Ratios & Proportional Relationships

Percent Increase & Decrease

Solving problems involving percent increase, decrease, tax, and tip.

Intermediate15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Constant of Proportionality

When a Quantity Changes by a Percent

A jacket's price increases by 20%, and later that same new price decreases by 20%. Before reading on, predict: does the jacket end up back at its original price? Many people expect yes — think carefully about what each 20% is actually a percent of.

The two 20% changes apply to different amounts. The increase is 20% of the original price; the decrease is 20% of the already-increased price, which is a larger number. Since 20% of a larger amount is more than 20% of the smaller original amount, the final price ends up lower than where it started — the two changes don't cancel out.

Definition — Percent Increase and Decrease

A percent increase adds a percentage of the original amount to itself; a percent decrease subtracts a percentage of the original amount. Both can be found in one step using a multiplier: new amount = original × (1 + rate) for an increase, or original × (1 − rate) for a decrease.

Worked Example — Verifying the Opening Question

A $50 jacket increases by 20%: 50 × 1.20 = $60. That $60 price then decreases by 20%: 60 × 0.80 = $48. The final price, $48, is less than the original $50 — confirming the two changes don't cancel.

Worked Example — A Multi-Step Percent Problem with Tax and Tip

A restaurant bill is $40 before tax and tip. Add 8% sales tax, then a 15% tip on the tax-included total. After tax: 40 × 1.08 = $43.20. After tip: 43.20 × 1.15 = $49.68. The final total is $49.68.

\text{new amount} = \text{original} \times (1 \pm \text{rate})

Tip

Convert a percent change into a single multiplier — 1 plus the rate for an increase, 1 minus the rate for a decrease — to combine the original amount and its change into one multiplication step instead of two separate ones.

Common Mistakes

Assuming a percent increase followed by the same percent decrease returns to the original amount.
The two percent changes apply to different base amounts — the decrease applies to the already-larger increased amount, so the final result ends up lower than the original.
Applying a percent tip or tax to the original amount when the problem describes applying it after an earlier change, such as tipping on the pre-tax total instead of the post-tax total.
Read carefully which amount each percent change applies to — a tip described as being on the total after tax must be calculated using that already-adjusted amount, not the original bill.

Key Takeaways

A percent increase or decrease can be found in one step using a multiplier: 1 + rate or 1 − rate.
An increase followed by the same percent decrease doesn't return to the original amount, since the decrease applies to a larger base.
In a multi-step percent problem, each successive percent applies to the most recently adjusted amount, not always the original.

Summary

Percent increase and decrease apply proportional reasoning to real changes in price, tax, and tip. The next unit turns to a deeper look at the number system, extending arithmetic with negative numbers to fractions and decimals.

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