Top 10 Percentage Formulas
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01
Basic Percentage & Finding the Part
$$ \text{Percentage} = \dfrac{\text{Part}}{\text{Whole}} \times 100 $$
Step 1
$$ \dfrac{\text{Part}}{\text{Whole}} $$
Divide the part by the whole to get the fraction.
Step 2
$$ \text{Fraction} \times 100 $$
Multiply the fraction by 100 to convert it into a percentage.
12.5%
Fraction: 25 / 200 = 0.125
02
Percentage Increase or Decrease
$$ \text{Percentage Change} = \dfrac{\text{New} - \text{Old}}{\text{Old}} \times 100 $$
Step 1
$$ \text{Difference} = \text{New} - \text{Old} $$
Find the absolute change between the two values.
Step 2
$$ \dfrac{\text{Difference}}{\text{Old}} $$
Divide the difference by the original (old) value.
Step 3
$$ \text{Result} \times 100 $$
Multiply by 100 to express the change as a percentage.
+25% ↑
Absolute difference: 20
03
Successive Percentage Change (Net Change)
$$ \text{Net Change} = A + B + \dfrac{A \times B}{100} $$
Step 1
$$ \text{Start with base } 100 $$
Assume an original value of 100 for easy calculation.
Step 2
$$ 100 + A = 100 \times \left(1 + \dfrac{A}{100}\right) $$
Apply the first percentage change.
Step 3
$$ \text{Result after A} \times \left(1 + \dfrac{B}{100}\right) $$
Apply the second percentage change on the new value.
Step 4
$$ \text{Cross term} = \dfrac{A \times B}{100} $$
The extra amount comes from the product of the two rates.
Step 5
$$ \text{Net} = A + B + \dfrac{A \times B}{100} $$
Add the individual changes and the cross term to get the net percentage.
+32%
Cross term (A×B)/100 = 2
100 → 110 → 132
04
Price and Consumption (Constant Expenditure)
$$ \text{Reduction \%} = \dfrac{R}{100 + R} \times 100 $$
Step 1
$$ \text{Expenditure is constant} $$
Price × Consumption remains the same before and after the change.
Step 2
$$ \text{New Price} = \text{Old Price} \times \left(1 + \dfrac{R}{100}\right) $$
Price rises by R%.
Step 3
$$ \text{New Consumption} = \dfrac{\text{Old Expenditure}}{\text{New Price}} $$
Consumption must fall so that the product stays constant.
Step 4
$$ \text{Reduction factor} = \dfrac{100}{100 + R} $$
The new consumption is this fraction of the old consumption.
Step 5
$$ \text{Reduction \%} = \left(1 - \dfrac{100}{100 + R}\right) \times 100 = \dfrac{R}{100 + R} \times 100 $$
For a decrease in price of R%, change the denominator to 100 − R.
20% reduction in consumption
Original
Price × 100 × Consumption 1 = 100
New
Price × 125 × Consumption 0.8 = 100
05
Reverse Percentage (Finding the Original Value)
$$ \text{Original} = \dfrac{\text{Final} \times 100}{100 + R} $$
Step 1
$$ \text{Final} = \text{Original} \times \left(1 + \dfrac{R}{100}\right) $$
The final value is the original increased (or decreased) by R%.
Step 2
$$ \text{Original} = \dfrac{\text{Final}}{1 + \frac{R}{100}} = \dfrac{\text{Final} \times 100}{100 + R} $$
Algebraically rearrange to isolate the original value.
Original = 100
Subtracting R% from the final value is incorrect.
Wrong method (just subtract 20%): 120 − 24 = 96 ← incorrect
06
Comparing Two Values (More / Less Than)
$$ \% \text{ Less} = \dfrac{R}{100 + R} \times 100 $$
Step 1
$$ \text{Let smaller value } B = 100 $$
Set the smaller quantity as the base of 100.
Step 2
$$ A = 100 + R $$
If A is R% more than B, then A equals 100 + R.
Step 3
$$ \text{Difference} = R $$
The absolute difference is R units.
Step 4
$$ \% \text{ less of A} = \dfrac{R}{A} \times 100 = \dfrac{R}{100 + R} \times 100 $$
Express the difference as a percentage of the larger value.
Step 5
$$ B \text{ is } \dfrac{R}{100+R}\% \text{ less than } A $$
This is the required “less than” percentage.
A is 25% more than B
B is 20% less than A
Ratio A : B ≈ 125 : 100
07
Population Growth & Depreciation
$$ \text{Value after } n \text{ years} = P \times \left(1 \pm \dfrac{R}{100}\right)^n $$
Step 1
$$ \text{Growth factor} = 1 + \dfrac{R}{100} $$
For growth the sign is positive; for depreciation it is negative.
Step 2
$$ \text{Year 1} = P \times \text{factor} $$
Multiply the principal by the factor once.
Step 3
$$ \text{Year 2} = \text{Year 1} \times \text{factor} $$
Compound by multiplying the previous result again.
Step 4
$$ \text{After } n \text{ years} = P \times (\text{factor})^n $$
Raise the factor to the power n for the final value.
Final value = 12,597.12
| Year | Value |
|---|
08
The Commutative Property (x% of y)
$$ x\% \text{ of } y = y\% \text{ of } x $$
Step 1
$$ x\% \text{ of } y = \dfrac{x}{100} \times y = \dfrac{x y}{100} $$
Expand the left side.
Step 2
$$ y\% \text{ of } x = \dfrac{y}{100} \times x = \dfrac{y x}{100} $$
Expand the right side.
Step 3
$$ \dfrac{x y}{100} = \dfrac{y x}{100} $$
Both expressions are identical, proving the equality.
20% of 50 = 10
50% of 20 = 10
=
09
Ratio to Percentage
$$ A\text{'s \% Share} = \dfrac{A}{A + B} \times 100 $$
Step 1
$$ \text{Total parts} = A + B $$
Add the two ratio parts to get the whole.
Step 2
$$ \text{Fraction of A} = \dfrac{A}{A + B} $$
Form the fraction that belongs to A.
Step 3
$$ \text{Percentage} = \text{Fraction} \times 100 $$
Convert the fraction into a percentage. B’s share is the remainder.
A: 60% | B: 40%
Shares always sum to 100%
10
Quick Mental Math Bases
$$ x\% \text{ of } y $$
Step 1
$$ 10\% = \text{move decimal one place left} $$
10% of any number is the number with the decimal point shifted one place left.
Step 2
$$ 5\% = \text{half of the 10\% value} $$
5% is simply half of the 10% amount.
Step 3
$$ 1\% = \text{move decimal two places left} $$
1% is the number with the decimal point shifted two places left.
Step 4
$$ \text{Combine the needed blocks} $$
Add the required 10%, 5% and 1% pieces to reach the exact percentage.
16% of 250 = 40