Ratios and Proportions

Writing Ratios

Write and simplify a ratio by dividing both terms by their GCF to express in lowest terms.

write-ratio To write and simplify a ratio a to b: - Find the GCF of a and b. - Divide both terms by the GCF.

Example

Write 12 to 18 as a simplified ratio. A ratio compares two quantities by division, and like a fraction it can be reduced. Dividing both parts by their greatest common factor keeps the comparison identical while expressing it in the smallest whole numbers, which makes the relationship between the quantities easiest to read.

highlighted = computed this step

Step 1 — Set up

Set up the relationship.

12:1812 : 18
Count in order: 12 blue to 18 green.first partsecond part12181218part-to-part: 12:18whole: 12 + 18Part-to-whole view: 12 blue + 18 green objects.

Step 2 — Find GCF

Find the GCF: gcd(12, 18) = 6.

gcd⁡(12,18)=6\gcd( 12 , 18 )= \hl{6}
Group each row by 6; the order stays blue to green.first partsecond part12181218part-to-part: 12:18whole: 12 + 18Each outlined group has exactly 6 objects.

Step 3 — Divide both terms

Divide both terms: 12 / 6 = 2 and 18 / 6 = 3.

12÷6=218÷6=312 \div 6 = \hl{2} \quad 18 \div 6 = \hl{3}
Divide both counts by 6: 12 -> 2 groups and 18 -> 3 groups.first partsecond part12181218part-to-part: 12:18whole: 12 + 1812 / 6 = 218 / 6 = 3The same-size groups keep the comparison honest.

Step 4 — Result

Read the final result.

2:32 : 3
The simplified part-to-part ratio is 2:3.first partsecond part12181218part-to-part: 12:18whole: 12 + 1812 / 6 = 218 / 6 = 32:3Whole view: 2 blue groups + 3 green groups, from 12 + 18 objects.