---
title: "Percentages and Ratios Cheatsheet | FindUtils"
description: "Calculate percentage shares, increases, discounts, and ratios. Use worked examples to distinguish percentage points from relative percentage changes."
url: https://findutils.com/cheatsheets/percentages-ratios/
---

# Percentages and Ratios Cheatsheet

Calculate percentage shares, increases, discounts, and ratios. Use worked examples to distinguish percentage points from relative percentage changes. 37 commands in 7 sections, filed under [Student & Academic](https://findutils.com/cheatsheets/student/).

## Start with the right quantity

- `p% = p / 100`: A percentage expresses an amount per hundred
- `Base value`: Use the original total that the percentage refers to
- `Part / whole`: The whole must be nonzero
- `Same units`: Convert both quantities to the same unit before a ratio
- `Round at the end`: Keep intermediate precision until the final result

## Convert percentages, decimals, and fractions

- `25% = 0.25 = 1/4`: Divide the percentage number by 100
- `50% = 0.5 = 1/2`: Half of the whole
- `75% = 0.75 = 3/4`: Three quarters of the whole
- `0.08 × 100% = 8%`: Convert a decimal to a percentage
- `3/5 × 100% = 60%`: Divide the numerator by the denominator, then multiply by 100%

## Find a part or a whole

- `Part = whole × p / 100`: Find a percentage of a total
- `15% of 80 = 80 × 0.15 = 12`: Multiply the total by the decimal rate
- `Percentage = part / whole × 100%`: Find the share of a nonzero total
- `18 out of 60 = 30%`: Calculate 18 / 60 × 100%
- `Whole = part / (p / 100)`: Recover the total when the percentage is nonzero
- `12 is 15% of 80`: Calculate 12 / 0.15

## Increase, discount, and reverse a change

- `New = original × (1 + p / 100)`: Increase a value by p percent
- `80 increased by 15% = 92`: Calculate 80 × 1.15
- `New = original × (1 - p / 100)`: Decrease a value by p percent
- `80 reduced by 15% = 68`: Calculate 80 × 0.85
- `Original = final / (1 + p / 100)`: Reverse an increase when the divisor is nonzero
- `Original = final / (1 - p / 100)`: Reverse a decrease below 100%

## Compare changes correctly

- `Change % = (new - old) / old × 100%`: Use a positive original value for the usual increase interpretation
- `From 40 to 50 = 25% increase`: Calculate (50 - 40) / 40 × 100%
- `From 50 to 40 = 20% decrease`: Use 50 as the original value for the reverse change
- `From 20% to 25% = 5 percentage points`: Subtract the two percentage values
- `From 20% to 25% = 25% relative increase`: Calculate (25 - 20) / 20 × 100%
- `20% off, then 10% off = 28% off`: Multiply 0.8 × 0.9; successive discounts do not add

## Ratios and shares

- `12:18 = 2:3`: Divide both quantities by their common factor of 6
- `2:3 has 5 total parts`: Add the ratio terms before dividing a total
- `Share A = total × 2/5`: Split a total in the ratio 2:3
- `Split 60 in the ratio 2:3 → 24 and 36`: Check that both shares add to the total
- `Scale 2:3 by 4 → 8:12`: Multiply both ratio terms by the same number

## Sources and related references

- `https://openstax.org/books/prealgebra-2e/pages/6-1-understand-percent`: Read the percent, fraction, and decimal definitions
- `https://openstax.org/books/prealgebra-2e/pages/6-3-solve-general-applications-of-percent`: Check percent change and application methods
- `/cheatsheets/spreadsheet-formulas/`: Use percentages in Excel and Google Sheets
- `/cheatsheets/everyday-unit-conversions/`: Convert units before comparing quantities

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- Guide: [Percentage Calculator Guide: Increase, Decrease and Change](https://findutils.com/guides/percentage-calculator-guide/)
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