Learning goals
- Compute percent increase and percent decrease from an old value to a new value
- Compute percent difference when neither value is a natural baseline
- Choose the correct formula for sales growth, price gaps, and lab comparisons
Two related percent ideas, different baselines
Percent change answers: “How much did this value move relative to where it started?” The starting value is the whole (100%). Sales from $100.00 to $120.00 are a 20% increase; from $120.00 back to $100.00 is a 16.7% decrease — same dollar move, different percents, because the baseline changed.
Percent difference answers: “How far apart are two values relative to their size?” when neither is clearly “before.” Scientists often divide the absolute gap by the average of the two values. That comparison is symmetric: swapping A and B does not flip the sign story the way change does.
Use percent change for time series and “increased/decreased by.” Use percent difference for comparing two measurements, two quotes, or two experimental results side by side.
Percent change
percent change = ((new − old) ÷ old) × 100
Positive means an increase; negative means a decrease. The absolute change is |new − old|; the percent uses the old value as the denominator.
Percent difference (symmetric)
percent difference = (|A − B| ÷ ((A + B) ÷ 2)) × 100
Divide the absolute gap by the average of the two values, then multiply by 100. Some fields use a different reference (for example, a gold-standard measurement); always state which denominator you used.
Example 1 — Percent increase
Given
A product’s price rises from $40.00 to $50.00. What is the percent change?
Steps
- Old = 40, new = 50.
- Change = 50 − 40 = 10.
- Percent change = (10 ÷ 40) × 100 = 25%.
Answer
The price increased by 25%.
Tip: Going back from $50.00 to $40.00 is (40 − 50) ÷ 50 × 100 = −20%, not −25%.
Example 2 — Percent decrease
Given
Website visits fall from 8,000 to 6,000. What is the percent decrease?
Steps
- Old = 8,000, new = 6,000.
- Change = 6,000 − 8,000 = −2,000.
- Percent change = (−2,000 ÷ 8,000) × 100 = −25% (a 25% decrease).
Answer
Visits decreased by 25%.
Example 3 — Percent difference between two quotes
Given
Quote A is $180.00 and quote B is $220.00. What is the percent difference?
Steps
- Absolute gap = |220 − 180| = 40.
- Average = (180 + 220) ÷ 2 = 200.
- Percent difference = (40 ÷ 200) × 100 = 20%.
Answer
The quotes differ by 20%.
Tip: If you wrongly used A as a baseline: (40 ÷ 180) × 100 ≈ 22.2% — that is percent change from A to B, not the symmetric difference.
Using the new value as the baseline for “change”
Common mistake
Price $40.00 → $50.00: compute (10 ÷ 50) × 100 = 20% and call it the increase.
Better approach
Percent change divides by the original: (10 ÷ 40) × 100 = 25%. The new value is not the whole unless the question asks something else.
Expecting up and down percents to match
Common mistake
“Up 25% then down 25% returns to the start.”
Better approach
Up 25% from 100 → 125; down 25% of 125 → 93.75. Equal percent moves in opposite directions do not cancel unless the baseline is the same.
Check your understanding
1.Sales grow from 200 to 260. Percent change?
Answer: ((260 − 200) ÷ 200) × 100 = 30%.
2.A stock falls from $50.00 to $35.00. Percent decrease?
Answer: ((35 − 50) ÷ 50) × 100 = −30% (30% decrease).
3.Measurements 9.0 and 11.0. Percent difference?
Answer: |11−9| / 10 × 100 = 20%.
4.Why is a 10% raise then 10% cut not a wash?
Answer: 100 → 110 → 99; the second percent uses a larger baseline.
Key takeaways
- Percent change = ((new − old) ÷ old) × 100; the old value is the baseline.
- Percent difference usually divides |A − B| by the average of A and B.
- Equal dollar (or unit) moves can be different percents when the baseline changes.
- Always name whether you are reporting change from a start value or a symmetric difference.