Mathematics, three units

Why does a 10 percent rise followed by a 10 percent fall not return the original price?

Quick answer

A 10 percent rise followed by a 10 percent fall does not return the original price because each change applies to a different base. The 10 percent increase is calculated on the original amount, but the 10 percent decrease is calculated on the new, larger total. Consequently, the decrease removes more value than the increase added, leaving the final price 1 percent lower.

The topic explained

Different bases produce different amounts

A 10 percent rise followed by a 10 percent fall does not balance out because the two percentages are calculated from different starting values. When a price rises by 10 percent, that 10 percent is calculated on the lower original price. When the price subsequently falls by 10 percent, that deduction is calculated on the new, higher price. Because 10 percent of a larger number is bigger than 10 percent of a smaller number, you subtract more than you added.

Consider an item priced at 100 dollars. A 10 percent increase adds 10 dollars, bringing the price to 110 dollars. A 10 percent decrease on 110 dollars subtracts 11 dollars. The final price is 99 dollars. You end up 1 percent lower than where you started.

Multiply factors instead of adding percentages

Every percentage change corresponds to multiplication by a single decimal factor. To increase a value by 10 percent, multiply by 1.10. To decrease a value by 10 percent, multiply by 0.90. Successive percentage changes require multiplying these decimal multipliers together rather than adding or subtracting the percentages.

Multiplying 1.10 by 0.90 yields 0.99. This represents 99 percent of the starting price, which confirms a 1 percent net decrease. The order of operations does not matter: a 10 percent decrease followed by a 10 percent increase produces the exact same result because 0.90 times 1.10 is also 0.99.

Common mistakes and how to avoid them

The most frequent error is assuming that opposite percentages cancel each other out. They never do. In fact, the greater the percentage change, the larger the final deficit. For instance, a 50 percent rise followed by a 50 percent fall results in 1.50 times 0.50, which equals 0.75—leaving you 25 percent below your starting amount.

This same principle extends to compound interest and exponential decay. In compound interest, each period's percentage gain multiplies the new total rather than the original principal. Always convert each percentage change into its decimal multiplier and multiply sequentially.

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Successive Percentage Changes and Multipliers

Convert each percentage change into a decimal multiplier, then multiply sequentially to find the final value or net percentage change. Show all working.

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Successive Percentage Changes and MultipliersName:

An item priced at $50 increases by 10% and then decreases by 10%. What is the final price?
A shirt priced at $80 decreases by 20% and then increases by 20%. What is the final price?
Calculate the single overall multiplier for a 20% increase followed by a 30% increase.
A stock valued at $200 drops by 50% on Monday and rises by 50% on Tuesday. What is its value on Tuesday evening?
A bicycle costs $800. It increases in price by 25% and subsequently drops by 20%. What is the final price?
A savings account with $2,000 earns 4% compound interest annually. Write the expression and calculate the balance after 5 years.
A patient receives 400 mg of a medication with a half-life of 6 hours. How much medication remains in the blood after 18 hours?

Common questions

Does the order of the percentage changes matter?

No. Because multiplication is commutative, a 10 percent rise followed by a 10 percent fall produces the same final result as a 10 percent fall followed by a 10 percent rise. In both cases, the multiplier is 1.10 times 0.90, which equals 0.99.

What percentage increase is needed to recover from a 20 percent drop?

You need a 25 percent increase. After a 20 percent drop, an item retains 80 percent of its value (a factor of 0.80). To return to 1.00, divide 1 by 0.80, which equals 1.25, or a 25 percent increase.

Why can you not simply add or subtract successive percentages?

Percentages cannot be added directly because each percentage is evaluated against a different base value. Adding percentages assumes that both rates apply to the identical starting number, which ignores the change that already occurred in the first step.

How does this concept connect to compound interest?

Compound interest uses the same successive multiplier mechanism. Each period, interest is applied to the accumulated balance rather than solely the original deposit. The formula multiplies the principal by the growth factor raised to the number of periods.

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