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Algebra Difficulty 5.8 AIME, harder Find the answer

Let's assume that savings banks give as much interest to depositors annually as the rate of inflation. The state deducts 20%20 \% of the interest as tax. By what percentage does the real value of the state's interest tax revenue decrease if the inflation rate falls from 25%25 \% to 16%16 \%, and the real value of the deposit stock remains unchanged?

A number or a short expression. Spacing and $ signs are ignored.

Solution

At the beginning of the first year, let the nominal value of the deposit be 100,000 units, and the real value be 100,000 "reals." At the end of the first year, the interest on the deposit is 25,000 units. Of this, 20,000 units belong to the deposit owners, and 5,000 units go to the state. In real value terms, the former is 200001.25=16000\frac{20000}{1.25}=16000 reals, and the latter is 50001.25=4000\frac{5000}{1.25}=4000 reals.

At the beginning of the second year, the deposit amount is still 100,000 reals, but its nominal value is now 125,000 units. (New savings cover the interest tax.) At the end of the second year, the interest on the deposit is 1250000.16=20000125000 \cdot 0.16=20000 units, of which 16,000 units belong to the deposit owners, and 4,000 units go to the state. The real value of the interest tax is 40001.251.162759\frac{4000}{1.25 \cdot 1.16} \approx 2759 reals. The state receives 1241 reals less compared to the previous year's 4000 reals. The decrease is approximately 31%31\%.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.