# If the price of the commodity is increased by 50% by what fraction must its consumption be reduced so as to keep the same expenditure on its consumption?

Let’s use the following variables:

• P = initial price of the commodity
• Q = initial consumption of the commodity
• E = initial expenditure on the commodity

From the problem statement, we know that the price of the commodity is increased by 50%, which means the new price is 1.5P. We want to find the fraction by which consumption must be reduced to keep the same expenditure, which means the new consumption is some fraction of the initial consumption, or xQ, where 0 < x < 1.

The new expenditure is the product of the new price and the new consumption, or 1.5P * xQ, which must be equal to the initial expenditure E. Therefore, we can write:

1.5P * xQ = E

Solving for x, we get:

x = E / (1.5P * Q)

Substituting the expression for E, we get:

x = (P * Q) / (1.5P * Q)

Simplifying, we get:

x = 2/3

Therefore, the consumption must be reduced by 1/3, or by a fraction of 2/3, in order to keep the same expenditure on the commodity after the price is increased by 50%.

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