Percent as Ratio Formula

The Formula

p\% = \frac{p}{100}; equivalently, \text{decimal} = \frac{\text{percent}}{100}

When to use: 'Per cent' means 'per hundred'—25\% means 25 out of every 100.

Quick Example

75\% = \frac{75}{100} = 0.75 = \frac{3}{4}. 'Got 80\%' means 80 out of 100 points.

Notation

\% means 'per hundred'; p\% is read as 'p percent'

What This Formula Means

A ratio comparing a quantity to 100, written with the % symbol; 'per cent' literally means 'per hundred'.

'Per cent' means 'per hundred'—25\% means 25 out of every 100.

Formal View

p\% = \frac{p}{100}. Conversions: fraction \frac{a}{b} = \frac{100a}{b}\%; decimal d = 100d\%. Percent of a quantity: p\% of x = \frac{p}{100} \cdot x.

Worked Examples

Example 1

easy
A shirt originally costs \45. It is on sale for 20\%$ off. Find the sale price.

Solution

  1. 1
    Interpret 20\% as a ratio: 20\% = \dfrac{20}{100} = 0.20.
  2. 2
    Calculate the discount: 0.20 \times 45 = \9.00$.
  3. 3
    Sale price = 45 - 9 = \36.00$.

Answer

The sale price is \36.00$.
Percent means 'per hundred' — it is a ratio with denominator 100. Converting the percent to a decimal ratio makes multiplication straightforward. The discount amount is then subtracted from the original price.

Example 2

medium
A student scores 34 out of 40 on a test. Express this as a percentage. Then determine what score out of 40 corresponds to 90\%.

Common Mistakes

  • Moving the decimal point the wrong direction — to convert 0.35 to percent, multiply by 100 to get 35\%, not 0.0035\%
  • Thinking 50\% of 80 is 50 — 50\% means half, so 50\% of 80 is 40
  • Writing \frac{25}{100} as 25\% but failing to simplify to \frac{1}{4} — not recognizing that percent, decimal, and fraction forms are interchangeable

Why This Formula Matters

Universal language for sales, statistics, probability, and comparison.

Frequently Asked Questions

What is the Percent as Ratio formula?

A ratio comparing a quantity to 100, written with the % symbol; 'per cent' literally means 'per hundred'.

How do you use the Percent as Ratio formula?

'Per cent' means 'per hundred'—25\% means 25 out of every 100.

What do the symbols mean in the Percent as Ratio formula?

\% means 'per hundred'; p\% is read as 'p percent'

Why is the Percent as Ratio formula important in Math?

Universal language for sales, statistics, probability, and comparison.

What do students get wrong about Percent as Ratio?

Converting between percent, decimal, and fraction forms—remember: \% \div 100 = \text{decimal}, e.g., 45\% = 0.45.

What should I learn before the Percent as Ratio formula?

Before studying the Percent as Ratio formula, you should understand: fractions, decimal representation.

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