In spite of an increase in the price of a commodity by 20%, the overall expenditure on it increases by 12%. What is the percentage decrease in the quantity of commodities consumed?

Let's assume the number of commodities consumed initially is 10 and the price of each commodity is 10.

Total expenditure initially = $$10\times10$$ = 100

In spite of an increase in the price of a commodity by 20%, the overall expenditure on it increases by 12%.

Price of each commodity after increase = 10 of (100+20)%

= 10 of 120%

= 12

Total expenditure after the increase in the price = 100 of (100+12)%

= 100 of 112%

= 112

number of commodities consumed after the increase in the price = $$\frac{112}{12}$$

= $$\frac{28}{3}$$

Percentage decrease in the quantity of commodities consumed = $$\frac{\left(10-\frac{28}{3}\right)}{10}\times\ 100$$

= $$\frac{\left(\frac{30}{3}-\frac{28}{3}\right)}{10}\times\ 100$$

= $$\frac{\frac{2}{3}}{10}\times\ 100$$

= $$\frac{2}{30}\times\ 100$$

= $$\frac{20}{3}$$

= $$6\frac{2}{3}$$ %