The 3 Core Percentage Formulas
All percentage calculations reduce to a triangle of three variables: the Part (a portion of something), the Whole (the total or base), and the Percentage (the ratio expressed per 100). Every percentage formula is a rearrangement of this triangle.
Formula 1 — Find the Part (X% of a Number)
What is 30% of 200? → 200 × (30 ÷ 100) = 200 × 0.30 =60
What is 7% of $850? → $850 × 0.07 =$59.50
What is 12.5% of 480? → 480 × 0.125 =60
Formula 2 — Find the Percentage (What % is A of B?)
45 is what % of 180? → (45 ÷ 180) × 100 =25%
72 out of 90 on a test? → (72 ÷ 90) × 100 =80%
$37.50 spent out of $150 budget? → (37.50 ÷ 150) × 100 =25%
Formula 3 — Find the Whole (Reverse Percentage)
30 is 25% of what number? → 30 ÷ 0.25 =120
15 students = 12% of class. How many total? → 15 ÷ 0.12 =125
$56 is 70% of what? (after 30% discount) → $56 ÷ 0.70 =$80
The main Percentage Calculator solves all three formula types instantly — just enter any two values and it finds the third with full step-by-step working.
Percentage Increase Formula
Use this formula when a value has grown and you want to express that growth as a percentage of the original. The key rule: always divide by the old (original) value.
Step-by-Step
- Subtract Old from New to get the absolute increase: New − Old
- Divide by the Old value: Increase ÷ Old
- Multiply by 100 to get the percentage
Price: $40 → $52 → ((52−40) ÷ 40) × 100 = (12 ÷ 40) × 100 =30% increase
Salary: $48,000 → $54,000 → ((54,000−48,000) ÷ 48,000) × 100 =12.5% raise
Visits: 8,200 → 11,070 → (2,870 ÷ 8,200) × 100 =35% increase
Find new value: $200 + 15% increase → $200 × 1.15 =$230
Use the Percentage Increase Calculator for instant results with a visual increase indicator.
Percentage Decrease Formula
The mirror of percentage increase. Use when a value has fallen. The formula is identical except you subtract New from Old. The result should always be positive (0–100%).
Price: $80 → $60 → ((80−60) ÷ 80) × 100 = (20 ÷ 80) × 100 =25% decrease
Headcount: 450 → 378 → (72 ÷ 450) × 100 =16% reduction
Bill: $95 → $76/month → (19 ÷ 95) × 100 =20% decrease
Find new value: $150 − 20% → $150 × (1 − 0.20) = $150 × 0.80 =$120
Use the Percentage Decrease Calculator — enter old and new values for instant % decrease and a visual bar.
Percentage Change Formula
Percentage change is the unified formula covering both increases and decreases. A positive result = increase; a negative result = decrease. It is the standard formula used in finance, analytics, and science for before-and-after comparisons.
Revenue $120k → $156k → ((156−120) ÷ 120) × 100 =+30%
Errors 24/day → 18/day → ((18−24) ÷ 24) × 100 =−25%
Stock $45 → $45 → ((45−45) ÷ 45) × 100 =0%
Population 2.0M → 2.3M → ((2.3−2.0) ÷ 2.0) × 100 =+15%
A 50% increase then a 50% decrease does NOT return to the original. $100 × 1.50 = $150 → $150 × 0.50 = $75. Net change = −25%, not 0%. Always apply changes multiplicatively.
Use the Percentage Change Calculator for any two values — shows direction (↑ or ↓) and magnitude instantly.
Percentage Difference Formula
Percentage difference compares two values without a reference direction — neither value is treated as the "original." The denominator is the average of both values. Use this when comparing two peers, not a before/after situation.
Percentage Change vs. Percentage Difference
Compare prices: Product A = $40, Product B = $50
|40−50| = 10 · Average = (40+50)÷2 = 45 · % Diff = (10 ÷ 45) × 100 =22.22%
Two test scores: 78 and 92
|78−92| = 14 · Average = 85 · % Diff = (14 ÷ 85) × 100 =16.47%
Two measurements: 1,200 and 1,350
|1200−1350| = 150 · Average = 1,275 · % Diff = (150 ÷ 1,275) × 100 =11.76%
Use the Percentage Difference Calculator — enter any two values and see the difference, change, and ratio all at once.
Reverse Percentage Formula
A reverse percentage recovers the original (whole) value from a known part and the percentage that part represents. The core insight: if a value has already had a percentage applied (e.g., discounted or taxed), you divide by the factor — you don't subtract the percentage.
60 is 75% of what? → 60 ÷ 0.75 =80
Paid $63 after 30% off — original price?
$63 is 70% of original → $63 ÷ 0.70 =$90
£120 price includes 20% VAT — pre-VAT price?
£120 is 120% of pre-VAT → £120 ÷ 1.20 =£100
$35 saved = 7% of income — what is the income?
$35 ÷ 0.07 =$500
If a price is $63 after 30% off, do NOT calculate: $63 + 30% of $63 = $81.90. That is wrong. You must divide by (1 − 0.30 = 0.70): $63 ÷ 0.70 = $90. Adding back 30% of the reduced price does not give the original because 30% of $63 ≠ 30% of $90.
Percent Error Formula
Percent error is used in science and engineering to express how far an observed or measured value deviates from the accepted true (theoretical) value. It quantifies the accuracy of a measurement.
