### Brain Teasers

# Increasing Order

The following numbers are in increasing order: 3/5 < 7/a < 9/4 < 5/b < 16/(a+b), where a and b are positive integers. What is the value of a+b?

### Answer

3/5<7/a implies 3a<35, or a<=11;7/a<9/4 implies 28<9a, or a>=4;

9/4<5/b implies 9b<20, or b<=2;

5/b<16/(a+b) implies 5(a+b)<16b, or 5a<11b.

We see that a is at least 4, but b can be only 1 or 2. Since 5a<11b, a=4 and b=2, whose sum is 6.

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