a common denominator, then work out how you might get a fraction with a value between the two. 
June 3rd, 2015, 07:05 PM  #3 
Senior Member Joined: Nov 2010 From: Indonesia Posts: 2,001 Thanks: 132 Math Focus: Trigonometry  $\displaystyle \frac{5}{12}<\frac{p}{q}<\frac{3}{7}$ $\displaystyle \frac{5\times7}{12\times7}<\frac{p}{q}<\frac{3 \times12}{7\times12}$ $\displaystyle \frac{35}{84}<\frac{p}{q}<\frac{36}{84}$ $\displaystyle \frac{35\times2}{84\times2}<\frac{p}{q}<\frac{36 \times2}{84\times2}$ $\displaystyle \frac{70}{168}<\frac{p}{q}<\frac{72}{168}$ Can you continue? 
June 4th, 2015, 07:44 AM  #4  
Senior Member Joined: Jan 2015 From: USA Posts: 107 Thanks: 2  Quote:
Thanx but no, i cant. I dont understand why you follow that procedure. Could you explain it to me? Thanx!  
June 4th, 2015, 08:11 AM  #5 
Senior Member Joined: Nov 2010 From: Indonesia Posts: 2,001 Thanks: 132 Math Focus: Trigonometry  
June 4th, 2015, 05:59 PM  #6 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond 
(1) $\displaystyle s+t=5p2q\quad[1]$ $\displaystyle st=19p8q\quad[2]$ $\displaystyle 4\cdot[1][2]\Rightarrow p=3s+5t$ $\displaystyle 19\cdot[1]5\cdot[2]\Rightarrow q=7s+12t$ (2) $\displaystyle \frac pq=\frac{3s+5t}{7s+12t}\Rightarrow\min(q)=19$ 

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