Measured: 9.75, Theoretical: 10.00
|9.75 − 10.00| = 0.25 · (0.25 ÷ 10.00) × 100 =2.5% error
Experiment result: 148 g, Expected: 150 g
|148−150| = 2 · (2 ÷ 150) × 100 =1.33% error
Measured speed: 32 m/s, Accepted: 30 m/s
|32−30| = 2 · (2 ÷ 30) × 100 =6.67% error
Absolute error = |Experimental − Theoretical| (in same units as measurement).
Relative error = Absolute Error ÷ Theoretical (a plain ratio, no %).
Percent error = Relative Error × 100 (expressed as a %).
Use the Percent Error Calculator — enter your experimental and theoretical values for instant calculation.
Discount & Sale Price Formulas
These are the most-used money percentage formulas for shopping, retail, and finance. All four variations are covered — finding sale price, savings, original price, and the discount percentage.
30% off $120 jacket: $120 × 0.70 =$84 sale price · Saved: $120 × 0.30 = $36
Original price: paid $56 after 30% off → $56 ÷ 0.70 =$80 original
Was $120, now $90 — what % off? → (30 ÷ 120) × 100 =25% off
Double: $200 with 20% then 10%: $200 × 0.80 × 0.90 =$144 (effective 28% off, not 30%)
Discount Calculator — all 4 modes (sale price, original, discount %, double discount).
Percent Off Calculator — quick sale price from any % off.
Compound Interest Formula
Compound interest applies a percentage growth repeatedly, with each period's interest earning interest in subsequent periods. It is the foundation of investment growth, loan repayment, and inflation calculations.
$1,000 at 8% annual rate, 5 years, annual compounding:
A = $1,000 × (1.08)⁵ = $1,000 × 1.4693 =$1,469.33
$5,000 at 6% compounded monthly for 3 years:
A = $5,000 × (1 + 0.06/12)^(12×3) = $5,000 × (1.005)^36 = $5,000 × 1.1967 =$5,983.40
Simple vs Compound (why compounding matters):
$10,000 at 7% for 10 years → Simple: $10,000 + $7,000 = $17,000 · Compound: $10,000 × (1.07)¹⁰ =$19,671.51
| Principal | Rate | Years | Compounding | Final Amount | Total Earned |
|---|---|---|---|---|---|
| $1,000 | 5% | 10 | Annual | $1,628.89 | $628.89 |
| $1,000 | 5% | 10 | Monthly | $1,647.01 | $647.01 |
| $1,000 | 10% | 10 | Annual | $2,593.74 | $1,593.74 |
| $10,000 | 7% | 20 | Annual | $38,696.84 | $28,696.84 |
Master Percentage Formula Reference Table
Every percentage formula at a glance — formula, example result, and a direct link to the matching free calculator.
| Formula Name | Formula | Example | Calculator |
|---|---|---|---|
| Find the Part | Part = Whole × (% ÷ 100) | 20% of 80 = 16 | Try → |
| Find the Percentage | % = (Part ÷ Whole) × 100 | 16 of 80 = 20% | Try → |
| Find the Whole | Whole = Part ÷ (% ÷ 100) | 16 is 20% of 80 | Try → |
| % Increase | ((New − Old) ÷ Old) × 100 | $40→$52 = 30%↑ | Try → |
| % Decrease | ((Old − New) ÷ Old) × 100 | $80→$60 = 25%↓ | Try → |
| % Change | ((New − Old) ÷ Old) × 100 | 120→156 = +30% | Try → |
| % Difference | (|V1−V2| ÷ avg(V1,V2)) × 100 | 40 vs 50 = 22.2% | Try → |
| Reverse % | Whole = Part ÷ (% ÷ 100) | 60 ÷ 0.75 = 80 | Try → |
| % Error | (|Exp−Theo| ÷ Theo) × 100 | 9.75 vs 10 = 2.5% | Try → |
| Sale Price | Original × (1 − Disc% ÷ 100) | $100 − 25% = $75 | Try → |
| Original Price | Sale ÷ (1 − Disc% ÷ 100) | $63 ÷ 0.70 = $90 | Try → |
| Discount % | ((Orig − Sale) ÷ Orig) × 100 | $120→$90 = 25% | Try → |
| Double Discount | Orig × (1−D1÷100) × (1−D2÷100) | 20%+10% = 28% eff. | Try → |
| Add Tax | Price × (1 + Tax% ÷ 100) | $50 × 1.08 = $54 | Try → |
| Remove Tax | Total ÷ (1 + Tax% ÷ 100) | $108 ÷ 1.08 = $100 | Try → |
| Tip Amount | Bill × (Tip% ÷ 100) | $65 × 0.18 = $11.70 | Try → |
| Salary Raise | Salary × (1 + Raise% ÷ 100) | $50k × 1.07 = $53.5k | Try → |
| Compound Interest | P × (1 + r/n)^(n×t) | $1k @ 8%, 5yr = $1,469 | Try → |
🔗 All Free Percentage Calculators
- Percentage Calculator (all 3 types)
- Percentage Increase Calculator
- Percentage Decrease Calculator
- Percentage Change Calculator
- Percentage Difference Calculator
- Discount Calculator (sale price, original, %, double)
- Percent Off Calculator
- Money % Calculator (tip, tax, salary, discount)
- Percent Error Calculator
- Fraction to Percentage Calculator