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Nested Percentages (P₁% of P₂%)

When one percentage applies to another, multiply them as decimals: P₁% of P₂% = (P₁/100)×(P₂/100)×100. Order doesn't matter—10% of 50% = 50% of 10% = 5%. Used for tax on discounts, commission splits, and cascading shares.

Concept Fundamentals
5%
10% of 50%
2% of base
8% tax on 25% off
28% total
20% then 10% off
9%
15% of 60%

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20% off then 10% off = 0.8 × 0.9 = 0.72, so 28% total discount—not 30%. 8% tax on 25% discount of $100: 8% of 25% = 2%, so $2 tax on the discounted portion. Commission: 15% of 60% = 9% of total sale goes to the agent.

Key quantities
5%
10% of 50%
Key relation
2% of base
8% tax on 25% off
Key relation
28% total
20% then 10% off
Key relation
9%
15% of 60%
Key relation

Ready to run the numbers?

Why: Retail uses nested discounts (20% off, then 10% off). Tax applies to discounted prices. Commission splits cascade (agent gets 15% of broker's 60%). Profit sharing has multiple tiers.

How: Convert each % to decimal (÷100), multiply, then ×100 for result. With base: Base × (P₁/100)×(P₂/100). For 3-chain, extend: (P₁/100)×(P₂/100)×(P₃/100)×100.

20% off then 10% off = 0.8 × 0.9 = 0.72, so 28% total discount—not 30%.8% tax on 25% discount of $100: 8% of 25% = 2%, so $2 tax on the discounted portion.
Sources:Compound PercentagesRetail Mathematics

Run the calculator when you are ready.

Multiply, Don't AddSequential percentages compound multiplicatively. 20% + 10% off ≠ 30% off.

Quick Examples — Click to Load

Input Values

%
%
%
percentage_of_percentage
CALCULATED
$ calc --p1="8%" --p2="25%" --base=100
Combined %
2.00%
Intermediate (P₂% of base)
25.00
Result
2.00
3-Chain %
0.20%
3-Chain Result
0.20
Share:

Waterfall: Base → P₂% → P₁% of P₂%

Comparison: Original vs After Each %

Step-by-Step Breakdown

CALCULATION
Formula
Combined % = (P₁/100) × (P₂/100) × 100
ext{Multiply} ext{percentages} ext{as} ext{decimals}
Convert & multiply
(8/100) × (25/100) × 100
0.0800 × 0.2500 × 100
ANSWER
Combined percentage
2.00%
P_{1}% ext{of} P_{2}%
Intermediate (P₂% of base)
25.00
100 × (25/100)
Result (P₁% of P₂% of base)
2.00
100 × (8/100) × (25/100)
3-chain combined %
0.20%
(P_{1}/100) imes (P_{2}/100) imes (P_{3}/100) imes 100
3-chain result
0.20
ext{Base} imes (P_{1}/100) imes (P_{2}/100) imes (P_{3}/100)

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

— Commutativity

— Extended formula

Key Takeaways

  • • Compounding percentages: P₁% of P₂% = (P₁/100)×(P₂/100)×100 — multiply as decimals, then convert back.
  • • Order matters: 10% of 50% = 5%, but 50% of 10% is also 5% — multiplication is commutative.
  • • Applied to base: P₁% of P₂% of Base = Base × (P₁/100)×(P₂/100).
  • • Nested discounts: Multiple percentage layers compound multiplicatively, not additively.
  • • 3-chain: Extend to P₁% of P₂% of P₃% for three-level cascading calculations.

Did You Know?

🛒A 20% discount followed by 10% off is not 30% off — it's 28% total (0.8 × 0.9 = 0.72)Source: Retail
💰Commission splits: 15% of 60% = 9% — the agent gets 9% of the total saleSource: Sales
📊Tax on taxable portion: 8% tax on 75% taxable = 6% effective rate on the wholeSource: Tax
📈Profit sharing: 40% of 25% of revenue = 10% of total revenue goes to that tierSource: Finance
🎓Grade curves: 10% boost on 85% = 8.5 points added, giving 93.5%Source: Education
🔬Concentration of concentration: 5% of 20% solution = 1% final concentrationSource: Chemistry

How It Works

Step 1: Convert each percentage to decimal (divide by 100).

Step 2: Multiply the decimals: (P₁/100) × (P₂/100).

Step 3: Convert back to percentage: multiply by 100.

Step 4: If you have a base value, multiply: Base × (P₁/100)×(P₂/100).

Example: 8% of 25% of $100

8% × 25% = 0.08 × 0.25 × 100 = 2%. Result: $100 × 0.02 = $2.

Expert Tips: Nested Discounts

Don't Add Percentages

20% + 10% ≠ 30% when applied sequentially. Multiply: 0.8 × 0.9 = 0.72 (28% off).

Tax on Discounted Price

8% tax on 25% off: apply discount first, then tax. Or use P₁% of P₂% for the tax-on-discount portion.

Commission Cascades

Agent gets 15% of broker's 60%. That's 9% of the deal. Use this calculator to verify splits.

3-Chain for Complex Cases

Enable advanced mode for P₁% of P₂% of P₃% — e.g., regional share of department share of total.

Frequently Asked Questions

What is 10% of 50%?

10% of 50% = (0.10 × 0.50) × 100 = 5%. So 5% of any base value.

Does order matter for percentage of percentage?

No. P₁% of P₂% = P₂% of P₁%. Multiplication is commutative.

How do I calculate 8% tax on 25% discount of $100?

8% of 25% = 2%. So 2% of $100 = $2. That's the tax on the discounted portion. For final price: $100 × 0.75 × 1.08 = $81.

Why isn't 20% off then 10% off equal to 30% off?

Because the second discount applies to the already-reduced price. 20% off → 80% remains. 10% of that → 72% remains. So 28% total discount.

What is a 3-chain percentage?

P₁% of P₂% of P₃% = (P₁/100)×(P₂/100)×(P₃/100)×100. Use advanced mode for cascading splits.

Can I use this for profit sharing?

Yes. E.g., 40% of 25% of revenue = 10% of total. Enter 40, 25, and your revenue as base.

Quick Reference

(P₁/100)×(P₂/100)×100
Combined %
Base × decimals
Result
10% of 50% = 5%
Example
Multiply, not add
Nested

Disclaimer: This calculator provides mathematical results for educational and practical purposes. For financial, tax, or business applications, verify with qualified professionals. Rounding may affect precision.

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