Basic ratio notation and partitioning explained

Ratio, proportion and rates • Ratio and proportion

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Convert a ratio to simplest form when parts are decimals

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Multiply parts by a common factor to remove decimals, then divide by the greatest common divisor to simplify.

Key concepts

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Definition of ratio notation

Ratio notation records a multiplicative relationship between quantities using the colon symbol or as a fraction. For example, 3:2 reads as three parts to two parts and 3/2 reads as three divided by two. Ratios compare relative sizes and do not include units unless parts represent different units; parts remain proportional rather than absolute. Ratios operate on integer counts of parts. A ratio 3:2 implies six equal sub-units if each part is taken as one unit, so any actual quantities that follow 3:2 maintain the same relative counts when scaled.

Part:part versus part:whole

Part:part notation describes how two parts compare directly, for example 3:2 means the first part is three parts and the second part is two parts. Part:whole notation expresses one part compared with the entire quantity, for example 3:5 as 3 out of 5 parts of the whole. Conversion uses the sum of parts. For a ratio a:b, the whole equals a + b parts. The value of the first part equals (a ÷ (a + b)) × whole and the second part equals (b ÷ (a + b)) × whole. The cause (sum of parts) produces the effect (size of each part as a fraction of the whole).

Partitioning a quantity using a given ratio

Partitioning a quantity into two parts in ratio a:b requires adding parts to find the total parts, then dividing the quantity by total parts and allocating parts accordingly. For example, splitting 84 into ratio 3:2 uses total parts 5, so each part equals 84 ÷ 5 = 16.8; the parts equal 3 × 16.8 = 50.4 and 2 × 16.8 = 33.6. Integer results occur when the total quantity is divisible by the total parts; otherwise parts become fractional. The method remains identical regardless of whether parts are whole numbers or decimals because the ratio fixes relative sizes.

Simplifying ratios

Simplifying a ratio uses the greatest common divisor to reduce each part to smallest whole numbers that retain the same relative sizes. For example, 8:12 reduces by 4 to 2:3 because dividing both parts by 4 preserves the multiplicative relationship. Simplified ratios remove redundant scaling factors and make partitioning and comparison easier. Limiting factor: simplification requires integer parts; if parts are non-integers, multiply by a common factor to obtain whole-number parts before simplifying.

Ratios as multiplicative relationships and fractions

A ratio a:b represents the multiplicative relationship a ÷ b when expressed as a fraction a/b. That fraction describes how many times larger one quantity is compared with the other. For example, a ratio 4:1 corresponds to the fraction 4/1 = 4, meaning the first quantity is four times the second. Expressing ratios as fractions supports algebraic manipulation and comparison. If one quantity is greater than the other, the fraction exceeds 1; if it is smaller, the fraction is less than 1. The fraction equals 1 exactly when the two quantities are equal.

Expressing one quantity as a fraction of another

One quantity expressed as a fraction of another uses the form part ÷ whole or first ÷ second. For example, if two quantities are 7 and 5, then the first as a fraction of the second is 7/5, which is greater than 1. If quantities are 3 and 8, the first as a fraction of the second is 3/8, which is less than 1. The fraction gives a precise multiplicative comparison and converts directly to ratio notation: 7/5 corresponds to ratio 7:5 and 3/8 corresponds to 3:8. Conversion allows calculation of percentages or scaling by multiplication.

Key notes

Important points to keep in mind

Interpret ratios as counts of equal parts, not absolute amounts.

Convert a:b to part:whole by using a + b as the whole number of parts.

Partition by dividing the whole by total parts, then multiply by each part count.

Simplify ratios using the greatest common divisor to get smallest whole-number form.

Express a ratio a:b as a fraction a/b to show how many times larger one is than the other.

A fraction greater than 1 indicates the numerator is larger than the denominator; less than 1 indicates it is smaller.

Multiply by a common factor to remove decimals before simplifying ratios with non-integer parts.

Check answers by recombining parts to the original total to verify correctness.

When parts are not integers, allow fractional or decimal part sizes; proportional relationships remain unchanged.

Conversion between ratios, fractions and percentages uses the same multiplicative relationships.