Fractions and percentages as operators

Number • Fractions, decimals and percentages

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How to handle successive percentage increases of 10% then 20%?

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Handle successive increases by multiplying multipliers: 1.10 × 1.20 = 1.32, causing a 32% overall increase.

Key concepts

What you'll likely be quizzed about

Fractions as operators

A fraction a/b interpreted as an operator means multiply by a/b. Because multiplication by a/b scales the original quantity, the result represents a proportional part of the amount. Limiting factors include a non-zero denominator and the sign of a and b; negative values produce a negative result.

Percentages as operators

A percentage p% interpreted as an operator means multiply by p/100. Because p% converts to the decimal multiplier p ÷ 100, applying p% of an amount scales the amount by that multiplier. Percentages greater than 100 produce a larger result, percentages between 0 and 100 reduce the amount, and negative percentages invert the sign and scale.

The 'of' keyword means multiply

The word 'of' in expressions such as '1/2 of 30' indicates multiplication, causing evaluation as (1/2) × 30. Because 'of' triggers multiplication, expressions with 'of' do not indicate division or subtraction by default. The consequence is direct calculation by converting the fraction or percentage to a multiplier and multiplying the amount.

Percent change as an operator

A percentage increase or decrease acts as a multiplier of the form 1 ± p/100. Because adding or subtracting the fractional part produces the multiplier, successive percentage changes produce multiplication of multipliers. The effect of successive changes compounds rather than adds: applying +20% then +10% results in a net multiplier of 1.2 × 1.1 = 1.32, causing a 32% increase overall.

Reversing an operator to find original amount

Finding the original amount before a fraction or percentage operator requires dividing by the multiplier. Because forward application multiplies by m, reversing requires division by m. For example, if the result after applying 70% is 84, the original amount equals 84 ÷ 0.7 = 120. Division by zero is not permitted; multipliers must be non-zero.

Combining and simplifying operators

Multiple fractions and percentages combine through multiplication of their multipliers. Because multiplication is commutative, the order of operators does not change the final product, but the effect compounds. Converting every operator to a decimal multiplier first and then multiplying produces the final scaling factor, which simplifies calculation and reduces error.

Key notes

Important points to keep in mind

Convert fractions and percentages to decimal multipliers before calculating.

Interpret the word 'of' as multiplication, not division.

Check that fraction denominators are non-zero before using as operators.

Percent increases and decreases use multipliers 1 ± p/100 and compound when successive.

Reverse an operator by dividing by its multiplier to find the original amount.

Multiply successive operators to find the combined effect; order does not change the product.

Percentages over 100 produce multipliers greater than 1 and increase amounts.

Negative fractions or percentages produce negative results and require sign attention.