【Analysis】According to the conditions, substitute and verify to find $a, b$, and draw the conclusion.
【Solution】Solution: According to the problem, if $a=7$, then take $b=1, \frac{a}{5}+\frac{b}{7}=1.4+0.143 \approx 1.54$, which does not meet the requirement; if $a=6$, then take $b=3, \frac{a}{5}+\frac{b}{7}=1.2+0.429 \approx 1.63$, which does not meet the requirement; if $a=5$, then take $b=4, \frac{a}{5}+\frac{b}{7}=1+0.571 \approx 1.57$, which does not meet the requirement; if $a=4$, then take $b=5, \frac{a}{5}+\frac{b}{7}=0.8+0.714 \approx 1.51$, which meets the requirement; $\therefore a+b=9$.
11. (10 points) $a, b$ are positive integers, after rounding to the 3rd decimal place, the expression $\frac{\mathrm{a}}{5}+\frac{b}{7}=1.51$, find $a+b=$ ?

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