Decline and Repeated Percentage Change

A television bought for ₹21,000 loses 5% of its value over the first year in a simple depreciation model. The reduction is ₹1050, leaving ₹19,950. If the same 5% rate applies the following year, should you remove another ₹1050? Only if the reduction were based again on the original purchase price. When the problem says the current value falls by 5% each year, the second reduction must use the smaller ₹19,950 as its base.

Repeated decline uses the same updated-base reasoning as compounding. The multiplier is below one instead of above one. Each step keeps a fixed fraction of the current amount, so the absolute reductions gradually become smaller.

We can also mix increases and decreases. The safest method is to process them in sequence, writing the new amount after every step or multiplying the corresponding factors.

A 5% reduction leaves 100% − 5% = 95% of the current value. The remaining-value multiplier is 0.95. For the television, 0.95 × ₹21,000 = ₹19,950. This calculation already includes the subtraction; do not subtract another 5% afterward.

After the second year, multiply the updated value by 0.95 again. The new value is 0.95 × ₹19,950 = ₹18,952.50. The second reduction is ₹997.50, smaller than the first ₹1050, because five per cent acts on a smaller base.

A fixed percentage does not mean a fixed amount. This distinction explains why repeated decreases cannot generally be combined by adding their percentage numbers.

The exponent counts repetitions, just as in compound growth. A 10% decrease each decade leaves a multiplier of 0.9 each decade. Three decades produce 0.9 × 0.9 × 0.9 = 0.729. Thus 72.9% remains and the overall decrease is 27.1%.

The rate and period must match. Three decades means three applications of a per-decade rate, not thirty applications. Read the time label before choosing the exponent.

Rounding at every stage changes the later bases. If you round 1012.5 to 1013 before the final step, you obtain 911.7 and might report 912. Keeping the unrounded model value produces 911.25 and rounds to 911. Both displays may look close, but they follow different rounding procedures.

Unless a question explicitly requires whole-number rounding after each stage, preserve exact or sufficiently precise intermediate values. State the final precision so that “911”, “911.25” and “about 910” are not presented as conflicting exact answers.

For changing rates, write a separate factor for each period. A 5% increase gives 1.05. A 2% decrease gives 0.98. A later 3% decrease gives 0.97. Multiplying the starting amount by all three factors follows the evolving base automatically.

Adding +5 − 2 − 3 gives zero, but that addition hides the different amounts used as bases. The second percentage applies after the first change, and the third applies after the second.

Equal percentage increases and decreases usually do not cancel. Starting with 100, a 10% increase produces 110. A 10% decrease of 110 removes 11, leaving 99. The larger second base makes the reduction larger than the original increase.

Reversing a percentage change requires the reciprocal multiplier. After a 25% increase, the multiplier is 1.25. Returning to the original needs multiplication by 1/1.25 = 0.8, which is a 20% decrease. This extends the reverse-percentage reasoning from a single change.