BEIJING, Nov 10 (Reuters) - China’s factory-gate prices declined at a sharper-than-expected rate in October year-on-year, while consumer inflation eased to its slowest since 2009, data showed on Tuesday, underscoring the fragile nature of the economic recovery. The producer price index fell 2.1% from a year earlier, the National Bureau of Statistics said in a statement, unchanged from a 2.1% drop in September and missing expectations for a 2.0% decline tipped by the median forecast from a Reuters survey of analysts. The bureau said in a separate statement that the consumer price index rose 0.5% from a year earlier, the slowest since October 2009 and compared with a 0.8% rise tipped by the Reuters poll and a 1.7% rise in September.